Business Statistics Business Statistics
Business Statistics:
Revealing Facts From Figures

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Interactive Online Version

Europe Mirror Site

I am always happy to help students who are not enrolled in my courses with questions and problems. But unfortunately, I don't have enough time to respond to everyone. Thank you for your understanding.

Professor Hossein Arsham   


  • Introduction

    Towards Statistical Thinking For Decision Making Under Uncertainties

    The Birth of Statistics
    What is Business Statistics
    Belief, Opinion, and Fact
    Kinds of Lies: Lies, Damned Lies and Statistics

    Probability for Statistical Inference

    Different Schools of Thought in Inferential Statistics
    Bayesian, Frequentist, and Classical Methods
    Probability, Chance, Likelihood, and Odds
    How to Assign Probabilities
    General Laws of Probability
    Mutually Exclusive versus Independent Events
    Entropy Measure
    Applications of and Conditions for Using Statistical Tables
    Relationships Among Distributions and Unification of Statistical Tables

    Topics in Business Statistics

    Greek Letters Commonly Used in Statistics
    Type of Data and Levels of Measurement
    Sampling Methods
    Number of Class Intervals in a Histogram
    How to Construct a Box Plot
    Outlier Removal
    Statistical Summaries
    What Is So Important About the Normal Distributions
    What Is a Sampling Distribution
    What Is Central Limit Theorem
    What Is "Degrees of Freedom"

    Parameters' Estimation and Quality of a 'Good' Estimate
    Procedures for Statistical Decision Making
    Statistics with Confidence and Determining Sample Size
    Hypothesis Testing: Rejecting a Claim
    The Classical Approach to the Test of Hypotheses
    The Meaning and Interpretation of P-values (what the data say)
    Blending the Classical and the P-value Based Approaches in Test of Hypotheses
    Conditions Under Which Most Statistical Testings Apply Statistical Tests for Equality of Populations Characteristics Power of a Test
    Parametric vs. Non-Parametric vs. Distribution-free Tests
    Chi-square Tests
    Bonferroni Method
    Goodness-of-fit Test for Discrete Random Variables
    When We Should Pool Variance Estimates
    Resampling Techniques: Jackknifing, and Bootstrapping
    What is a Linear Least Squares Model
    Pearson's and Spearman's Correlations
    How to Compare Two Correlations Coefficients
    Independence vs. Correlated
    Correlation, and Level of Significance
    Regression Analysis: Planning, Development, and Maintenance
    Predicting Market Response
    Warranties: Statistical Planning and Analysis
    Factor Analysis

    Interesting and Useful Sites (topical category)

    Selected Reciprocal Web Sites
    Review of Statistical Tools on the Internet
    General References
    Statistical Societies & Organizations
    Statistics References
    Statistics Resources
    Statistical Data Analysis
    Probability Resources
    Data and Data Analysis
    Computational Probability and Statistics Resources
    Questionnaire Design, Surveys Sampling and Analysis
    Statistical Software
    Learning Statistics
    Econometric and Forecasting
    Selected Topics
    Glossary Collections Sites
    Statistical Tables


    This Web site is a course in statistics appreciation, i.e. to acquire a feel for the statistical way of thinking. An introductory course in statistics designed to provide you with the basic concepts and methods of statistical analysis for processes and products. Materials in this Web site are tailored to meet your needs in business decision making. It promotes think statistically. The cardinal objective for this Web site is to increase the extent to which statistical thinking is embedded in management thinking for decision making under uncertainties. It is already an accepted fact that "Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write." So, let's be ahead of our time.

    To be competitive, business must design quality into products and processes. Further, they must facilitate a process of never-ending improvement at all stages of manufacturing. A strategy employing statistical methods, particularly statistically designed experiments, produces processes that provide high yield and products that seldom fail. Moreover, it facilitates development of robust products that are insensitive to changes in the environment and internal component variation. Carefully planned statistical studies remove hindrances to high quality and productivity at every stage of production, saving time and money. It is well recognized that quality must be engineered into products as early as possible in the design process. One must know how to use carefully planned, cost-effective experiments to improve, optimize and make robust products and processes.

    Business Statistics is a science assisting you to make business decisions under uncertainties based on some numerical and measurable scales. Decision making process must be based on data neither on personal opinion nor on belief.

    Know that data are only crude information and not knowledge by themselves. The sequence from data to knowledge is: from Data to Information, from Information to Facts, and finally, from Facts to Knowledge. Data becomes information when it becomes relevant to your decision problem. Information becomes fact when the data can support it. Fact becomes knowledge when it is used in the successful completion of decision process. The following figure illustrates the statistical thinking process based on data in constructing statistical models for decision making under uncertainties.

    From Data to Knowledge

    Knowledge is more than knowing something technical. Knowledge needs wisdom, and wisdom comes with age and experience. Wisdom is about knowing how something technical can be best used to meet the needs of the decision-maker. Wisdom, for example, creates statistical software that is useful, rather than technically brilliant.

    The Devil is in the Deviations: Variation is an inevitability in life! Every process has variation. Every measurement. Every sample! Managers need to understand variation for two key reasons. First, so that they can lead others to apply statistical thinking in day to day activities and secondly, to apply the concept for the purpose of continuous improvement. This course will provide you with hands-on experience to promote the use of statistical thinking and techniques to apply them to make educated decisions whenever you encounter variation in business data. You will learn techniques to intelligently assess and manage the risks inherent in decision-making. Therefore, remember that:

    Just like weather, if you cannot control something, you should learn how to measure and analyze, in order to predict it, effectively.

    If you have taken statistics before, and have a feeling of inability to grasp concepts, it is largely due to your former non-statistician instructors teaching statistics. Their deficiencies lead students to develop phobias for the sweet science of statistics. In this respect, the following remark is made by Professor Herman Chernoff, in Statistical Science, Vol. 11, No. 4, 335-350, 1996:

    "Since everybody in the world thinks he can teach statistics even though he does not know any, I shall put myself in the position of teaching biology even though I do not know any"

    Plugging numbers in the formulas and crunching them has no value by themselves. You should continue to put effort into the concepts and concentrate on interpreting the results.

    Even, when you solve a small size problem by hand, I would like you to use the available computer software and Web-based computation to do the dirty work for you.

    You must be able to read off the logical secrete in any formulas not memorizing them. For example, in computing the variance, consider its formula. Instead of memorizing, you should start with some whys:

    i. Why we square the deviations from the mean.
    Because, if we add up all deviations we get always zero. So to get away from this problem, we square the deviations. Why not raising to the power of four (three will not work)? Since squaring does the trick why should we make life more complicated than it is. Notice also that squaring also magnifies the deviations, therefore it works to our advantage to measure the quality of the data.

    ii. Why there is a summation notation in the formula.
    To add up the squared deviation of each data point to compute the total sum of squared deviations.

    iii. Why we divide the sum of squares by n-1.
    The amount of deviation should reflects also how large is the sample, so we must bring in the sample size. That is, in general larger sample size have larger sum of square deviation from the mean. Okay. Why n-1 and not n. The reason for it is that when you divide by n-1 the sample's variance provide a much closer to the population variance than when you divide by n, on average. You note that for large sample size n (say over 30) it really does not matter whether you divide by n or n-1. The results are almost the same and acceptable. The factor n-1 is the so called the "degrees of freedom".

    This was just an example for you to show as how to question the formulas rather than memorizing them. If fact when you try to understand the formulas you do not need to remember them, they are parts of your brain connectivity. Clear thinking is always more important than the ability to do a lot of arithmetic.

    When you look at a statistical formula the formula should talk to you, as when a musician looks at a piece of musical-notes he/she hears the music. How to become a statistician who is also a musician?

    The objectives for this course are to learn statistical thinking; to emphasize more data and concepts, less theory and fewer recipes; and finally to foster active learning using, e.g., the useful and interesting Web-sites.

    Some Topics in Business Statistics

    Greek Letters Commonly Used as Statistical Notations

    We use Greek letters in statistics and other scientific areas to honor the ancient Greek philosophers who invented science (such as Socrates, the inventor of dialectic reasoning).

    Greek Letters Commonly Used as Statistical Notations
    alphabetaki-sqredeltamunupi rhosigmatautheta
    c 2

    Note: ki-square (ki-sqre, Chi-square), c2, is not the square of anything, its name imply Chi-square (read, ki-square). Ki does not exist in statistics. I'm glad that you're overcoming all the confusions that exist in learning statistics.

    The Birth of Statistics

    The original idea of "statistics" was the collection of information about and for the "State".

    The birth of statistics occurred in mid-17th century. A commoner, named John Graunt, who was a native of London, begin reviewing a weekly church publication issued by the local parish clerk that listed the number of births, christenings, and deaths in each parish. These so called Bills of Mortality also listed the causes of death. Graunt who was a shopkeeper organized this data in the forms we call descriptive statistics, which was published as Natural and Political Observation Made upon the Bills of Mortality. Shortly thereafter, he was elected as a member of Royal Society. Thus, statistics has to borrow some concepts from sociology, such as the concept of "Population". It has been argued that since statistics usually involves the study of human behavior, it cannot claim the precision of the physical sciences.

    Probability has much longer history. It originated from the study of games of chance and gambling during the sixteenth century. Probability theory was a branch of mathematics studied by Blaise Pascal and Pierre de Fermat in the seventeenth century. Currently, in 21st centuray, probabilistic modeling are used to control the flow of traffic through a highway system, a telephone interchange, or a computer processor; find the genetic makeup of individuals or populations; quality control; insurance; investment; and other sectors of business and industry.

    New and ever growing diverse fields of human activities are using statistics, however, it seems that this field itself remains obscure to the public. Professor Bradley Efron expressed this fact nicely:

    During the 20th Century statistical thinking and methodology have become the scientific framework for literally dozens of fields including education, agriculture, economics, biology, and medicine, and with increasing influence recently on the hard sciences such as astronomy, geology, and physics. In other words, we have grown from a small obscure field into a big obscure field.

    For the history of probability, and history of statistics, visit History of Statistics Material. I also recommend the following books.

    Further Readings:
    Daston L., Classical Probability in the Enlightenment, Princeton University Press, 1988.
    The book points out that early Enlightenment thinkers could not face uncertainty. A mechanistic, deterministic machine, was the Enlightenment view of the world.

    Gillies D., Philosophical Theories of Probability, Routledge, 2000. Covers the classical, logical, subjective, frequency, and propensity views.

    Hacking I., The Emergence of Probability, Cambridge University Press, London, 1975.
    A philosophical study of early ideas about probability, induction and statistical inference.

    Peters W., Counting for Something: Statistical Principles and Personalities, Springer, New York, 1987.
    It teaches the principles of applied economic and social statistics in a historical context. Featured topics include public opinion polls, industrial quality control, factor analysis, Bayesian methods, program evaluation, non-parametric and robust methods, and exploratory data analysis.

    Porter T., The Rise of Statistical Thinking, 1820-1900, Princeton University Press, 1986.
    The author states that statistics has become known in the twentieth century as the mathematical tool for analyzing experimental and observational data. Enshrined by public policy as the only reliable basis for judgments as the efficacy of medical procedures or the safety of chemicals, and adopted by business for such uses as industrial quality control, it is evidently among the products of science whose influence on public and private life has been most pervasive. Statistical analysis has also come to be seen in many scientific disciplines as indispensable for drawing reliable conclusions from empirical results.This new field of mathematics found so extensive a domain of applications.

    Stigler S., The History of Statistics: The Measurement of Uncertainty Before 1900, U. of Chicago Press, 1990.
    It covers the people, ideas, and events underlying the birth and development of early statistics.

    Tankard J., The Statistical Pioneers, Schenkman Books, New York, 1984.
    This work provides the detailed lives and times of theorists whose work continues to shape much of the modern statistics.

    What is Business Statistics?

    In this diverse world of ours, no two things are exactly the same. A statistician is interested in both the differences and the similarities, i.e. both patterns and departures.

    The actuarial tables published by insurance companies reflect their statistical analysis of the average life expectancy of men and women at any given age. From these numbers, the insurance companies then calculate the appropriate premiums for a particular individual to purchase a given amount of insurance.

    Exploratory analysis of data makes use of numerical and graphical techniques to study patterns and departures from patterns. The widely used descriptive statistical techniques are: Frequency Distribution Histograms; Box & Whisker and Spread plots; Normal plots; Cochrane (odds ratio) plots; Scattergrams and Error Bar plots; Ladder, Agreement and Survival plots; Residual, ROC and diagnostic plots; and Population pyramid. Graphical modeling is a collection of powerful and practical techniques for simplifying and describing inter-relationships between many variables, based on the remarkable correspondence between the statistical concept of conditional independence and the graph-theoretic concept of separation.

    The controversial "Million Man March on Washington" was in 1995 demonstrated the size of a rally can have important political consequences. March organizers steadfastly maintained the official attendance estimates offered by the U. S. Park Service (300,000) were too low. Is it?

    In examining distributions of data, you should be able to detect important characteristics, such as shape, location, variability, and unusual values. From careful observations of patterns in data, you can generate conjectures about relationships among variables. The notion of how one variable may be associated with another permeates almost all of statistics, from simple comparisons of proportions through linear regression. The difference between association and causation must accompany this conceptual development.

    Data must be collected according to a well-developed plan if valid information on a conjecture is to be obtained. The plan must identify important variables related to the conjecture and specify how they are to be measured. From the data collection plan, a statistical model can be formulated from which inferences can be drawn.

    Statistical models are currently used in various fields of business and science. However, the terminology differs from field to field. For example, the fitting of models to data, called calibration, history matching, and data assimilation, are all synonymous with parameter estimation.

    Know that data are only crude information and not knowledge by themselves. The sequence from data to knowledge is: from Data to Information, from Information to Facts, and finally, from Facts to Knowledge. Data becomes information when it becomes relevant to your decision problem. Information becomes fact when the data can support it. Fact becomes knowledge when it is used in the successful completion of decision process. The following figure illustrates the statistical thinking process based on data in constructing statistical models for decision making under uncertainties.

    From Data to Knowledge

    That's why we need Business Statistics. Statistics arose from the need to place knowledge on a systematic evidence base. This required a study of the laws of probability, the development of measures of data properties and relationships, and so on.

    The main objective of Business Statistics is to make inference (prediction, making decisions) about certain characteristics of a population based on information contained in a random sample from the entire population, as depicted below:

    Business Statistics is the science of ‘good' decision making in the face of uncertainty and is used in many disciplines such as financial analysis, econometrics, auditing, production and operations including services improvement, and marketing research. It provides knowledge and skills to interpret and use statistical techniques in a variety of business applications. A typical Business Statistics course is intended for business majors, and covers statistical study, descriptive statistics (collection, description, analysis, and summary of data), probability, and the binomial and normal distributions, test of hypotheses and confidence intervals, linear regression, and correlation.

    what is statistics?

    The following discussion refers to the above chart. Statistics is a science of making decisions with respect to the characteristics of a group of persons or objects on the basis of numerical information obtained from a randomly selected sample of the group.

    At the planning stage of a statistical investigation the question of sample size (n) is critical. This course provides a practical introduction to sample size determination in the context of some commonly used significance tests.

    Population: A population is any entire collection of people, animals, plants or things from which we may collect data. It is the entire group we are interested in, which we wish to describe or draw conclusions about. In the above figure the life of the light bulbs manufactured say by GE, is the concerned population.

    Statistical Experiment

    In order to make any generalization about a population, a random sample from the entire population, that is meant to be representative of the population, is often studied. For each population there are many possible samples. A sample statistic gives information about a corresponding population parameter. For example, the sample mean for a set of data would give information about the overall population mean m .

    It is important that the investigator carefully and completely defines the population before collecting the sample, including a description of the members to be included.

    Example: The population for a study of infant health might be all children born in the U.S.A. in the 1980's. The sample might be all babies born on 7th May in any of the years.

    An experiment is any process or study which results in the collection of data, the outcome of which is unknown. In statistics, the term is usually restricted to situations in which the researcher has control over some of the conditions under which the experiment takes place.

    Example: Before introducing a new drug treatment to reduce high blood pressure, the manufacturer carries out an experiment to compare the effectiveness of the new drug with that of one currently prescribed. Newly diagnosed subjects are recruited from a group of local general practices. Half of them are chosen at random to receive the new drug, the remainder receive the present one. So, the researcher has control over the type of subject recruited and the way in which they are allocated to treatment.

    Experimental (or Sampling) Unit: A unit is a person, animal, plant or thing which is actually studied by a researcher; the basic objects upon which the study or experiment is carried out. For example, a person; a monkey; a sample of soil; a pot of seedlings; a postcode area; a doctor's practice.

    Design of experiments is a key tool for increasing the rate of acquiring new knowledge–knowledge that in turn can be used to gain competitive advantage, shorten the product development cycle, and produce new products and processes which will meet and exceed your customer's expectations.

    The major task of statistics is to study the characteristics of populations whether these populations are people, objects, or collections of information. For two major reasons, it is often impossible to study an entire population:

    The process would be too expensive or time consuming.
    The process would be destructive.

    In either case, we would resort to looking at a sample chosen from the population and trying to infer information about the entire population by only examining the smaller sample. Very often the numbers which interest us most about the population are the mean m and standard deviation s. Any number -- like the mean or standard deviation -- which is calculated from an entire population is called a Parameter. If the very same numbers are derived only from the data of a sample, then the resulting numbers are called Statistics. Frequently, parameters are represented by Greek letters and statistics by Latin letters (as shown in the above Figure). The step function in this figure is the Empirical Distribution Function (EDF), known also as Ogive, which is used to graph cumulative frequency. An EDF is constructed by placing a point corresponding to the middle point of each class at a height equal to the cumulative frequency of the class. EDF represents the distribution function Fx.


    A parameter is a value, usually unknown (and therefore has to be estimated), used to represent a certain population characteristic. For example, the population mean m is a parameter that is often used to indicate the average value of a quantity.

    Within a population, a parameter is a fixed value which does not vary. Each sample drawn from the population has its own value of any statistic that is used to estimate this parameter. For example, the mean of the data in a sample is used to give information about the overall mean m in the population from which that sample was drawn.

    Statistic: A statistic is a quantity that is calculated from a sample of data. It is used to give information about unknown values in the corresponding population. For example, the average of the data in a sample is used to give information about the overall average in the population from which that sample was drawn.

    It is possible to draw more than one sample from the same population and the value of a statistic will in general vary from sample to sample. For example, the average value in a sample is a statistic. The average values in more than one sample, drawn from the same population, will not necessarily be equal.

    Statistics are often assigned Roman letters (e.g. and s), whereas the equivalent unknown values in the population (parameters ) are assigned Greek letters (e.g. µ, s).

    The word estimate means to esteem, that is giving a value to something. A statistical estimate is an indication of the value of an unknown quantity based on observed data.

    More formally, an estimate is the particular value of an estimator that is obtained from a particular sample of data and used to indicate the value of a parameter.

    Example: Suppose the manager of a shop wanted to know m , the mean expenditure of customers in her shop in the last year. She could calculate the average expenditure of the hundreds (or perhaps thousands) of customers who bought goods in her shop, that is, the population mean m . Instead she could use an estimate of this population mean m by calculating the mean of a representative sample of customers. If this value was found to be $25, then $25 would be her estimate.

    There are two broad subdivisions of statistics: Descriptive statistics and Inferential statistics.

    The principal descriptive quantity derived from sample data is the mean ( ), which is the arithmetic average of the sample data. It serves as the most reliable single measure of the value of a typical member of the sample. If the sample contains a few values that are so large or so small that they have an exaggerated effect on the value of the mean, the sample is more accurately represented by the median -- the value where half the sample values fall below and half above.

    The quantities most commonly used to measure the dispersion of the values about their mean are the variance s2 and its square root , the standard deviation s. The variance is calculated by determining the mean, subtracting it from each of the sample values (yielding the deviation of the samples), and then averaging the squares of these deviations. The mean and standard deviation of the sample are used as estimates of the corresponding characteristics of the entire group from which the sample was drawn. They do not, in general, completely describe the distribution (Fx) of values within either the sample or the parent group; indeed, different distributions may have the same mean and standard deviation. They do, however, provide a complete description of the Normal Distribution, in which positive and negative deviations from the mean are equally common and small deviations are much more common than large ones. For a normally distributed set of values, a graph showing the dependence of the frequency of the deviations upon their magnitudes is a bell-shaped curve. About 68 percent of the values will differ from the mean by less than the standard deviation, and almost 100 percent will differ by less than three times the standard deviation.

    Statistical inference refers to extending your knowledge obtained from a random sample from the entire population to the whole population. This is known in mathematics as Inductive Reasoning. That is, knowledge of the whole from a particular. Its main application is in hypotheses testing about a given population.

    Inferential statistics is concerned with making inferences from samples about the populations from which they have been drawn. In other words, if we find a difference between two samples, we would like to know, is this a "real" difference (i.e., is it present in the population) or just a "chance" difference (i.e. it could just be the result of random sampling error). That's what tests of statistical significance are all about.

    Statistical inference guides the selection of appropriate statistical models. Models and data interact in statistical work. Models are used to draw conclusions from data, while the data are allowed to criticize, and even falsify the model through inferential and diagnostic methods. Inference from data can be thought of as the process of selecting a reasonable model, including a statement in probability language of how confident one can be about the selection.

    Inferences made in statistics are of two types. The first is estimation, which involves the determination, with a possible error due to sampling, of the unknown value of a population characteristic, such as the proportion having a specific attribute or the average value m of some numerical measurement. To express the accuracy of the estimates of population characteristics, one must also compute the "standard errors" of the estimates; these are margins that determine the possible errors arising from the fact that the estimates are based on random samples from the entire population and not on a complete population census. The second type of inference is hypothesis testing. It involves the definitions of a "hypothesis" as one set of possible population values and an "alternative," a different set. There are many statistical procedures for determining, on the basis of a sample, whether the true population characteristic belongs to the set of values in the hypothesis or the alternative.

    The statistical inference is grounded in probability, idealized concepts of the group under study, called the population, and the sample. The statistician may view the population as a set of balls from which the sample is selected at random, that is, in such a way that each ball has the same chance as every other one for inclusion in the sample.

    Notice that to be able to estimate the population parameters, the sample size n most be greater than one. For example, with a sample size of one the variation (s2) within the sample is 0/1 = 0. An estimate for the variation (s2) within the population would be 0/0, which is indeterminate quantity, meaning impossible. For working with zero correctly, visit the Web site The Zero Saga & Confusions With Numbers.

    Probability is the tool used for anticipating what the distribution of data should look like under a given model. Random phenomena are not haphazard: they display an order that emerges only in the long run and is described by a distribution. The mathematical description of variation is central to statistics. The probability required for statistical inference is not primarily axiomatic or combinatorial, but is oriented toward describing data distributions.

    Statistics is a tool that enables us to impose order on the disorganized cacophony of the real world of modern society. The business world has grown both in size and competition. Corporations must perform risky businesses, hence the growth in popularity and need for business statistics.

    Business statistics has grown out of the art of constructing charts and tables! It is a science of basing decisions on numerical data in the face of uncertainty.

    Business statistics is a scientific approach to decision making under risk. In practicing business statistics, we search for an insight, not the solution. Our search is for the one solution that meets all the business's needs with the lowest level of risk. Business statistics can take a normal business situation and with the proper data gathering, analysis, and re-search for a solution, turn it into an opportunity.

    While business statistics cannot replace the knowledge and experience of the decision maker, it is a valuable tool that the manager can employ to assist in the decision making process in order to reduce the inherent risk.

    Business Statistics provides justifiable answers to the following concerns for every consumer and producer:

    1. What is your or your customer's Expectation of the product/service you buy or that you sell? That is, what is a good estimate for m ?
    2. Given the information about your or your customer's expectation, what is the Quality of the product/service you buy or you sell. That is, what is a good estimate for s ?
    3. Given the information about your or your customer's expectation, and the quality of the product/service you buy or you sell, does the product/servive Compare with other existing similar types? That is, comparing several m 's.

    Visit also the following Web sites:
    What is Statistics?
    How to Study Statistics
    Decision Analysis

    Kinds of Lies: Lies, Damned Lies and Statistics

    "There are three kinds of lies -- lies, damned lies, and statistics." quoted in Mark Twain's autobiography.

    It is already an accepted fact that "Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write."

    The following are some examples as how statistics could be misused in advertising, which can be described as the science of arresting human unintelligence long enough to get money from it. The founder of Revlon says "In factory we make cosmetics; in the store we sell hope."

    In most cases, the deception of advertising is achieved by omission:

    1. The Incredible Expansion Toyota: "How can it be that an automobile that's a mere nine inches longer on the outside give you over two feet more room on the inside? May be it's the new math!" Toyota Camry Ad.

      Where is the fallacy in this statement? Taking volume as length! For example : 3x6x4=72 feet (cubic), 3x6x4.75=85.5 feet (cubic). It could be even more than 2 feet!

    2. Pepsi Cola Ad.: " In recent side-by-side blind taste tests, nationwide, more people preferred Pepsi over Coca-Cola".

      The questions are: Was it just some of taste tests, what was the sample size? It does not say "In all recent…"

    3. Correlation? Consortium of Electric Companies Ad. "96% of streets in the US are under-lit and, moreover, 88% of crimes take place on under-lit streets".
    4. Dependent or Independent Events? "If the probability of someone carrying a bomb on a plane is .001, then the chance of two people carrying a bomb is .000001. Therefore, I should start carrying a bomb on every flight."
    5. Paperboard Packaging Council's concerns: "University studies show paper milk cartons give you more vitamins to the gallon."

      How was the design of experiment? The research was sponsored by the council! Paperboard sales is declining!

    6. All the vitamins or just one? "You'd have to eat four bowls of Raisin Bran to get the vitamin nutrition in one bowl of Total".
    7. Six Times as Safe: "Last year 35 people drowned in boating accidents. Only 5 were wearing life jackets. The rest were not. Always wear life jacket when boating".

      What percentage of boaters wear life jackets? Conditional probability.

    8. A Tax Accountant Firm Ad.: "One of our officers would accompany you in the case of Audit".

      This sounds like a unique selling proposition, but it conceals the fact that the statement is a US Law.

    9. Dunkin Donuts Ad.: "Free 3 muffins when you buy three at the regular 1/2 dozen price."

    References and Further Readings:
    200% of Nothing, by A. Dewdney, John Wiley, New York, 1993. Based on his articles about math abuse in Scientific American, Dewdney lists the many ways we are manipulated with fancy mathematical footwork and faulty thinking in print ads, the news, company reports and product labels. He shows how to detect the full range of math abuses and defend against them.
    The Informed Citizen: Argument and Analysis for Today, by W. Schindley, Harcourt Brace, 1996. This rhetoric/reader explores the study and practice of writing argumentative prose. The focus is on exploring current issues in communities, from the classroom to cyberspace. The "interacting in communities" theme and the high-interest readings engage students, while helping them develop informed opinions, effective arguments, and polished writing.

    Visit also the Web site: Glossary of Mathematical Mistakes.

    Belief, Opinion, and Fact

    The letters in your course number: OPRE 504, stand for OPerations RE-search. OPRE is a science of making decisions (based on some numerical and measurable scales) by searching, and re-searching for a solution. I refer you to What Is OR/MS? for a deeper understanding of what OPRE is all about. Decision making under uncertainty must be based on facts not on personal opinion nor on belief.

    Belief, Opinion, and Fact
    Belief Opinion Fact
    Self says I'm right This is my view This is a fact
    Says to others You're wrong That is yours I can prove it to you

    Sensible decisions are always based on facts. We should not confuse facts with beliefs or opinions. Beliefs are defined as someone's own understanding or needs. In belief, "I am" always right and "you" are wrong. There is nothing that can be done to convince the person that what they believe in is wrong. Opinions are slightly less extreme than beliefs. An opinion means that a person has certain views that they think are right. They also know that others are entitled to their own opinions. People respect other's opinions and in turn expect the same. Contrary to beliefs and opinions are facts. Facts are the basis of decisions. A fact is something that is right, and one can prove it to be true based on evidence and logical arguments.

    Examples for belief, opinion, and facts can be found in religion, economics, and econometrics, respectively.

    With respect to belief, Henri Poincaré said "Doubt everything or believe everything: these are two equally convenient strategies. With either we dispense with the need to think."

    How to Assign Probabilities?

    Probability is an instrument to measure the likelihood of the occurrence of an event. There are three major approaches of assigning probabilities as follows:
    1. Classical Approach: Classical probability is predicated on the condition that the outcomes of an experiment are equally likely to happen. The classical probability utilizes the idea that the lack of knowledge implies that all possibilities are equally likely. The classical probability is applied when the events have the same chance of occurring (called equally likely events), and the set of events are mutually exclusive and collectively exhaustive. The classical probability is defined as:

      P(X) = Number of favorable outcomes / Total number of possible outcomes

    2. Relative Frequency Approach: Relative probability is based on accumulated historical or experimental data. Frequency-based probability is defined as:

      P(X) = Number of times an event occurred / Total number of opportunities for the event to occur.

      Note that relative probability is based on the ideas that what has happened in the past will hold.

    3. Subjective Approach: The subjective probability is based on personal judgment and experience. For example, medical doctors sometimes assign subjective probability to the length of life expectancy for a person who has cancer.

    General Laws of Probability

    1. General Law of Addition: When two or more events will happen at the same time, and the events are not mutually exclusive, then:

      P(X or Y) = P(X) + P(Y) - P(X and Y)

    2. Special Law of Addition: When two or more events will happen at the same time, and the events are mutually exclusive, then:

      P(X or Y) = P(X) + P(Y)

    3. General Law of Multiplication: When two or more events will happen at the same time, and the events are dependent, then the general rule of multiplicative law is used to find the joint probability:

      P(X and Y) = P(X) . P(Y|X),

      where P(X|Y) is a conditional probability.

    4. Special Law of Multiplicative: When two or more events will happen at the same time, and the events are independent, then the special rule of multiplication law is used to find the joint probability:

      P(X and Y) = P(X) . P(Y)

    5. Conditional Probability Law: A conditional probability is denoted by P(X|Y). This phrase is read: the probability that X will occur given that Y is known to have occurred.

      Conditional probabilities are based on knowledge of one of the variables. The conditional probability of an event, such as X, occurring given that another event, such as Y, has occurred is expressed as:

      P(X|Y) = P(X and Y) / P(Y)

      Provided P(y) is not zero. Note that when using the conditional law of probability, you always divide the joint probability by the probability of the event after the word given. Thus, to get P(X given Y), you divide the joint probability of X and Y by the unconditional probability of Y. In other words, the above equation is used to find the conditional probability for any two dependent events.

      A special case of the Bayes Theorem is:

      P(X|Y) = P(Y|X). P(X) / P(Y)

      If two events, such as X and Y, are independent then:

      P(X|Y) = P(X),

      P(Y|X) = P(Y)

    Mutually Exclusive versus Independent Events

    Mutually Exclusive (ME): Event A and B are M.E if both cannot occur simultaneously. That is, P[A and B] = 0.

    Independency (Ind.): Events A and B are independent if having the information that B already occurred does not change the probability that A will occur. That is P[A given B occurred] = P[A].

    If two events are ME they are also Dependent: P(A given B) = P[A and B]/P[B], and since P[A and B] = 0 (by ME), then P[A given B] = 0. Similarly,

    If two events are Dependent then they are also not ME.

    If two events are Dependent then they may or may not be ME.

    If two events are not ME, then they may or may not be Independent.

    The following Figure contains all possibilities. The notations used in this table are as follows: X means does not imply, question mark ? means it may or may not imply, while the check mark means it implies.

    Mutual Exclusive vs. Independent Events
    Bernstein was the first to discovere that (probabilistic) pairwise independency and mutual independency for a collection of events A1,..., An are different notions.

    Different Schools of Thought in Inferential Statistics

    There are few different schools of thoughts in statistics. They are introduced sequentially in time by necessity.

    The Birth Process of a New School of Thought

    The process of devising a new school of thought in any field has always taken a natural path. Birth of new schools of thought in statistics is not an exception. The birth process is outlined below:

    Given an already established school, one must work within the defined framework.

    A crisis appears, i.e., some inconsistencies in the framework result from its own laws.

    Response behavior:

    1. Reluctance to consider the crisis.
    2. Try to accommodate and explain the crisis within the existing framework.
    3. Conversion of some well-known scientists attracts followers in the new school.

    The following Figure illustrates the three major schools of thought; namely, the Classical (attributed to Laplace), Relative Frequency (attributed to Fisher), and Bayesian (attributed to Savage). The arrows in this figure represent some of the main criticisms among Objective, Frequentist, and Subjective schools of thought. To which school do you belong? Read the conclusion in this figure.

    what type of statistician are you?

    Bayesian, Frequentist, and Classical Methods

    The problem with the Classical Approach is that what constitutes an outcome is not objectively determined. One person's simple event is another person's compound event. One researcher may ask, of a newly discovered planet, "what is the probability that life exists on the new planet?" while another may ask "what is the probability that carbon-based life exists on it?"

    Bruno de Finetti, in the introduction to his two-volume treatise on Bayesian ideas, clearly states that "Probabilities Do not Exist". By this he means that probabilities are not located in coins or dice; they are not characteristics of things like mass, density, etc.

    Some Bayesian approaches consider probability theory as an extension of deductive logic to handle uncertainty. It purports to deduce from first principles the uniquely correct way of representing your beliefs about the state of things, and updating them in the light of the evidence. The laws of probability have the same status as the laws of logic. These Bayesian approahe is explicitly "subjective" in the sense that it deals with the plausibility which a rational agent ought to attach to the propositions she considers, "given her current state of knowledge and experience." By contrast, at least some non-Bayesian approaches consider probabilities as "objective" attributes of things (or situations) which are really out there (availability of data).

    A Bayesian and a classical statistician analyzing the same data will generally reach the same conclusion. However, the Bayesian is better able to quantify the true uncertainty in his analysis, particularly when substantial prior information is available. Bayesians are willing to assign probability distribution function(s) to the population's parameter(s) while frequentists are not.

    From a scientist's perspective, there are good grounds to reject Bayesian reasoning. The problem is that Bayesian reasoning deals not with objective, but subjective probabilities. The result is that any reasoning using a Bayesian approach cannot be publicly checked -- something that makes it, in effect, worthless to science, like non replicative experiments.

    Bayesian perspectives often shed a helpful light on classical procedures. It is necessary to go into a Bayesian framework to give confidence intervals the probabilistic interpretation which practitioners often want to place on them. This insight is helpful in drawing attention to the point that another prior distribution would lead to a different interval.

    A Bayesian may cheat by basing the prior distribution on the data; a Frequentist can base the hypothesis to be tested on the data. For example, the role of a protocol in clinical trials is to prevent this from happening by requiring the hypothesis to be specified before the data are collected. In the same way, a Bayesian could be obliged to specify the prior in a public protocol before beginning a study. In a collective scientific study, this would be somewhat more complex than for Frequentist hypotheses because priors must be personal for coherence to hold.

    A suitable quantity that has been proposed to measure inferential uncertainty; i.e., to handle the a priori unexpected, is the likelihood function itself.

    If you perform a series of identical random experiments (e.g., coin tosses), the underlying probability distribution that maximizes the probability of the outcome you observed is the probability distribution proportional to the results of the experiment.

    This has the direct interpretation of telling how (relatively) well each possible explanation (model), whether obtained from the data or not, predicts the observed data. If the data happen to be extreme ("atypical") in some way, so that the likelihood points to a poor set of models, this will soon be picked up in the next rounds of scientific investigation by the scientific community. No long run frequency guarantee nor personal opinions are required.

    There is a sense in which the Bayesian approach is oriented toward making decisions and the frequentist hypothesis testing approach is oriented toward science. For example, there may not be enough evidence to show scientifically that agent X is harmful to human beings, but one may be justified in deciding to avoid it in one's diet.

    Since the probability (or the distribution of possible probabilities) is continuous, the probability that the probability is any specific point estimate is really zero. This means that in a vacuum of information, we can make no guess about the probability. Even if we have information, we can really only guess at a range for the probability.

    Further Readings:
    Land F., Operational Subjective Statistical Methods, Wiley, 1996. Presents a systematic treatment of subjectivist methods along with a good discussion of the historical and philosophical backgrounds of the major approaches to probability and statistics.
    Plato, Jan von, Creating Modern Probability, Cambridge University Press, 1994. This book provides a historical point of view on subjectivist and objectivist probability school of thoughts.
    Weatherson B., Begging the question and Bayesians, Studies in History and Philosophy of Science, 30(4), 687-697, 1999.
    Zimmerman H., Fuzzy Set Theory, Kluwer Academic Publishers, 1991. Fuzzy logic approaches to probability (based on L.A. Zadeh and his followers) present a difference between "possibility theory" and probability theory.

    For more information, visit the Web sites Bayesian Inference for the Physical Sciences, Bayesians vs. Non-Bayesians, Society for Bayesian Analysis, Probability Theory As Extended Logic, and Bayesians worldwide.

    Type of Data and Levels of Measurement

    Information can be collected in statistics using qualitative or quantitative data.

    Qualitative data, such as eye color of a group of individuals, is not computable by arithmetic relations. They are labels that advise in which category or class an individual, object, or process fall. They are called categorical variables.

    Quantitative data sets consist of measures that take numerical values for which descriptions such as means and standard deviations are meaningful. They can be put into an order and further divided into two groups: discrete data or continuous data. Discrete data are countable data, for example, the number of defective items produced during a day's production. Continuous data, when the parameters (variables) are measurable, are expressed on a continuous scale. For example, measuring the height of a person.

    The first activity in statistics is to measure or count. Measurement/counting theory is concerned with the connection between data and reality. A set of data is a representation (i.e., a model) of the reality based on a numerical and mensurable scales. Data are called "primary type" data if the analyst has been involved in collecting the data relevant to his/her investigation. Otherwise, it is called "secondary type" data.

    Data come in the forms of Nominal, Ordinal, Interval and Ratio (remember the French word NOIR for color black). Data can be either continuous or discrete.

    Level of Measurements
    Nominal Ordinal Interval/Ratio
    Ranking? no yes yes
    Numerical difference no no yes

    Zero and unit of measurement are arbitrary in the Interval scale. While the unit of measurement is arbitrary in Ratio scale, its zero point is a natural attribute. The categorical variable is measured on an ordinal or nominal scale.

    Measurement theory is concerned with the connection between data and reality. Both statistical theory and measurement theory are necessary to make inferences about reality.

    Since statisticians live for precision, they prefer Interval/Ratio levels of measurement.

    Visit the Web site Measurement theory: Frequently Asked Questions

    Number of Class Intervals in a Histogram

    Before we can construct our frequency distribution we must determine how many classes we should use. This is purely arbitrary, but too few classes or too many classes will not provide as clear a picture as can be obtained with some more nearly optimum number. An empirical relationship, known as Sturge's Rule, may be used as a useful guide to determine the optimal number of classes (k) is given by

    k = the smallest integer greater than or equal to 1 + 3.332 Log(n)

    where k is the number of classes, Log is in base 10, n is the total number of the numerical values which comprise the data set.

    Therefore, class width is:

    (highest value - lowest value) / (1 + 3.332 Logn)

    where n is the total number of items in the data set.

    To have an "optimum" you need some measure of quality -- presumably in this case, the "best" way to display whatever information is available in the data. The sample size contributes to this; so the usual guidelines are to use between 5 and 15 classes, with more classes possible if you have a larger sample. You should take into account a preference for tidy class widths, preferably a multiple of 5 or 10, because this makes it easier to understand.

    Beyond this it becomes a matter of judgement. Try out a range of class widths, and choose the one that works best. (This assumes you have a computer and can generate alternative histograms fairly readily.)

    There are often management issues that come into play as well. For example, if your data is to be compared to similar data -- such as prior studies, or from other countries -- you are restricted to the intervals used therein.

    If the histogram is very skewed, then unequal classes should be considered. Use narrow classes where the class frequencies are high, wide classes where they are low.

    The following approaches are common:

    Let n be the sample size, then the number of class intervals could be

    MIN {n, 10 Log(n) }.

    The Log is the logarithm in base 10. Thus for 200 observations you would use 14 intervals but for 2000 you would use 33.


    1. Find the range (highest value - lowest value).
    2. Divide the range by a reasonable interval size: 2, 3, 5, 10 or a multiple of 10.
    3. Aim for no fewer than 5 intervals and no more than 15.

    Visit also the Web site Histogram Applet, and Histogram Generator

    How to Construct a BoxPlot

    A BoxPlot is a graphical display that has many characteristics. It includes the presence of possible outliers. It illustrates the range of data. It shows a measure of dispersion such as the upper quartile, lower quartile and interquartile range (IQR) of the data set as well as the median as a measure of central location which is useful for comparing sets of data. It also gives an indication of the symmetry or skewness of the distribution. The main reason for the popularity of boxplots is that they offer a lot of information in a compact way.

    Steps to Construct a BoxPlot:

    1. Horizontal lines are drawn at the median and at the upper and lower quartiles. These horizontal lines are joined by vertical lines to produce the box.
    2. A vertical lines is drawn up from the upper quartile to the most extreme data point that is within a distance of 1.5 (IQR) of the upper quartile. A similar defined vertical line is drawn from the lower quartile.
    3. Each data point beyond the end of the vertical line is marked with and asterisk (*).

    Probability, Chance, Likelihood, and Odds

    "Probability" has an exact technical meaning -- well, in fact it has several, and there is still debate as to which term ought to be used. However, for most events for which probability is easily computed e.g. rolling of a die the probability of getting a four [::], almost all agree on the actual value (1/6), if not the philosophical interpretation. A probability is always a number between 0 [not "quite" the same thing as impossibility: it is possible that "if" a coin were flipped infinitely many times, it would never show "tails", but the probability of an infinite run of heads is 0] and 1 [again, not "quite" the same thing as certainty but close enough].

    The word "chance" or "chances" is often used as an approximate synonym of "probability", either for variety or to save syllables. It would be better practice to leave "chance" for informal use, and say "probability" if that is what is meant.

    In cases where the probability of an observation is described by a parametric model, the "likelihood" of a parameter value given the data is defined to be the probability of the data given the parameter. One occasionally sees "likely" and "likelihood", however, these terms are used casually as synonyms for "probable" and "probability".

    "Odds" is a probabilistic concept related to probability. It is the ratio of the probability (p) of an event to the probability (1-p) that it does not happen: p/(1-p). It is often expressed as a ratio, often of whole numbers; e.g., "odds" of 1 to 5 in the die example above, but for technical purposes the division may be carried out to yield a positive real number (here 0.2). The logarithm of the odds ratio is useful for technical purposes, as it maps the range of probabilities onto the (extended) real numbers in a way that preserves symmetry between the probability that an event occurs and the probability that it does not occur.

    Odds are a ratio of nonevents to events. If the event rate for a disease is 0.1 (10 per cent), its nonevent rate is 0.9 and therefore its odds are 9:1. Note that this is not the same expression as the inverse of event rate.

    Another way to compare probabilities and odds is using "part-whole thinking" with a binary (dichotomous) split in a group. A probability is often a ratio of a part to a whole; e.g., the ratio of the part [those who survived 5 years after being diagnosed with a disease] to the whole [those who were diagnosed with the disease]. Odds are often a ratio of a part to a part; e.g., the odds against dying are the ratio of the part that succeeded [those who survived 5 years after being diagnosed with a disease] to the part that 'failed' [those who did not survive 5 years after being diagnosed with a disease].

    Obviously, probability and odds are intimately related: Odds = p / (1-p). Note that probability is always between zero and one, whereas odds range from zero to infinity.

    Aside from their value in betting, odds allow one to specify a small probability (near zero) or a large probability (near one) using large whole numbers (1,000 to 1 or a million to one). Odds magnify small probabilities (or large probabilities) so as to make the relative differences visible. Consider two probabilities: 0.01 and 0.005. They are both small. An untrained observer might not realize that one is twice as much as the other. But if expressed as odds (99 to 1 versus 199 to 1) it may be easier to compare the two situations by focusing on large whole numbers (199 versus 99) rather than on small ratios or fractions.

    Visit also the Web site Counting and Combinatorial

    What Is "Degrees of Freedom"

    Recall that in estimating the population's variance, we used (n-1) rather than n, in the denominator. The factor (n-1) is called "degrees of freedom."

    Estimation of the Population Variance: Variance in a population is defined as the average of squared deviations from the population mean. If we draw a random sample of n cases from a population where the mean is known, we can estimate the population variance in an intuitive way. We sum the deviations of scores from the population mean and divide this sum by n. This estimate is based on n independent pieces of information and we have n degrees of freedom. Each of the n observations, including the last one, is unconstrained ('free' to vary).

    When we do not know the population mean, we can still estimate the population variance, but now we compute deviations around the sample mean. This introduces an important constraint because the sum of the deviations around the sample mean is known to be zero. If we know the value for the first (n-1) deviations, the last one is known. There are only n-1 independent pieces of information in this estimate of variance.

    If you study a system with n parameters xi, i =,1..., n you can represent it in a n-dimension space. Any point of this space shall represent a potential state of your system. If your n parameters could vary independently, then your system would be fully described in a n-dimension hyper-volume. Now, imagine you've got one constraint between the parameters (an equation relying your n parameters), then your system would be described by a (n-1)-dimension hyper-surface. For example, in three dimensional space, a linear relationship means a plane which is 2-dimensional.

    In statistics, your n parameters are your n data. To evaluate variance, you first need to infer the mean E(X). So when you evaluate the variance, you've got one constraint on your system (which is the expression of the mean), and it only remains (n-1) degrees of freedom to your system.

    Therefore, we divide the sum of squared deviations by n-1 rather than by n when we have sample data. On average, deviations around the sample mean are smaller than deviations around the population mean. This is because our sample mean is always in the middle of our sample scores; in fact the minimum possible sum of squared deviations for any sample of numbers is around the mean for that sample of numbers. Thus, if we sum the squared deviations from the sample mean and divide by n, we have an underestimate of the variance in the population (which is based on deviations around the population mean).

    If we divide the sum of squared deviations by n-1 instead of n, our estimate is a bit larger, and it can be shown that this adjustment gives us an unbiased estimate of the population variance. However, for large n, say, over 30, it does not make too much of difference if we divide by n, or n-1.

    Degrees of Freedom in ANOVA: You will see the key parse "degrees of freedom" also appearing in the Analysis of Variance (ANOVA) tables. If I tell you about 4 numbers, but don't say what they are, the average could be anything. I have 4 degrees of freedom in the data set. If I tell you 3 of those numbers, and the average, you can guess the fourth number. The data set, given the average, has 3 degrees of freedom. If I tell you the average and the standard deviation of the numbers, I have given you 2 pieces of information, and reduced the degrees of freedom to from 4 to 2. You only need to know 2 of the numbers' values to guess the other 2.

    In an ANOVA table, degree of freedom (df) is the divisor in SS/df which will result in an unbiased estimate of the variance of a population.

    df = N - k, where N is the sample size, and k is a small number, equal to the number of "constraints", the number of "bits of information" already "used up". Degree of freedom is an additive quantity; total amounts of it can be "partitioned" into various components.

    For example, suppose we have a sample of size 13 and calculate its mean, and then the deviations from the mean, only 12 of the deviations are free to vary: once one has found 12 of the deviations, the thirteenth one is determined. Therefore, if one is estimating a population variance from a sample, k = 1.

    In bivariate correlation or regression situations, k = 2: the calculation of the sample means of each variable "uses up" two bits of information, leaving N - 2 independent bits of information.

    In a one-way analysis of variance (ANOVA) with g groups, there are three ways of using the data to estimate the population variance. If all the data are pooled, the conventional SST/(n-1) would provide an estimate of the population variance.

    If the treatment groups are considered separately, the sample means can also be considered as estimates of the population mean, and thus SSb/(g - 1) can be used as an estimate. The remaining ("within-group", "error") variance can be estimated from SSw/(n - g). This example demonstrates the partitioning of df: df total = n - 1 = df(between) + df(within) = (g - 1) + (n - g).

    Therefore, the simple 'working definition' of df is ‘sample size minus the number of estimated parameters'. A fuller answer would have to explain why there are situations in which the degrees of freedom is not an integer. After, we said all this, the best explanation, is mathematical in that we use df to obtain an unbiased estimate.

    In summary, the concept of degrees of freedom is used for the following two different purposes:

    • Parameter(s) of certain distributions, such as F, and t-distribution are called degrees of freedom. Therefore, degrees of freedom could be positive non-integer number(s).

    • Degrees of freedom is used to obtain unbiased estimate for the population parameters.

    Outlier Removal

    Because of the potentially large variance, outliers could be the outcome of sampling. It's perfectly correct to have such an observation that legitimately belongs to the study group by definition. Lognormally distributed data (such as international exchange rate), for instance, will frequently exhibit such values.

    Therefore, you must be very careful and cautious: before declaring an observation "an outlier," find out why and how such observation occurred. It could even be an error at the data entering stage.

    First, construct the BoxPlot of your data. Form the Q1, Q2, and Q3 points which divide the samples into four equally sized groups. (Q2 = median) Let IQR = Q3 - Q1. Outliers are defined as those points outside the values Q3+k*IQR and Q1-k*IQR. For most case one sets k=1.5.

    Another alternative is the following algorithm

    a) Compute s of whole sample.
    b) Define a set of limits off the mean: mean + ks, mean - ks sigma (Allow user to enter k. A typical value for k is 2.)
    c) Remove all sample values outside the limits.

    Now, iterate N times through the algorithm, each time replacing the sample set with the reduced samples after applying step (c).

    Usually we need to iterate through this algorithm 4 times.

    As mentioned earlier, a common "standard" is any observation falling beyond 1.5 (interquartile range) i.e., (1.5 IQRs) ranges above the third quartile or below the first quartile. The following SPSS program, helps you in determining the outliers.

        DATA LIST          FREE FILE='A' / X1
        VAR LABLE
                    X1 'INPUT DATA'
        SORT CASES BY ZX1(A)

    Statistical Summaries

    Representative of a Sample: Measures of Central Tendency Summaries

    How do you describe the "average" or "typical" piece of information in a set of data? Different procedures are used to summarize the most representative information depending of the type of question asked and the nature of the data being summarized.

    Measures of location give information about the location of the central tendency within a group of numbers. The measures of location presented in this unit for ungrouped (raw) data are the mean, the median, and the mode.

    Mean: The arithmetic mean (or the average or simple mean) is computed by summing all numbers in an array of numbers (xi) and then dividing by the number of observations (n) in the array.

    the Mean

    The mean uses all of the observations, and each observation affects the mean. Even though the mean is sensitive to extreme values, i.e., extremely large or small data can cause the mean to be pulled toward the extreme data, it is still the most widely used measure of location. This is due to the fact that the mean has valuable mathematical properties that make it convenient for use with inferential statistical analysis. For example, the sum of the deviations of the numbers in a set of data from the mean is zero, and the sum of the squared deviations of the numbers in a set of data from the mean is the minimum value.

    Weighted Mean: In some cases, the data in the sample or population should not be weighted equally, rather each value should be weighted according to its importance.

    Median: The median is the middle value in an ordered array of observations. If there is an even number of observations in the array, the median is the average of the two middle numbers. If there is an odd number of data in the array, the median is the middle number.

    The median is often used to summarize the distribution of an outcome. If the distribution is skewed, the median and the IQR may be better than other measures to indicate where the observed data are concentrated.

    Generally, the median provides a better measure of location than the mean when there are some extremely large or small observations; i.e., when the data are skewed to the right or to the left. For this reason, median income is used as the measure of location for the U.S. household income. Note that if the median is less than the mean, the data set is skewed to the right. If the median is greater than the mean, the data set is skewed to the left.

    Mode: The mode is the most frequently occurring value in a set of observations. Why use the mode? The classic example is the shirt/shoe manufacturer who wants to decide what sizes to introduce. Data may have two modes. In this case, we say the data are bimodal, and sets of observations with more than two modes are referred to as multimodal. Note that the mode does not have important mathematical properties for future use. Also, the mode is not a helpful measure of location, because there can be more than one mode or even no mode.

    Whenever, more than one mode exist, then the population from which the sample came is a mixture of more than one population. Almost all standard statistical analyses assume that the population is homogeneous, meaning that its density is unimodal.

    Notice that Excel is a very limited statistical software. For example, it displays only one mode, the first one. Unfortunately, this is very misleading. However, you may find out if there are others by inspection only, as follow: Create a frequency distribution, invoke the menu sequence: Tools, Data analysis, Frequency and follow instructions on the screen. You will see the frequency distribution and then find the mode visually. Unfortunately, Excel does not draw a Stem and Leaf diagram. All commercial off-the-shelf software, such as SAS and SPSS display a Stem and Leaf diagram which is a frequency distribution of a given data set.

    Quartiles & Percentiles: Quantiles are values that separate a ranked data set into four equal classes. Whereas percentiles are values that separate a ranked the data into 100 equal classes. The widely used quartiles are the 25th, 50th, and 75th percentiles.

    Selecting Among the Mean, Median, and Mode

    It is a common mistake to specify the wrong index for central tenancy.
    The first consideration is the type of data, if the variable is categorical, the mode is the single measure that best describes that data.

    The second consideration in selecting the index is to ask whether the total of all observations is of any interest. If the answer is yes, then the mean is the proper index of central tendency.

    If the total is of no interest, then depending on whether the histogram is symmetric or skewed one must use either mean or median, respectively.

    In all cases the histogram must be unimodal.

    Suppose that four people want to get together to play poker. They live on 1st Street, 3rd Street, 7th Street, and 15th Street. They want to select a house that involves the minimum amount of driving for all parties concerned.

    Let's suppose that they decide to minimize the absolute amount of driving. If they met at 1st Street, the amount of driving would be 0 + 2 + 6 + 14 = 22 blocks. If they met at 3rd Street, the amount of driving would be 2 + 0+ 4 + 12 = 18 blocks. If they met at 7th Street, 6 + 4 + 0 + 8 = 18 blocks. Finally, at 15th Street, 14 + 12 + 8 + 0 = 34 blocks.

    So the two houses that would minimize the amount of driving would be 3rd or 7th Street. Actually, if they wanted a neutral site, any place on 4th, 5th, or 6th Street would also work.

    Note that any value between 3 and 7 could be defined as the median of 1, 3, 7, and 15. So the median is the value that minimizes the absolute distance to the data points.

    Now the person at 15th is upset at always having to do more driving. So the group agrees to consider a different rule. The decide to minimize the square of the distance driving. This is the least squares principle. By squaring, we give more weight to a single very long commute than to a bunch of shorter commutes. With this rule, the 7th Street house (36 + 16 + 0 + 64 = 116 square blocks) is preferred to the 3rd Street house (4 + 0 + 16 + 144 = 164 square blocks). If you consider any location, and not just the houses themselves, then 9th Street is the location that minimizes the square of the distances driven.

    Find the value of x that minimizes
    (1 - x)2 + (3 - x)2 +(7 - x)2 + (15 - x)2.

    The value that minimizes the sum of squared values is 6.5 which is also equal to the arithmetic mean of 1, 3, 7, and 15. With calculus, it's easy to show that this holds in general.

    For moderately asymmetrical distributions the mode, median and mean satisfy the formula: mode=3 (median) - 2(mean).

    Consider a small sample of scores with an even number of cases, for example, 1, 2, 4, 7, 10, and 12. The median is 5.5, the midpoint of the interval between the scores of 4 and 7.

    As we discussed above, it is true that the median is a point around which the sum absolute deviations is minimized. In this example the sum of absolute deviation is 22. However, it is not a unique point. Any point in the 4 to 7 region will have the same value of 22 for the sum of the absolute deviations.

    Indeed, medians are tricky. The 50%-50% (above-below) is not quite correct. For example, 1, 1, 1, 1, 1, 1, 8 has no median. The convention says that, the median is 1, however about 14% of the data lie strictly above it, 100% of the data is the median. This generalizes to other percentiles.

    We will make use of this idea in regression analysis. In an analogous argument, the regression line is a unique line which minimizes the sum of the squared deviations from it. There is no unique line which minimizes the sum of the absolute deviations from it.

    Quality of a Sample: Measures of Dispersion

    Average by itself is not a good indication of quality. You need to know the variance to make any educated assessment. We are reminded of the dilemma of the six-foot tall statistician who drowned in a stream that had an average depth of three feet.

    These are statistical procedures for describing the nature and extent of differences among the information in the distribution. A measure of variability is generally reported with a measure of central tendency.

    Statistical measures of variation are numerical values that indicate the variability inherent in a set of data measurements. Note that a small value for a measure of dispersion indicates that the data are concentrated around the mean; therefore, the mean is a good representative of the data set. On the other hand, a large measure of dispersion indicates that the mean is not a good representative of the data set. Also, measures of dispersion can be used when we want to compare the distributions of two or more sets of data. Quality of a data set is measured by its variability: Larger variability indicates lower quality. That is why high variation makes the manager very worried. Your job, as a statistician is to measure the variation, and if it is too high and unacceptable, then it is the job of the technical staff, such as engineers, to fix the process.

    The decision situations with flat uncertainty have the largest risk. For simplicity, consider the case when there are only two outcomes one with probability of p. Then, the variation in the outcomes is p(1-p). This variation is the largest if we set p = 50%. That is, equal chance for each outcome. In such a case, the quality of information is at its lowest level. Remember, quality of information and variation are inversely related. Larger the variation in the data, the lower the quality of the data (i.e., information). Remember that the Devil is in the Deviations.

    The four most common measures of variation are the range, variance, standard deviation, and coefficient of variation.

    Range: The range of a set of observations is the absolute value of the difference between the largest and smallest values in the set. It measures the size of the smallest contiguous interval of real numbers that encompasses all of the data values. It is not useful when extreme values are present. It is based solely on two values, not on the entire data set. In addition, it cannot be defined for open-ended distributions such as Normal distribution.

    Normal distribution does not have a range. A student said "since the tails of normal density function never touch the x-axis, at the same time since for an observation to contribute to forming the such a curve, very large positive and negative values must exist" Yet such remote values are always possible, but increasingly improbable. This encapsulates the asymptotic behavior of normal density very well.

    Variance: An important measure of variability is variance. Variance is the average of the squared deviations of each observation in the set from the arithmetic mean of all of observations.

    Variance = S (xi - ) 2 / (n - 1), n 2.

    The variance is a measure of spread or dispersion among values in a data set. Therefore, the greater the variance, the lower the quality.

    The variance is not expressed in the same units as the observations. In other words, the variance is hard to understand because the deviations from the mean are squared, making it too large for logical explanation. This problem can be solved by working with the square root of the variance, which is called the standard deviation.

    Standard Deviation: Both variance and standard deviation provide the same information; one can always be obtained from the other. In other words, the process of computing a standard deviation always involves computing a variance. Since standard deviation is the square root of the variance, it is always expressed in the same units as the raw data:

    Standard Deviation

    For large data set (more than 30, say), approximately 68% of the data will fall within one standard deviation of the mean, 95% fall within two standard deviations, and 97.7% (or almost 100% ) fall within three standard deviations (S) from the mean.

    Standard Error: Standard error is a statistic indicating the accuracy of an estimate. That is, it tells us to assess how different the estimate ( such as ) is from the population parameter (such as m). It is therefore, the standard deviation of a sampling distribution of the estimator such as 's.

    Coefficient of Variation: Coefficient of Variation (CV) is the relative deviation with respect to size :

    Coefficient of Variation

    CV is independent of the unit of measurement. In estimation of a parameter when CV is less than say 10%, the estimate is assumed acceptable. The inverse of CV; namely 1/CV is called the Signal-to-noise Ratio.

    The coefficient of variation is used to represent the relationship of the standard deviation to the mean, telling how much representative the mean is of the numbers from which it came. It expresses the standard deviation as a percentage of the mean; i.e., it reflects the variation in a distribution relative to the mean.

    Z Score: how many standard deviations a given point (i.e. observations) is above or below the mean. In other words, a Z score represents the number of standard deviations an observation (x) is above or below the mean. The larger the Z value, the further away a value will be from the mean. Note that values beyond three standard deviations are very unlikely. Note that if a Z score is negative, the observation (x) is below the mean. If the Z score is positive, the observation (x) is above the mean. The Z score is found as:

    Z = (x - ) / standard deviation of X

    The Z score is a measure of the number of standard deviations that an observation is above or below the mean. Since the standard deviation is never negative, a positive Z score indicates that the observation is above the mean, a negative Z score indicate that the observation is below the mean. Note that Z is a dimensionless value, and is therefore a useful measure by which to compare data values from two different populations even those measured by different units.

    Z-Transformation: Applying the formula z = (X - m) / s will always produce a transformed variable with a mean of zero and a standard deviation of one. However, the shape of the distribution will not be affected by the transformation. If X is not normal then the transformed distribution will not be normal either. In the following SPSS command variable x is transformed to zx.

    descriptives variables=x(zx)

    You have heard the terms z value, z test, z transformation, and z score. Do all of these terms mean the same thing? Certainly not:

    The z value is refereed to the critical value (a point on the horizontal axes) of the Normal (o, 1) density function, for a given area to the left of that z-value.

    The z test is refereed to the procedures for testing the equality of mean (s) of one (or two) population(s).

    z score of a given observation x in a sample of size n, is simply (x - average of the sample) divided by the standard deviation of the sample.

    The z transformation of a set of observations of size n is simply (each observation - average of all observation) divided by the standard deviation among all observations. The aim is to produce a transformed data set with a mean of zero and a standard deviation of one. This makes the transformed set dimensionless and manageable with respect to its magnitudes. It also used in comparing several data sets measured using different scales of measurements.

    Pearson coined the term "standard deviation" sometime near 1900. The idea of using squared deviations goes back to Laplace in the early 1800's.

    Finally, notice again, that the trandforming raw scores to z scores does NOT normalize the data.

    Guess a Distribution to Fit Your Data: Skewness & Kurtosis

    A pair of statistical measures skewness and kurtosis is a measuring tool which is used in selecting a distribution(s) to fit your data. To make an inference with respect to the population distribution, you may first compute skewness and kurtosis from your random sample from the entire population. Then, locating a point with these coordinates on some widely used Skewness-Kurtosis Charts (available from your instructor upon request), guess a couple of possible distributions to fit your data. Finally, you might use the goodness-of-fit test to rigorously come up with the best candidate fitting your data. Removing outliers improves both skewness and kurtosis.

    Skewness: Skewness is a measure of the degree to which the sample population deviates from symmetry with the mean at the center.

    Skewness = S (xi - ) 3 / [ (n - 1) S 3 ], n 2.

    Skewness will take on a value of zero when the distribution is a symmetrical curve. A positive value indicates the observations are clustered more to the left of the mean with most of the extreme values to the right of the mean. A negative skewness indicates clustering to the right. In this case we have: Mean Median Mode. The reverse order holds for the observations with positive skewness.

    Kurtosis: Kurtosis is a measure of the relative peakedness of the curve defined by the distribution of the observations.

    Kurtosis = S (xi - ) 4 / [ (n - 1) S 4 ], n 2.

    Standard normal distribution has kurtosis of +3. A kurtosis larger than 3 indicates the distribution is more peaked than the standard normal distribution.

    Coefficient of Excess Kurtosis = Kurtosis - 3.

    A less than 3 kurtosis value means that the distribution is flatter than the standard normal distribution.

    Skewness and kurtosis can be used to check for normality via the the Jarque-Bera test. For large n, under the normality condition the quantity

    n {Skewness2 / 6 +((Kurtosis - 3)2) / 24)}

    follows a chi-square distribution with d.f. = 2.

    Further Reading:
    Tabachnick B., and L. Fidell, Using Multivariate Statistics, HarperCollins, 1996. Has a good discussion on applications and significance tests for skewness and kurtosis.

    Numerical Example & Discussions

    A Numerical Example: Given the following, small (n = 4) data set, compute the descriptive statistics: x1 = 1, x2 = 2, x3 = 3, and x4 = 6.

    (xi- )
    (xi - ) 2
    (xi - ) 3
    (xi - )4

    The mean is 12 / 4 = 3, the variance is s2 = 14 / 3 = 4.67, the standard deviation is s = (14/3) 0.5 = 2.16, the skewness is 18 / [3 (2.16) 3 ] = 0.5952, and finally, the kurtosis is 18 / [3 (2.16) 4 ] = 1.5.

    A Short Discussion

    Deviations about the mean m of a distribution is the basis for most of the statistical tests we will learn. Since we are measuring how much a set of scores is dispersed about the mean m , we are measuring variability. We can calculate the deviations about the mean m and express it as variance s2 or standard deviation s. It is very important to have a firm grasp of this concept because it will be a central concept throughout your statistics course.

    Both variance s2 and standard deviation s measure variability within a distribution. Standard deviation s is a number that indicates how much on average each of the values in the distribution deviates from the mean m (or center) of the distribution. Keep in mind that variance s2 measures the same thing as standard deviation s (dispersion of scores in a distribution). Variance s2, however, is the average squared deviations about the mean m . Thus, variance s2 is the square of the standard deviation s.

    Expected value and variance of are m and s2/n, respectively.

    Expected value and variance of S2 are s2 and 2s4 / (n-1), respectively.

    and S2 are the best estimators for m and s2. They are Unbiased (you may update your estimate); Efficient (they have the smallest variation among other estimators); Consistent (increasing sample size provides a better estimate); and Sufficient (you do not need to have the whole data set; what you need are Sxi and Sxi2 for estimations). Note also that the above variance for of S2 is justified only in the case where the population distribution tends to be normal, otherwise one may use bootstrapping techniques.

    In general, it is believed that the pattern of mode, median, and mean go from lower to higher in positive skewed data sets, and just the opposite pattern in negative skewed data sets. However, for example, in the following 23 numbers, mean=2.87, median=3, but the data is positively skewed:

    4 2 7 6 4 3 5 3 1 3 1 2 4 3 1 2 1 1 5 2 2 3 1

    and, the following 10 numbers have mean=median=mode=4, but the data set is left skewed:

    1 2 3 4 4 4 5 5 6 6

    Note also that, most commercial software donot correctly compute skewness and kurtosis. There is no easy way to determine confidence intervals about a computed skewness or kurtosis value from a small to medium sample. The literature gives tables based on asymptotic methods for sample sets larger than 100 for normal distributions only.

    You may have noticed that using the above numerical example on some computer packages such as SPSS, the skewness and the kurtosis are different from what we have computed. For example, the SPSS output for the skewness is 1.190. However, for large a sample size n, the results are identical.

    Reference and Further Readings:
    David H., Early Sample Measures of Variability, Statistical Science, 13, 1998, 368-377. This article provides a good historical accounts of statistical measures.
    Groeneveld R., A class of quantile measures for kurtosis, The American Statistician, 325, Nov. 1998.
    Hosking J., M, Moments or L moments? An example comparing two measures of distributional shape, The American Statistician, Vo.l 46, 186-189, 1992.

    Parameters' Estimation and Quality of a 'Good' Estimate

    Estimation is the process by which sample data are used to indicate the value of an unknown quantity in a population.

    Results of estimation can be expressed as a single value, known as a point estimate; or a range of values, known as a confidence interval.

    Whenever we use point estimation, we calculate the margin of error associated with that point estimation. For example, for the estimation of the population mean m, the margin of errors calculated as follows: ±1.96 SE().

    In newspapers and television reports on public opinion pools, the margin of error is the margin of "sampling error". There are many nonsampling errors that can and do affect the accuracy of polls. Here we talk about sampling error. The fact that subgroups have larger sampling error than one must include the following statement: "Other sources of error include but are not limited to, individuals refusing to participate in the interview and inability to connect with the selected number. Every feasible effort is made to obtain a response and reduce the error, but the reader (or the viewer) should be aware that some error is inherent in all research."

    To estimate means to esteem (to give value to). An estimator is any quantity calculated from the sample data which is used to give information about an unknown quantity in the population. For example, the sample mean is an estimator of the population mean m.

    Estimators of population parameters are sometimes distinguished from the true value by using the symbol 'hat'. For example, true population standard deviation s is estimated (from a sample) population standard deviation.

    Example: The usual estimator of the population mean is = Sxi / n, where n is the size of the sample and x1, x2, x3,.......,xn are the values of the sample. If the value of the estimator in a particular sample is found to be 5, then 5 is the estimate of the population mean µ.

    A "Good" estimator is the one which provides an estimate with the following qualities:

    Unbiasedness: An estimate is said to be an unbiased estimate of a given parameter when the expected value that of estimator can be shown to be equal to the parameter being estimated. For example, the mean of a sample is an unbiased estimate of the mean of the population from which the sample was drawn. Unbiasedness is a good quality for an estimate since in such a case, using weighted average of several estimates provides a better estimate than each one of those estimates. Therefore, unbiasedness allows us to upgrade our estimates. For example is your estimate of the population mean µ are say, 10, and 11.2 from two independent samples of equal sizes 20, and 30 respectively, then the estimate of the population mean µ based on both samples is [20 (10) + 30 (11.2)] (20 + 30) = 10.75.

    Consistency: The standard deviation of an estimate is called the standard error of that estimate. The larger the standard error means more error in your estimate. It is a commonly used index of the error entailed in estimating a population parameter based on the information in a random sample of size n from the entire population.

    An estimator is said to be "consistent" if increasing the sample size produces an estimate with smaller standard error. Therefore, your estimate is "consistent" with the sample size. That is, spending more money (to obtain a larger sample) produces a better estimate.

    Efficiency: An efficient estimate is the one which has the smallest standard error among all other estimators of equal size.

    Sufficiency: A sufficient estimator based on a statistic contains all the information which is present in the raw data. For example, the sum of your data is sufficient to estimate the mean of the population. You don't have to know the data set itself. This saves a lot of money if the data has to be transmitted by telecommunication network. Simply, send out the total, and the sample.

    A sufficient statistic t for a parameter q is a function of the sample data x1,...,xn, which contains all information in the sample about the parameter q . More formally, sufficiency is defined in terms of the likelihood function for q . For a sufficient statistic t, the Likelihood L(x1,...,xn| q ) can be written as

    g (t | q )*k(x1,...,xn)

    Since the second term does not depend on q , t is said to be a sufficient statistic for q .

    Another way of stating this for the usual problems is that one could construct a random process starting from the sufficient statistic, which will have exactly the same distribution as the full sample for all states of nature.

    To illustrate, let the observations be independent Bernoulli trials with the same probability of success. Suppose that there are n trials, and that person A observes which observations are successes, and person B only finds out the number of successes. Then if B places these successes at random points without replication, the probability that B will now get any given set of successes is exactly the same as the probability that A will see that set, no matter what the true probability of success happens to be.

    The widely used estimator of the population mean µ is = Sxi/n, where n is the size of the sample and x1, x2, x3,......., xn are the values of the sample that have all the above properties. Therefore, it is a "good" estimator.

    If you want an estimate of central tendency as a parameter for a test or for comparison, then small sample sizes are unlikely to yield any stable estimate. The mean is sensible in a symmetrical distribution, as a measure of central tendency, but, e.g., with ten cases you will not be able to judge whether you have a symmetrical distribution. However, the mean estimate is useful if you are trying to estimate the a population sum, or some other function of the expected value of the distribution. Would the median be a better measure? In some distributions (e.g., shirt size) the mode may be better. Box-plot will indicate outliers in the data set. If there are outliers, median is better than mean as a measure of the central tendency.

    If you have a yes/no question you probably want to calculate a proportion p of yeses (or noes). Under simple random sampling, the variance of p is p(1-p)/n, ignoring the finite population correction. Now a 95% confidence interval is 1.96 [p(1-p)/n]2. A conservative interval can be calculated assuming p(1-p) takes its maximum value, which it does when p = 1/2. Replace 1.96 by 2, put p = 1/2 and you have a 95% confidence interval of 1/n1/2. This approximation works well as long as p is not too close to 0 or 1. This useful approximation allows you to calculate approximate 95% confidence intervals.

    Conditions Under Which Most Statistical Testing Apply

    Don't just learn formulas and number-crunching: learn about the conditions under which statistical testing procedures apply. The following conditions are common to almost all tests:

    1. homogeneous population (see if there are more than one mode)
    2. sample must be random (to test this, perform the Runs Test).
    3. In addition to requirement No. 1, each population has a normal distribution (perform Test for Normality)
    4. homogeneity of variances. Variation in each population is almost the same as in the others.

      For 2 populations use the F-test. For 3 or more populations, there is a practical rule known as the "Rule of 2". In this rule one divides the highest variance of a sample to the lowest variance of the other sample. Given that the sample sizes are almost the same, and the value of this division is less than 2, then, the variations of the populations are almost the same.

      Notice: This important condition in analysis of variance (ANOVA and the t-test for mean differences) is commonly tested by the Levene or its modified test known as the Brown-Forsythe test. Unfortunately, both tests rely on the homogeneity of variances assumption!

    These assumptions are crucial, not for the method/computation, but for the testing using the resultant statistic. Otherwise, we can do, for example, ANOVA and regression without any assumptions, and the numbers come out the same -- simple computations give us least-square fits, partitions of variance, regression coefficients, and so on. Only when testing certain assumptions about independence, and homogeneous distribution of error terms known as residuals.

    Homogeneous Population

    Homogeneous Population: A homogeneous population is a statistical population which has a unique mode. To determine if a given population is homogeneous or not, construct the histogram of a random sample from the entire population. If there is more than one mode, then you have a mixture of population. Know that to perform any statistical testing, you need to make sure you are dealing with homogeneous population.

    Test for Randomness: The Runs Test

    A "run" is a maximal subsequence of like elements.

    Consider the following sequence (D for Defective, N for non-defective items) out of a production line: DDDNNDNDNDDD. Number of runs is R = 7, with n1 = 8, and n2 = 4 which are number of D's and N's (whichever).

    A sequence is random if it is neither "over-mixed" nor "under-mixed". An example of over-mixed sequence is DDDNDNDNDNDD, with R = 9 while under-mixed looks like DDDDDDDDNNNN with R = 2. There the above sequence seems to be random.

    The Runs Tests, which is also known as Wald-Wolfowitz Test, is designed to test the randomness of a given sample at 100(1- a)% confidence level. To conduct a runs test on a sample, perform the following steps:

    Step 1: compute the mean of the sample.

    Step 2: going through the sample sequence, replace any observation with +, or - depending on wether it is above or below the mean. Discard any ties.

    Step 3: compute R, n1, and n2.

    Step 4: compute the expected mean and variance of R, as follows:

    a =1 + 2n1n2/(n 1 + n2).

    s2 = 2n1n2(2n 1n2-n1- n2)/[[n1 + n2)2 (n1 + n2 -1)].

    Step 5: Compute z = (R-m)/ s.

    Step 6: Conclusion:

    If z > Za, then there might be cyclic, seasonality behavior (under-mixing).

    If z < - Za, then there might be a trend.

    If z < - Za/2, or z > Za/2, reject the randomness.

    Note: This test is valid for cases for which both n1 and n2 are large, say greater than 10. For small sample sizes special tables must be used.

    The SPSS command for the runs test:
    NPAR TEST RUNS(MEAN) X (the name of the variable).

    For example, suppose for a given sample of size 50, we have R = 24, n1 = 14 and n2 = 36. Test for randomness at a = 0.05.
    The Plugging these into the above formulas we have a = 16.95, s = 2.473, and z = -2.0 From Z-table, we have Z = 1.645. Therefore, there might be a trend, which means that the sample is not random.

    Visit the Web site Test for Randomness

    Lilliefors Test for Normality

    The following SPSS program computes the Kolmogrov-Smirinov-Lilliefors statistic called LS. It can easily be converted and run in any other platforms.

                X 'SAMPLE VALUES'

    The output is the statistic LS, which should be compared with the following critical values after setting a significance level a (as a function of the sample size n).

    Critical Values for the Lilliefors Test
    Significance Level
    Critical Value
    a = 0.15
    0.775 / ( n ˝ - 0.01 + 0.85 n )
    a = 0.10
    0.819 / ( n ˝ - 0.01 + 0.85 n )
    a = 0.05
    0.895 / ( n ˝ - 0.01 + 0.85 n )
    a = 0.025
    0.995 / ( n ˝ - 0.01 + 0.85 n )

    A normal probability plot will also help you distinguish between a systematic departure from normality when it shows up as a curve. In SAS do a PROC UNIVARIATE NORMAL PLOT. Bera-Jarque test, which is widely used by econometricians, might also be applicable.

    Further Reading
    Statistical inference by normal probability paper, by T. Takahashi, Computers & Industrial Engineering, Vol. 37, Iss. 1 - 2, pp 121-124, 1999.

    Bonferroni Method

    One may combine several t-tests by using the Bonferroni method. It works reasonably well when there are only a few tests, but as the number of comparisons increases above 8, the value of 't' required to conclude that a difference exists becomes much larger than it really needs to be and the method becomes over conservative.

    One way to make the Bonferroni t test less conservative is to use the estimate of the population variance computed from within the groups in the analysis of variance.

    t = ( 1 -2 )/ ( s2 / n1 + s2 / n2 )1/2

    where VW is the population variance computed from within the groups.

    Chi-square Tests

    The Chi-square is a distribution, as is the Normal and others. The Normal (or Gaussian or bell-shaped) often occurs naturally in real life. When we know the mean and variance of a Normal then it allows us to find probabilities. So if, for example, you knew some things about the average height of women in the nation (including the fact that heights are distributed normally, you could measure all the women in your extended family, find the average height, and determine a probability associated with your result; if the probability of getting your result, given your knowledge of women nationwide, is high, then your family's female height cannot be said to be different from average. If that probability is low, then your result is rare (given the knowledge about women nationwide), and you can say your family is different. You've just completed a test of the hypothesis that the average height of women in your family is different from the overall average.

    There are other (similar) tests where finding that probability means NOT using Normal distribution. One of these is a Chi-square test. For instance, if you tested the variance of your family's female heights (which is analogous to your previous test of the mean), you can't assume that the normal distribution is appropriate to use. This should make sense, since the Normal is bell-shaped, and variances have a lower limit of zero. So, while a variance could be any huge number, it gets bounded on the low side by zero. If you were to test whether the variance of heights in your family is different from the nation, a Chi-square test happens to be appropriate, given our original above conditions. The formula and procedure is in your textbook.

    Crosstables: The variance is not the only thing for which you use a Chi-square test for. Often times it is used to test relationship among two categorical type data, or independence of two variables, such as cigarette smoking and drug use. If you were to survey 1000 people on whether or not they smoke and whether or not they use drugs, you will get one of four answers: (no,no) (no,yes) (yes,no) (yes,yes).

    By compiling the number of people in each category, you can ultimately test whether drug usage is independent of cigarette smoking by using the Chi-square distribution (this is approximate, but works well). Again, the methodology for this is in your textbook. The degrees of freedom is equal to (number of rows-1)(number of columns -1). That is, these many figures needed to fill in the entire body of the crosstable, the rest will be determined by using the rows and columns sum figures.

    Don't forget the conditions for the validity of Chi-square test and related expected values greater than 5 in 80% or more of the cells. Otherwise, one could use an "exact" test, using either a permutation or resampling approach. Both SPSS and SAS are capable of doing this test.

    For a 2-by-2 table, you should use the Yates correction to the chi-square. Chi-square distribution is used as an approximation of the binomial distribution. By applying a continuity correction we get a better approximation to the binomial distribution for the purposes of calculating tail probabilities.

    Use a relative risk measure such as the risk ratio or odds ratio. In the table:

    a b
    c d

    The most usual measures are:

    Rate difference a/(a+c) - b/(b+d)
    Rate ratio (a/(a+c))/(b/(b+d))
    Odds ratio ad/bc

    The rate difference and rate ratio are appropriate when you are contrasting two groups, whose sizes (a+c and b+d) are given. The odds ratio is for when the issue is association rather than difference. Confidence interval methods are available for all of these - though not as well available in software as should be. If the hypothesis test is highly significant, the confidence interval will be well away from the null hypothesis value (0 for the rate difference, 1 for the rate ratio or odds ratio).

    The risk ratio is the ratio of the proportion (a/(a+b)) to the proportion (c/(c+d)):

    RR = (a / (a + b)) / (c / (c + d))

    RR is thus a measure of how much larger the proportion in the first row is compared to the second row and ranges from 0 to infinity with < 1.00 indicating a 'negative' association [a/(a+b) < c/(c+d)], 1.00 indicating no association [a/(a+b) = c/(c+d)], and >1.00 indicating a 'positive' association [a/(a+b) > c/(c+d)]. The further from 1.00, the stronger the association. Most stats packages will calculate the RR and confidence intervals for you. A related measure is the odds ratio (or cross product ratio) which is (a/b)/(c/d).

    You could also look at the f statistic which is:

    f = ( c2/N)˝

    where c2 is the Pearson's chi-square and N is the sample size. This statistic ranges between 0 and 1 and can be interpreted like the correlation coefficient.

    Visit Critical Values for the Chi- square Distribution

    Visit also, the Web sites Exact Unconditional Tests, Statistical tests

    Fleiss J., Statistical Methods for Rates and Proportions, Wiley, 1981.

    Goodness-of-fit Test for Discrete Random Variables

    There are other tests which might use the Chi-square, such as goodness-of-fit test for discrete random variables. Again don't forget the conditions for the validity of Chi-square test and related expected values greater than 5 in 80% or more of the cells. Therefore, Chi-square is a statistical test that measures "goodness-of-fit". In other words, it measures how much the observed or actual frequencies differ from the expected or predicted frequencies. Using a Chi-square table will enable you to discover how significant the difference is. A null hypothesis in the context of the Chi-square test is the model that you use to calculate your expected or predicted values. If the value you get from calculating the Chi-square statistic is sufficiently high (as compared to the values in the Chi-square table) it tells you that your null hypothesis is probably wrong.

    Let Y1, Y 2, . . ., Y n be a set of independent and identically distributed random variables. Assume that the probability distribution of the Y i's has the density function
    f o (y). We can divide the set of all possible values of Yi, i Î {1, 2, ..., n}, into m non-overlapping intervals D1, D2, ...., Dm. Define the probability values p1, p2 , ..., pm as;

    p1 = P(Yi Î D1)
    p2 = P(Yi Î D2)


    pm = P(Yi Î Dm)

    Since the union of the mutually exclusive intervals D1, D2, ...., Dm is the set of all possible values for the Yi's, (p1 + p2 + .... + pm) = 1. Define the set of discrete random variables X1, X2, ...., Xm, where

    X1= number of Yi's whose value Î D1
    X2= number of Yi's whose value Î D2


    Xm= number of Yi's whose value Î Dm

    and (X1+ X2+ .... + Xm) = n. Then the set of discrete random variables X1, X2, ...., Xmwill have a multinomial probability distribution with parameters n and the set of probabilities {p1, p2, ..., pm}. If the intervals D1, D2, ...., Dm are chosen such that npił 5 for i = 1, 2, ..., m, then;

    C = S (Xi - npi) 2/ npi. The sum is over i= 1, 2,..., m. The results is distributed as c2 m-1.

    For the goodness-of-fit sample test, we formulate the null and alternative hypothesis as

    Ho : fY(y) = fo(y)
    H1 : fY(y) ą fo(y)

    At the a level of significance, Ho will be rejected in favor of H1 if

    C = S (Xi - npi) 2/ npi is greater than c2 m

    However, it is possible that in a goodness-of-fit test, one or more of the parameters of fo(y) are unknown. Then the probability values p1, p2, ..., pm will have to be estimated by assuming that Ho is true and calculating their estimated values from the sample data. That is, another set of probability values p'1, p'2, ..., p'm will need to be computed so that the values (np'1, np'2, ..., np'm) are the estimated expected values of the multinomial random variable (X1, X2, ...., Xm). In this case, the random variable C will still have a chi-square distribution, but its degrees of freedom will be reduced. In particular, if the density function fo(y) has r unknown parameters,

    C = S (Xi - npi) 2/ npi is distributed as c2 m-1-r.

    For this goodness-of-fit test, we formulate the null and alternative hypothesis as

    Ho: fY(y) = fo(y)
    H1: fY(y) ą fo(y)

    At the a level of significance, Ho will be rejected in favor of H1 if C is greater than
    c2 m-1-r.

    Using chi-square in a 2x2 table requires the Yates's correction. One first subtracts 0.5 from the absolute differences between observed and expected frequencies for each of the 3 genotypes before squaring, dividing by the expected frequency, and summing. The formula for the chi-square value in a 2x2 table can be derived from the Normal Theory comparison of the two proportions in the table using the total incidence to produce the standard errors. The rationale of the correction is a better equivalence of the area under the normal curve and the probabilities obtained from the discrete frequencies. In other words, the simplest correction is to move the cut-off point for the continuous distribution from the observed value of the discrete distribution to midway between that and the next value in the direction of the null hypothesis expectation. Therefore, the correction essentially only applied to 1 df tests where the "square root" of the chi-square looks like a "normal/t-test" and where a direction can be attached to the 0.5 addition.

    For more, visit the Web sites Chi-Square Lesson, and Exact Unconditional Tests.

    Statistics with Confidence

    In practice, a confidence interval is used to express the uncertainty in a quantity being estimated. There is uncertainty because inferences are based on a random sample of finite size from the entire population or process of interest. To judge the statistical procedure we can ask what would happen if we were to repeat the same study, over and over, getting different data (and thus different confidence intervals) each time.

    In most studies investigators are usually interested in determining the size of difference of a measured outcome between groups, rather than a simple indication of whether or not it is statistically significant. Confidence intervals present a range of values, on the basis of the sample data, in which the population value for such a difference may lie.

    Know that a confidence interval computed from one sample will be different from a confidence interval computed from another sample.

    Understand the relationship between sample size and width of confidence interval.

    Know that sometimes the computed confidence interval does not contain the true mean value m (that is, it is incorrect) and understand how this coverage rate is related to confidence level.

    Just a word of interpretive caution. Let's say you compute a 95% confidence interval for a mean m . The way to interpret this is to imagine an infinite number of samples from the same population, 95% of the computed intervals will contain the population mean m . However, it is wrong to state, "I am 95% confident that the population mean m falls within the interval."

    Again, the usual definition of a 95% confidence interval is an interval constructed by a process such that the interval will contain the true value 95% of the time. This means that "95%" is a property of the process, not the interval.

    Is the probability of occurrence of the population mean greater in the confidence interval center and lowest at the boundaries? Does the probability of occurrence of the population mean in a confidence interval vary in a measurable way from the center to the boundaries? In a general sense, normality is assumed, and then the interval between CI limits is represented by a bell shaped t distribution. The expectation (E) of another value is highest at the calculated mean value, and decreases as the values approach the CI interval limits.

    An approximation for the single measurement tolerance interval is n times confidence interval of the mean.

    Determining sample size: At the planning stage of a statistical investigation the question of sample size (n) is critical. The above figure also provides a practical guide to sample size determination in the context of statistical estimations and statistical significance tests.

    The confidence level of conclusions drawn from a set of data depends on the size of data set. The larger the sample, the higher is the associated confidence. However, larger samples also require more effort and resources. Thus, your goal must be to find the smallest sample size that will provide the desirable confidence. In the above figure, formulas are presented for determining the sample size required to achieve a given level of accuracy and confidence.

    In estimating the sample size, when the standard deviation is not known, one may use 1/4 of the range for sample of size over 30 as a "good" estimate for the standard deviation. It is a good practice to compare the result with IQR/1.349.

    A Note on Multiple Comparison via the Individual Intervals: Notice that, if the confidence intervals from two samples do not overlap there is a statistically significant difference, say at 5%. However, the other way is not true two confidence intervals can overlap quite a lot yet there is a significant difference between them. One should examine the confidence interval for the difference explicitly. Even if the C.I.'s are overlapping it is hard to find the exact overall confidence level. However, the sum of individual confidence levels can serve as an upper limit upper limit. This is evident from the fact that: P(A and B) P(A) + P(B).

    Further Reading
    Hahn G. and W. Meeker, Statistical Intervals: A Guide for Practitioners, Wiley, 1991.
    Also visit the Web sites Confidence Interval Applet, statpage.

    Entropy Measure

    Inequality coefficients used in sociology, economy, biostatistics, ecology, physics, image analysis and information processing are analyzed in order to shed light on economic disparity world-wide. Variability of a categorical data is measured by the entropy function:

    E= - S pi ln(pi)

    where, sum is over all the categories and pi is the relative frequency of the ith category. It is interesting to note that this quantity is maximized when all pi's, are equal.

    For a rXc contingency table it is E= S S pij ln(pij) - S( S pij) ln( S(pij) - S( S pij) ln( S(pij)

    The sums are over all i and j, and j and i's.

    Another measure is the Kullback-Liebler distance (related to information theory):

    S((Pi - Qi)*log(Pi/Qi)) =
    S(Pi*log(Pi/Qi )) + S(Qi*log(Qi/Pi ))

    or the variation distance

    S( | Pi - Qi | )/2

    where Pi and Qi are the probabilities for the i-th category for the two populations.

    For more on entropy visit the Web sites Entropy on WWW, Entropy and Inequality Measures, and Biodiversity.

    What Is Central Limit Theorem?

    The central limit theorem (CLT) is a "limit" that is "central" to statistical practice. For practical purposes, the main idea of the CLT is that the average (center of data) of a sample of observations drawn from some population is approximately distributed as a normal distribution if certain conditions are met. In theoretical statistics there are several versions of the central limit theorem depending on how these conditions are specified. These are concerned with the types of conditions made about the distribution of the parent population (population from which the sample is drawn) and the actual sampling procedure.

    One of the simplest versions of the theorem says that if we take a random sample of size (n) from the entire population, then the sample mean which is a random variable defined by S xi / n has a histogram which converges to a normal distribution shape if n is large enough (say more than 30). Equivalently, the sample mean distribution approaches to a normal distribution as the sample size increases.

    In applications of the central limit theorem to practical problems in statistical inference, however, statisticians are more interested in how closely the approximate distribution of the sample mean follows a normal distribution for finite sample sizes, than the limiting distribution itself. Sufficiently close agreement with a normal distribution allows statisticians to use normal theory for making inferences about population parameters (such as the mean ) using the sample mean, irrespective of the actual form of the parent population.

    It can be shown that, if the parent population has mean m and finite standard deviation s, then the sample mean distribution has the same mean m but with smaller standard deviation which is s divided by n˝.

    You know by now that, whatever the parent population is, the standardized variable will have a distribution with a mean m = 0 and standard deviation s=1 under random sampling. Moreover, if the parent population is normal, then z is distributed exactly as a standard normal variable. The central limit theorem states the remarkable result that, even when the parent population is non-normal, the standardized variable is approximately normal if the sample size is large enough. It is generally not possible to state conditions under which the approximation given by the central limit theorem works and what sample sizes are needed before the approximation becomes good enough. As a general guideline, statisticians have used the prescription that if the parent distribution is symmetric and relatively short-tailed, then the sample mean reaches approximate normality for smaller samples than if the parent population is skewed or long-tailed.

    Under certain conditions, in large samples, the sampling distribution of the sample mean can be approximated by a normal distribution. The sample size needed for the approximation to be adequate depends strongly on the shape of the parent distribution. Symmetry (or lack thereof) is particularly important.

    For a symmetric parent distribution, even if very different from the shape of a normal distribution, an adequate approximation can be obtained with small samples (e.g., 10 or 12 for the uniform distribution). For symmetric short-tailed parent distributions, the sample mean reaches approximate normality for smaller samples than if the parent population is skewed and long-tailed. In some extreme cases (e.g. binomial with ) samples sizes far exceeding the typical guidelines (e.g., 30 or 60) are needed for an adequate approximation. For some distributions without first and second moments (e.g., Cauchy), the central limit theorem does not hold.

    For some distributions, extremely large (impractical) samples would be required to approach a normal distribution. In manufacturing, for example, when defects occur at a rate of less than 100 parts per million, using a Beta distribution yields an honest CI of total defects in the population.

    Review also Central Limit Theorem Applet, Sampling Distribution Simulation, and CLT.

    What Is a Sampling Distribution

    The sampling distribution describes probabilities associated with a statistic when a random sample is drawn from the entire population.

    The sampling distribution is the probability distribution or probability density function of the statistic.

    Derivation of the sampling distribution is the first step in calculating a confidence interval or carrying out a hypothesis test for a parameter.

    Example: Suppose that x1,.......,xn are a simple random sample from a normally distributed population with expected value m and known variance s2. Then the sample mean is a statistic used to give information about the population parameter is normally distributed with expected value m and variance s2/n.

    The main idea of statistical inference is to take a random sample from the entire population and then to use the information from the sample to make inferences about particular population characteristics such as the mean m (measure of central tendency), the standard deviation (measure of spread) s or the proportion of units in the population that have a certain characteristic. Sampling saves money, time, and effort. Additionally, a sample can, in some cases, provide as much or more accuracy than a corresponding study that would attempt to investigate an entire population-careful collection of data from a sample will often provide better information than a less careful study that tries to look at everything.

    One must also study the behavior of the mean of sample values from a different specified populations. Because a sample examines only part of a population, the sample mean will not exactly equal the corresponding mean of the population m . Thus, an important consideration for those planning and interpreting sampling results is the degree to which sample estimates, such as the sample mean, will agree with the corresponding population characteristic.

    In practice, only one sample is usually taken (in some cases a small "pilot sample" is used to test the data-gathering mechanisms and to get preliminary information for planning the main sampling scheme). However, for purposes of understanding the degree to which sample means will agree with the corresponding population mean m , it is useful to consider what would happen if 10, or 50, or 100 separate sampling studies, of the same type, were conducted. How consistent would the results be across these different studies? If we could see that the results from each of the samples would be nearly the same (and nearly correct!), then we would have confidence in the single sample that will actually be used. On the other hand, seeing that answers from the repeated samples were too variable for the needed accuracy would suggest that a different sampling plan (perhaps with a larger sample size) should be used.

    A sampling distribution is used to describe the distribution of outcomes that one would observe from replication of a particular sampling plan.

    Know that estimates computed from one sample will be different from estimates that would be computed from another sample.

    Understand that estimates are expected to differ from the population characteristics (parameters) that we are trying to estimate, but that the properties of sampling distributions allow us to quantify, based on probability, how they will differ.

    Understand that different statistics have different sampling distributions with distribution shape depending on (a) the specific statistic, (b) the sample size, and (c) the parent distribution.

    Understand the relationship between sample size and the distribution of sample estimates.

    Understand that the variability in a sampling distribution can be reduced by increasing the sample size.

    See that in large samples, many sampling distributions can be approximated with a normal distribution.

    To learn more, visit the Web sites Sample, and Sampling Distribution Applet

    Applications of and Conditions for Using Statistical Tables

    Some widely used applications of the popular statistical tables can be categorized as follows:

    Z - Table: Tests concerning µ for one or two-population based on their large size random sample(s), (say, > 30, to invoke the Central Limit Theorem).
    Test concerning proportions, with large size random sample size n (say, n> 50, to invoke a convergence theorem).

    Conditions for using this table: Test for randomness of the data is needed before using this table. Test for normality of the sample distribution is also needed if the sample size is small or it may not be possible to invoke the Central Limit Theorem.

    T - Table: Tests concerning µ for one or two-population based on small random sample size (s).
    Tests concerning regression coefficients (slope, and intercepts), df = n - 2.

    Notes: As you know by now, in test of hypotheses concerning m, and construction of confidence interval for it, we start with s known, since the critical value (and the p-value) of the Z-Table distribution can be used. Considering the more realistic situations when we don't know s the T-Table is used. In both cases we need to verify the normality of the population's distribution, however, if the sample size n is very large, we can in fact switch back to Z-Table by the virtue of the central limit theorem. For perfectly normal population, the t-distribution corrects for any errors introduced by estimating s with s when doing inference.

    Note also that, in hypothesis testing concerning the parameter of binomial and Poisson distributions for large sample sizes, the standard deviation is known under the null hypotheses. That's why you may use the normal approximations to both of these distributions.

    Conditions for using this table: Test for randomness of the data is needed before using this table. Test for normality of the sample distribution is also needed if the sample size is small or it may not be possible to invoke the Central Limit Theorem.

    Chi-Square - Table: Tests concerning s2 for one population based on a random sample from the entire population.
    Contingency tables (test for independency of categorical data).
    Goodness-of-fit test for discrete random variables.

    Conditions for using this table: Tests for randomness of the data and normality of the sample distribution are needed before using this table.

    F - Table: ANOVA: Tests concerning µ for three or more populations based on their random samples.
    Tests concerning s2 for two-population based on their random samples.
    Overall assessment in regression analysis using the F-value.

    Conditions for using this table: Tests for randomness of the data and normality of the sample distribution are needed before using this table for ANOVA. Same conditions must be satisfied for the residuals in regression analysis.

    The following chart summarizes statistical tables application with respect to test of hypotheses and construction of confidence intervals for meanm and variance s 2 in one pr comparison of two or more populations.

    Further Reading:
    Kagan. A., What students can learn from tables of basic distributions, Int. Journal of Mathematical Education in Science & Technology, 30(6), 1999.

    Selection of a Table

    Statistical Tables on the Web:

    The following Web sites provide critical values useful in statistical testing and construction of confidence intervals. The results are identical to those given in statistic textbooks. However, in most cases they are more extensive (therefore more accurate).

    Read also
    Kanji G., 100 Statistical Tests, Sage Publisher, 1995.

    Relationships Among Distributions and Unification of Statistical Tables

    Particular attention must be paid to a first course in statistics. When I first began studying statistics, it bothered me that there were different tables for different tests. It took me a while to learn that this is not as haphazard as it appeared. Binomial, Normal, Chi-square, t, and F distributions that you will learn about are actually closely connected.

    A problem with elementary statistical textbooks is that they not only don't provide information of this kind, to permit a useful understanding of the principles involved, but they usually don't provide these conceptual links. If you want to understand connections between statistical concepts, then you should practice in making these connections. Learning by doing statistics lends itself to active rather than passive learning. Statistics is a highly interrelated set of concepts, and to be successful at it, you must learn to make these links conscious in your mind.

    Students often ask: Why T- table values with d.f.=1 are much larger compared with other d.f. values? Some tables are limited, what should I do when the sample size is too large?, How can I get familiarity with tables and their differences. Is there any type of integration among tables? Is there any connections between test of hypotheses and confidence interval under different scenario, for example testing with respect to one, two more than two populations. And so on.

    Further Reading:
    Kagan. A., What students can learn from tables of basic distributions, Int. Journal of Mathematical Education in Science & Technology, 30(6), 1999.

    The following two Figures demonstrate useful relationships among distributions and a unification of statistical tables:

    Unification of Common Statistical Tables, needs Acrobat to view

    Relationship Among Commonly Used Distributions in Testing, needs Acrobat to view

    Normal Distribution

    Up to this point we have been concerned with how empirical scores are distributed and how best to describe the distribution. We have discussed several different measures, but the mean m will be the measure that we use to describe the center of the distribution and the standard deviation s will be the measure we use to describe the spread of the distribution. Knowing these two facts gives us ample information to make statements about the probability of observing a certain value within that distribution. If I know, for example, that the average I.Q. score is 100 with a standard deviation of s = 20, then I know that someone with an I.Q. of 140 is very smart. I know this because 140 deviates from the mean m by twice the average amount as the rest of the scores in the distribution. Thus, it is unlikely to see a score as extreme as 140 because most the I.Q. scores are clustered around 100 and on average only deviate 20 points from the mean m .

    Many applications arise from central limit theorem (average of values of n observations approaches normal distribution, irrespective of form of original distribution under quite general conditions). Consequently, appropriate model for many, but not all, physical phenomena.

    Distribution of physical measurements on living organisms, intelligence test scores, product dimensions, average temperatures, and so on.

    Know that the Normal distribution is to satisfy seven requirements: the graph should be bell shaped curve, mean, medial and mode equal and located at the center of the distribution, only has one mode, symmetric about mean, continuous, never touches x-axis and area under curve equals one.

    Many methods of statistical analysis presume normal distribution.

    Normal Curve Area Area.

    What Is So Important About the Normal Distributions?

    Normal Distribution (called also Gaussian) curves, which have a bell-shaped appearance (it is sometimes even referred to as the "bell-shaped curves") are very important in statistical analysis. In any normal distribution is observations are distributed symmetrically around the mean, 68% of all values under the curve lie within one standard deviation of the mean and 95% lie within two standard deviations.

    There are many reasons for their popularity. The following are the most important reasons for its applicability:

    1. One reason the normal distribution is important is that a wide variety of naturally occurring random variables such as heights and weights of all creatures are distributed evenly around a central value, average, or norm (hence, the name normal distribution). Although the distributions are only approximately normal, they are usually quite close.

      When there are many, too many factors influencing the outcome of a random outcome, then the underlying distribution is approximately normal. For example, the height of a tree is determined by the "sum" of such factors as rain, soil quality, sunshine, disease, etc.

      As Francis Galton wrote in 1889, "Whenever a large sample of chaotic elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along."

      Visit the Web sites Quincunx (with 5 influencing factors) influencing, Central Limit Theorem ( with 8 influencing factors), or BallDrop for demos.

    2. Almost all statistical tables are limited by the size of their parameters. However, when these parameters are large enough one may use normal distribution for calculating the critical values for these tables. Visit Relationship Among Statistical Tables and Their Applications (pdf version).
    3. If the mean and standard deviation of a normal distribution are known, it is easy to convert back and forth from raw scores to percentiles.
    4. It's characterized by two independent parameters--mean and standard deviation. Therefore many effective transformations can be applied to convert almost any shaped distribution into a normal one.
    5. The most important reason for popularity of normal distribution is the Central Limit Theorem (CLT). The distribution of the sample averages of a large number of independent random variables will be approximately normal regardless of the distributions of the individual random variables. Visit also the Web sites Central Limit Theorem Applet, Sampling Distribution Simulation, and CLT, for some demos.
    6. The other reason the normal distributions are so important is that the normality condition is required by almost all kinds of parametric statistical tests. The CLT is a useful tool when you are dealing with a population with unknown distribution. Often, you may analyze the mean (or the sum) of a sample of size n. For example instead of analyzing the weights of individual items you may analyze the batch of size n, that is, the packages each containing n items.

    What is a Linear Least Squares Model?

    Many problems in analyzing data involve describing how variables are related. The simplest of all models describing the relationship between two variables is a linear, or straight-line, model. Linear regression is always linear in the coefficients being estimated, not necessarily linear in the variables.

    The simplest method of drawing a linear model is to "eye-ball" a line through the data on a plot, but a more elegant, and conventional method is that of least squares, which finds the line minimizing the sum of distances between observed points and the fitted line. Realize that fitting the "best" line by eye is difficult, especially when there is a lot of residual variability in the data.

    Know that there is a simple connection between the numerical coefficients in the regression equation and the slope and intercept of regression line.

    Know that a single summary statistic like a correlation coefficient does not tell the whole story. A scatterplot is an essential complement to examining the relationship between the two variables.

    Again, the regression line is a group of estimates for the variable plotted on the Y-axis. It has a form of y = a + mx, m is the slope of the line. The slope is the rise over run. If a line goes up 2 for each 1 it goes over, then its slope is 2.


    • = S x(i)/n
      This is just the mean of the x values.
    • = S y(i)/n
      This is just the mean of the y values.
    • Sxx = S(x(i) - )2 = Sx(i)2 - [ Sx(i) ] 2 / n
    • Syy = S(y(i) - )2 = Sy(i)2 - [ Sy(i) ] 2 / n
    • Sxx = S(x(i) - )(y(i) - ) = Sx(i).y(i) - [ Sx(i) . Sy(i)] / n
    • Slope m = Sxy / Sxx
    • Intercept, b = - m .
    • The least squares regression line is:
      y-predicted = yhat = mx + b

    The regression line goes through a mean-mean point. That is the point at the mean of the x values and the mean of the y values. If you drew lines from the mean-mean point out to each of the data points on the scatter plot, each of the lines that you drew would have a slope. The regression slope is the weighted mean of those slopes, where the weights are the runs squared.

    If you put in each x, the regression line would spit out for you an estimate for each y. Each estimate makes an error. Some errors are positive and some are negative. The sum of squared of the errors plus the sum of squared of the estimates add up to the sum of squared of Y. The regression line is the line that minimizes the variance of the errors. (the mean error is zero, so this means that it minimizes the sum of the squared errors.)

    The reason for finding the best line is so that you can make a reasonable predictions of what y will be if x is known (not vise-versa).

    r2 is the variance of the estimates divided by the variance of Y. r is ± the square root of r2. r is the size of the slope of the regression line, in terms of standard deviations. In other words, it is the slope if we use the standardized X and Y. It is how many standard deviations of Y you would go up, when you go one standard deviation of X to the right.

    Visit also the Web sites Simple Regression, Linear Regression, Putting Points

    Coefficient of Determination

    Another measure of the closeness of the points to the regression line is the Coefficient of Determination.

    r2 = Syhat yhat / Syy

    which is the amount of the squared deviation which is explained by the points on the least squares regression line.

    When you have regression equations based on theory, you should compare:

    1. R squares, that is, the percentage of of variance [in fact, sum of squares] in Y accounted for variance in X captured by the model.
    2. When you want to compare models of different size (different numbers of independent variables (p) and/or different sample sizes n) you must use the Adjusted R-Squared, because the usual R-Squared tends to grow with the number of independent variables.

      R2 a = 1 - (n - 1)(1 - R2)/(n - p - 1)

    3. prediction error or standard error
    4. trends in error, 'observed-predicted' as a function of control variables such as time. Systematic trends are not uncommon
    5. extrapolations to interesting extreme conditions of theoretical significance
    6. t-stats on individual parameters
    7. values of the parameters and its content to content underpinnings.
    8. Fdf1 df2 value for overall assessment. Where df1 (numerator degrees of freedom) is the number of linearly independent predictors in the assumed model minus the number of linearly independent predictors in the restricted model (i.e.,the number of linearly independent restrictions imposed on the assumed model), and df2 (denominator degrees of freedom) is the number of observations minus the number of linearly independent predictors in the assumed model.

    Homoscedasticity and Heteroscedasiticy: Homoscedasticity (homo=same, skedasis=scattering) is a word used to describe the distribution of data points around the line of best fit. The opposite term is heteroscedasiticy. Briefly, homoscedasticity means that data points are distributed equally about the line of best fit. Therefore, homoscedasticity means constancy of variances for/over all the levels of factors. Heteroscedasiticy means that the data points cluster or clump above and below the line in a non-equal pattern. You should find a discussion of these terms in any decent statistics text that deals with least squares regression. See, e.g., Testing Research Hypothesis with the GLM, by McNeil, Newman and Kelly, 1996 pages 174-176.

    Regression Analysis Procedure

    Finally in statistics for business, there exists an opinion that with more that 4 parameters one can fit an elephant, so that if one attempts to fit a curve that depends on many parameters the result should not be regarded as very reliable.

    If m1 and m2 are the slopes of two regressions y on x and x on y respectively then R2=m1.m2

    Logistic regression: Standard logistic regression is a method for modeling binary data (e.g., does a person smoke or not, does a person survive a disease, or not). Polygamous logistic regression models more than two options (beg., does a person take the bus, drive a car or take the subway, does an office use WordPerfect, Word, or another package).

    Test for equality of two slopes: Let m1 represent the regression coefficient for explanatory variable X1 in sample 1 with size n1. Let m2 represent the regression coefficient for X1 in sample 2 with size n2. Let S1 and S2 represent the associated standard error estimates. Then, the quantity

    (m1 - m2) / SQRT(S1 2 + S2 2)

    has the t distribution with df = n1 + n2 - 4

    Regression when both X and Y are in error: Simple linear least-square regression has among its conditions that the data for the independent (X) variables are known without error. Infact, the estimated results are conditioned on whatever errors happened to be present in the independent dataset. When the X-data have an error associated with them the result is to bias the slope downwards. A procedure known as Deming regression can handle this problem quite well. Biased slope estimates (due to error in x) can be avoided using Deming regression.

    Cook and Weisberg, An Introduction to Regression Graphics, Wiley, 1994

    Regression Analysis: Planning, Development, and Maintenance

    I – Planning:
    1. Define the problem, select response, suggest variables

    2. Are the proposed variables fundamental to the problem, and are they variables? Measurable? Can one get a complete set of observations at the same time? Ordinary regression analysis does not assume that the independent variables are measured without error. However, they are conditioned on whatever errors happened to be present in the independent dataset.

    3. Is the problem potentially solvable?

    4. Correlation Matrix and first regression runs (for a subset of data).
      Find the basic statistics, correlation matrix.
      How difficult this problem may be?

      Compute the Variance Inflation Factor, VIF = 1/(1 -rij),, i=1, 2, 3, .., i j. For moderate VIF, say between 2 and 8 you might be able to come-up with a ‘good' model.

      Inspect rij's , one or two must be large. If all are small, perhaps the ranges of the X variables are too-small.

    5. Establish goals, prepare budget and time table.

      a - the final equation should have R2 = 0.8 (say).
      b - Coef. of Variation of say less than 0.10
      c – Nunmer of predictors should not exceed p (say, 3), (for example for p=3, we need at least 30 points).
      d – All estimated coefficients must be significant at m = 0.05 (say).
      e – No pattern in the residuals

    6. Are goals and budget acceptable?

    II – Development of the Model:

    1. 1 – Collect date, plot, try models, check the quality of date, check the assumptions.

    2. 2 – Consult experts for criticism.
      Plot new variable and examine same fitted model.
      Also transformed Predictor Variable may be used.

    3. 3 – Are goals met?
      Have you found "the best" model?

    III – Validation and Maintenance of the Model:

    1. 1 – Are parameters stable over the sample space?

    2. 2 – Is there lack of fit?
      Are the coefficients reasonable?
      Are any obvious variables missing?
      Is the equation usable for control or for prediction?

    3. 3 – Maintenance of the Model.
      Need to have control chart to check the model periodically by statistical techniques.

    Predicting Market Response

    As applied researchers in business and economics, faced with the task of predicting market response, we seldom know the functional form of the response. Perhaps market response is a nonlinear monotonic, or even a non-monotonic function of explanatory variables. Perhaps it is determined by interactions of explanatory variable. Interaction is logically independent of its components.

    When we try to represent complex market relationships within the context of a linear model, using appropriate transformations of explanatory and response variables, we learn how hard the work of statistics can be. Finding reasonable models is a challenge, and justifying our choice of models to our peers can be even more of a challenge. Alternative specifications abound.

    Modern regression methods, such as generalized additive models, multivariate adaptive regression splines, and regression trees, have one clear advantage: They can be used without specifying a functional form in advance. These data-adaptive, computer- intensive methods offer a more flexible approach to modeling than traditional statistical methods. How well modern regression methods perform in predicting market response? Some perform quite well based on the results of simulation studies.

    How to Compare Two Correlations Coefficients?

    The statistical test is the following for Ho: r 1 = r 2.


    t = (z1 - z2) / [ 1/(n1-3) + 1/(n2-3) ]˝ n1, n2 3.


    z1 = 0.5 ln( (1+r1)/(1-r1) ),

    z2 = 0.5 ln( (1+r2)/(1-r2) ) and

    n1= sample size associated with r1, and n2=sample size associated with r2

    The distribution of the statistic t is approximately N(0,1). So, you should reject Ho if |t|> 1.96 at the 95% confidence level.

    r is (positive) scale and (any) shift invariant. That is ax + c, and by + d, have same r as x and y, for any positive a and b.

    Procedures for Statistical Decision Making

    The two most widely used measuring tools and decision procedures in statistical decision making, are Classical and Bayesian Approaches.

    Classical Approach: Classical probability of finding this sample statistic -- or any statistic more unlikely-- assuming the null hypothesis is true. A small p-value is not sufficient evidence to reject the null hypothesis and to accept the alternate.

    As indicated in the above Figure, type-I error occurs when based on your data you reject the null hypothesis when in fact it is true. The probability of a type I error is the level of significance of the test of hypothesis, and is denoted by a .

    A type II error occurs when you do not reject the null hypothesis when it is in fact it is false. The probability of a type-II error is denoted by b . The quantity 1 - b is known as the Power of a Test. Type-II error can be evaluated for any specific alternative hypotheses stated in the form "Not Equal to" as a competing hypothesis.

    Bayesian Approach: Difference in expected gain (loss) associated with taking various actions each having an associated gain (loss) and a given Bayesian statistical significance. This is standard Min/Max decision theory using Bayesian strength of belief assessments in the truth of the alternate hypothesis. One would choose the action which minimizes expected loss or maximizes expected gain (the risk function).

    Hypothesis Testing: Rejecting a Claim

    To perform a hypothesis testing, one must be very specific about the test one wishes to perform. The null hypothesis must be clearly stated, and the data must be collected in a repeatable manner. Usually, the sampling design will involve random, stratified random, or regular distribution of study plots. If there is any subjectivity, the results are technically not valid. All of the analyses, including the sample size, significance level, the time, and the budget, must be planned in advance, or else the user runs the risk of "data diving"

    Hypothesis testing is mathematical proof by contradiction. For example, for a Student's t test comparing 2 groups, we assume that the two groups come from the same population (same means, standard deviations, and in general same distributions). Then we try like all get out to prove that this assumption is false. Rejecting H0 means either H0 is false, or a rare event such as has occurred.

    The real question in statistics not whether a null hypothesis is correct, but whether it is close enough to be used as an approximation.

  • Selecting Statistics

    Test of Hypotheses

    In most statistical tests concerning m, we start by assuming the s2 & higher moments (skewness, kurtosis) are equal. Then we hypothesize that the a's are equal. Null hypothesis.

    The "null" suggests no difference between group means, or no relationship between quantitative variables, and so on.

    Then we test with a calculated t-value. For simplicity, suppose we have a 2 sided test. If the calculated t is close to 0, we say good, as we expected. If the calculated t is far from 0, we say, "the chance of getting this value of t, given my assumption of equal populations, is so small that I will not believe the assumption. We will say that the populations are not equal, specifically the means are not equal."

    Sketch a normal distribution, with mean 1 - 2 and standard deviation s. If the null hypothesis is true, then the mean is 0. We calculate the 't' value, as per the equation. We look up a "critical" value of t. The probability of calculating a t value more extreme ( + or - ) than this, given that the null hypothesis is true, is equal or less than the a risk we used in pulling the critical value from the table. Mark the calculated t, and critical t (both sides) on the sketch of the distribution. Now. If the calculated t is more extreme than the critical value, we say, "the chance of getting this t, by shear chance, when the null hypothesis is true, is so small that I would rather say the null hypothesis is false, and accept the alternative, that the means are not equal." When the calculated value is less extreme than the calculated value, we say, "I could get this value of t by shear chance, often enough that I will not write home about it. I cannot detect a difference in the means of the two groups at the a significance level."

    In this test we need (among others) the condition that the population variances (i.e., treatment impacts on central tendency but not variability) are equal. However, this test is robust to violations of that condition if n's are large and almost the same size. A counter example would be to try a t-test between (11, 12, 13) and (20, 30, 40). The pooled and un pooled tests both give t statistics of 3.10, but the degrees of freedom are different: 4 (pooled) or about 2 (unpooled). Consequently the pooled test gives p = .036 and the unpooled p = .088. We could go down to n = 2 and get something still more extreme.

    The Classical Approach to the Test of Hypotheses

    In this treatment there are two parties, one party (or a person) sets out the null hypothesis (the claim), an alternative hypothesis is proposed by the other party , a significance level a and a sample size n are agreed upon by both parties. The second step is to compute the relevant statistic based on the null hypothesis and the random sample of size n. Finally, one determines the critical region (i.e. rejection region). The conclusion based on this approach is as follows:

    If the computed statistics falls within the rejection region, then Reject the null hypothesis. Otherwise Do Not Reject the null hypothesis (the claim).

    You may ask: How to determine the the critical value (such as z-value) for the rejection interval: for one and two-tailed hypotheses. What is the rule?

    First you have to choose a significance level a. Knowing that the null hypothesis is always in "equality" form, then, the alternative hypothesis has one three possible forms: "greater-than", "less-than", or "not equal to". The first two forms correspond to one-tail hypotheses while the last one corresponds to a two-tail hypothesis.

    • if your alternative is in the form of "greater-than", then z is the value that gives you an area to the right tail of distribution that is equal to a.

    • if your alternative is in the form of "less-than", then z is the value that gives you an area to the left tail of distribution that is equal to a.

    • if your alternative is in the form of "not equal to" then, there are two z values, one positive the other negative. The positive z is the value that gives you an a/2 area to the right tail of distribution. While, the negative z is the value that gives you an a/2 area to the left tail of distribution.

    This is a general rule, and to implement this process in determining the critical value, for any test of hypothesis, you must first master reading the statistical tables well, because, as you see, not all tables in your textbook are presented in a same format.

    The Meaning and Interpretation of P-values (what the data say?)

    The p-value, which directly depends on a given sample, attempts to provide a measure of the strength of the results of a test for the null hypotheses, in contrast to a simple reject or do not reject in the classical approach to the test of hypotheses. If the null hypothesis is true and the chance of random variation is the only reason for sample differences, then the p-value is a quantitative measure to feed into the decision making process as evidence. The following table provides a reasonable interpretation of p-values:

    P < 0.01 very strong evidence against H0
    0.01 P < 0.05 moderate evidence against H0
    0.05 P < 0.10 suggestive evidence against H0
    0.10 P little or no real evidence against H0

    This interpretation is widely accepted, and many scientific journals routinely publish papers using such an interpretation for the result of test of hypothesis.

    For the fixed-sample size, when the number of realizations is decided in advance, the distribution of p is uniform (assuming the null hypothesis). We would express this as P(p x) = x. That means the criterion of p 0.05 achieves a of 0.05.

    Understand that the distribution of p-values under null hypothesis H0 is uniform, and thus does not depend on a particular form of the statistical test. In a statistical hypothesis test, the P value is the probability of observing a test statistic at least as extreme as the value actually observed, assuming that the null hypothesis is true. The value of p is defined with respect to a distribution. Therefore, we could call it "model-distributional hypothesis" rather than "the null hypothesis".

    In short, it simply means that if the null had been true, the p value is the probability against the null in that case. The p-value is determined by the observed value, however, this makes it difficult to even state the inverse of p.

    Arsham H., Kuiper's P-value as a Measuring Tool and Decision Procedure for the Goodness-of-fit Test, Journal of Applied Statistics, Vol. 15, No.3, 131-135, 1988.

    Blending the Classical and the P-value Based Approaches in Test of Hypotheses

    A p-value is a measure of how much evidence you have against the null hypothesis. Notice that, the null hypothesis is always in = form, and does not contain any forms of inequalities. The smaller the p-value, the more evidence you have. In this setting the p-value is based on the hull hypothesis and has nothing to do with alternative hypothesis and therefore with the rejection region. In recent years, some authors try to use the mixture of classical approach (which is based the critical value obtained from given a, and the computed statistics based) and the p-value approach. This is a blend of two different school of thoughts. In this setting, some textbooks compare the p-value with the significance level to make decision on a given test of hypothesis. Larger the p-value is when compared with a (in one sided alternative hypothesis, and a/2 for the two sided alternative hypotheses), less evidence we have in rejecting the null hypothesis. In such a comparison, if the p-value is less than some threshold (usually 0.05, sometimes a bit larger like 0.1 or a bit smaller like 0.01) then you reject the null hypothesis.The following paragraph deal with such a combined approach.

    Use of P-value and a: In this setting, we must also consider the alternative hypothesis in drawing the rejection interval (region) . There is only one p-value to compare with a (or a/2). Know that, for any test of hypothesis, there is only one p-value. The following outlines the computation of the p-value and the decision process involving in a given test of hypothesis:

    1. P-value for One-side Alternative Hypotheses: The p-value is defined as the area to the right tail of distribution if the rejection region in on the right tail, if the rejection region is on the left tail, then the p-value is the area to the left tail (in one-sided alternative hypotheses).

    2. P-value for Two-side Alternative Hypotheses: If the alternative hypothesis is a two-sided (that is, rejection regions are both, on the left and on the right tails) then the p-value is the area to the right tail or to the left of distribution depending on whether the computed statistic is closer to the right rejection region or left rejection region. For symmetric densities (such as t) the left and right tails p-values are the same. However, for non-symmetric densities (such as Chi-square) used the smaller of the two (this makes the test more conservative). Notice that, for two sided-test alternative hypotheses, the p-value is never greater than 0.5.

    3. After finding the p-value as defined here, you compare it with a preset a value for one-sided tests, and with a/2 for two sided-test. Larger the p-value is when compared with a (in one sided alternative hypothesis, and a/2 for the two sided alternative hypotheses), less evidence we have for rejecting the null hypothesis.

    To avoid looking-up the p-values from the limited statistical tables given in your textbook, most professional statistical packages such as SPSS provide the two-tail p-value. Based on where the rejection region is, you must find out what p-value to use.

    Unfortunately, some textbooks have many misleading statements about p-value and its applications, for example in many textbooks you find the authors double the p-value to compare it with a when dealing with the the two-sided test of hypotheses. One wonders how they do it in the case when "their" p-vaue exceeds 0.5? Notice that, while it is correct to compare the p-value with a for one side test of hypotheses a, however, for two-sided hypotheses, one must compare the p-value with a/2, NOT a with 2 times p-value, as unfortunately some text book advise. While, the decision is the same, but there is a clear distinction here and an important difference which the careful reader will note.

    When We Should POOL Variance Estimates?

    Variance estimates should be pooled only if there is a good reason for doing so, and then (depending on that reason) the conclusions might have to be made explicitly conditional on the validity of the equal-variance model. There are several different good reasons for pooling:

    (a) to get a single stable estimate from several relatively small samples, where variance fluctuations seem not to be systematic; or

    (b) for convenience, when all the variance estimates are near enough to equality; or

    (c) when there is no choice but to model variance (as in simple linear regression with no replicated X values), and deviations from the constant-variance model do not seem systematic; or

    (d) when group sizes are large and nearly equal, so that there is essentially no difference between the pooled and unpooled estimates of standard errors of pairwise contrasts, and degrees of freedom are nearly asymptotic.

    Note that this last rationale can fall apart for contrasts other than pairwise ones. One is not really pooling variance in case (d), rather one is merely taking a shortcut in the computation of standard errors of pairwise contrasts.

    If you calculate the test without the assumption, you have to determine the degrees of freedom (or let the statistics package do it). The formula works in such a way that df will be less if the larger sample variance is in the group with the smaller number of observations. This is the case in which the two tests will differ considerably. A study of the formula for the df is most enlightening and one must understand the correspondence between the unfortunate design (having the most observations in the group with little variance) and the low df and accompanying large t-value.

    Example: When doing t tests for differences in means of populations (a classic independent samples case):

    1. Use the standard error formula for differences in means that does not make any assumption about equality of population variances [i.e., (VAR1/n1 + VAR2/n2)˝].

    2. Use the "regular" way to calculate df in a t test (n1-1)+(n2-1), n1, n2 2.

    3. If total N is less than 50 and one sample is 1/2 the size of the other (or less) and the smaller sample has a standard deviation at least twice as large as the other sample, then replace #2 with formula for adjusting df value. Otherwise, don't worry about the problem of having an actual a level that is much different than what you have set it to be.

    In the Statistics With Confidence Section we are concerned with the construction of confidence interval where the equality of variances condition is an important issue.

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    Remember that in the t tests for differences in means there is a condition of equal population variances that must be examined. One way to test for possible differences in variances is to do an F test. However, the F test is very sensitive to violations of the normality condition; i.e., if populations appear not to be normal, then the F test will tend to over reject the null of no differences in population variances.

    Two-Populations Independent Means

    SPSS program for T-test, Two-Population Independent Means:

    DATA LIST    FREE FILE='A.IN'/drug walk
                   DRUG 'DRUG OR PLACEBO'
                    WALK 'DIFFERENCE IN TWO WALKS'
    NPAR TESTS       M-W=WALK BY DRUG(1,2)/
    NPAR TESTS       K-S=WALK BY DRUG(1,2)/
    NPAR TESTS       K-W=WALK BY DRUG(1,2)/
    SAMPLE 10 FROM 20

    Two Dependent Means

    SPSS program for T-test, Two-Population Dependent Means:

    DATA LIST          FILE=MC/YEAR1,YEAR2,(F4.1,1X,F4.1)
    VAR LABELS         

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    Analysis of Variance (ANOVA)

    The tests we have learned up to this point allow us to test hypotheses that examine the difference between only two means. Analysis of Variance or ANOVA will allow us to test the difference between 2 or more means. ANOVA does this by examining the ratio of variability between two conditions and variability within each condition. For example, say we give a drug that we believe will improve memory to a group of people and give a placebo to another group of people. We might measure memory performance by the number of words recalled from a list we ask everyone to memorize. A t-test would compare the likelihood of observing the difference in the mean number of words recalled for each group. An ANOVA test, on the other hand, would compare the variability that we observe between the two conditions to the variability observed within each condition. Recall that we measure variability as the sum of the difference of each score from the mean. When we actually calculate an ANOVA we will use a short-cut formula

    Thus, when the variability that we predict (between the two groups) is much greater than the variability we don't predict (within each group), then we will conclude that our treatments produce different results.

    An Illustrative Numerical Example for ANOVA

    Introducing ANOVA in simplest forms by numerical illustration.

    Example: Consider the following (small, and integer, indeed for illustration while saving space) random samples from three different populations.

    With the null hypothesis H0: µ1 = µ2 = µ3, and the Ha: at least two of the means are not equal. At the significance level a = 0.05, the critical value from F-table is
    F 0.05, 2, 12 = 3.89.

    Sample 1 Sample 2 Sample 3
    2 3 5
    3 4 5
    1 3 5
    3 5 3
    1 0 2
    SUM 10 15 20
    Mean 2 3 4

    Demonstrate that, SST=SSB+SSW

    Computation of sample SST: With the grand mean = 3, first, start with taking the difference between each observation and the grand mean, and then square it for each data point.

    Sample 1 Sample 2 Sample 3
    1 0 4
    0 1 4
    4 0 4
    0 4 0
    4 9 1
    SUM 9 14 13

    Therefore SST=36 with d.f = 15-1 = 14

    Computation of sample SSB:

    Second, let all the data in each sample have the same value as the mean in that sample. This removes any variation WITHIN. Compute SS differences from the grand mean.

    Sample 1 Sample 2 Sample 3
    1 0 1
    1 0 1
    1 0 1
    1 0 1
    1 0 1
    SUM 5 0 5

    Therefore SSB = 10, with d.f = 3-1 = 2

    Computation of sample SSW:

    Third, compute the SS difference within each sample using their own sample means. This provides SS deviation WITHIN all samples.

    Sample 1 Sample 2 Sample 3
    0 0 1
    1 1 1
    1 0 1
    1 4 1
    1 9 4
    SUM 4 14 8

    SSW = 26 with d.f = 3(5-1) = 12

    Results are: SST = SSB + SSW, and d.fSST = d.fSSB + d.fSSW, as expected.

    Now, construct the ANOVA table for this numerical example by plugging the results of your computation in the ANOVA Table.

    The ANOVA Table
    Sources of Variation Sum of Squares Degrees of Freedom Mean Squares F-Statistic
    Between Samples 10 2 5 2.30
    Within Samples 26 12 2.17  
    Total 36 14    

    Conclusion: There is not enough evidence to reject the null hypothesis Ho.

    Logic Behind ANOVA: First, let us try to explain the logic and then illustrate it with a simple example. In performing ANOVA test, we are trying to determine if a certain number of population means are equal. To do that, we measure the difference of the sample means and compare that to the variability within the sample observations. That is why the test statistic is the ratio of the between-sample variation (MST) and the within-sample variation (MSE). If this ratio is close to 1, there is evidence that the population means are equal.

    Here's a hypothetical example: many people believe that men get paid more in the business world than women, simply because they are male. To justify or reject such a claim, you could look at the variation within each group (one group being women's salaries and the other being men salaries) and compare that to the variation between the means of randomly selected samples of each population. If the variation in the women's salaries is much larger than the variation between the men and women's mean salaries, one could say that because the variation is so large within the women's group that this may not be a gender-related problem.

    Now, getting back to our numerical example, we notice that: given the test conclusion and the ANOVA test's conditions, we may conclude that these three populations are in fact the same population. Therefore, the ANOVA technique could be used as a measuring tool and statistical routine for quality control as described below using our numerical example.

    Construction of the Control Chart for the Sample Means: Under the null hypothesis the ANOVA concludes that µ1 = µ2 = µ3; that is, we have a "hypothetical parent population." The question is, what is its variance? The estimated variance is 36 / 14 = 2.75. Thus, estimated standard deviation is = 1.60 and estimated standard deviation for the means is 1.6 / 5 = 0.71. Under the conditions of ANOVA, we can construct a control chart with the warning limits = 3 ± 2(0.71); the action limits = 3 ± 3(0.71). The following figure depicts the control chart.

    Control Chart

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    More Than Two Independent Means

    Bartlett's Test: The Analysis of Variance requires certain conditions be met if the statistical tests are to be valid. One of the conditions we make is that the errors (residuals) all come from the same normal distribution. Thus we have to test not only for normality, but we must also test homogeneity of the variances. We can do this by subdividing the data into appropriate groups, computing the variances in each of the groups and testing that they are consistent with being sampled from a Normal distribution. The statistical test for homogeneity of variance is due to Bartlett which is a modification of the Neyman-Pearson likelihood ratio test.

    Bartlett's Test of Homogeneity of Variances for r Independent Samples is a test to check for equal variances between independent samples of data. The subgroups sizes do not have to be equal. This tests assumes that each sample was randomly and independently drawn from a normal population.

    SPSS program for ANOVA: More Than Two Independent Means:

    NPAR TESTS K-W Y BY GP(1,5)/

    ANOVA like two population t-test can go wrong when the equality of variances condition is not met.

    Homogeneity of Variance: Checking the equality of variances For 3 or more populations, there is a practical rule known as the "Rule of 2". According to this rule, one divides the highest variance of a sample by the lowest variance of the other sample. Given that the sample sizes are almost the same, and the value of this division is less than 2, then, the variations of the populations are almost the same.

    Example: Consider the following three random samples from three populations, P1, P2, P2

          P1     P2     P3
          25     17      8
          25     21     10
          20     17     14
          18     25     16
          13     19     12
           6     21     14
           5     15      6
          22     16     16
          25     24     13
          10     23      6

    The summary statistics and the ANOVA table are computed to be:

    Variable             N       Mean           St.Dev    SE Mean
    P1                  10      16.90           7.87       2.49
    P2                  10      19.80           3.52       1.11
    P3                  10      11.50           3.81       1.20
                     Analysis of Variance
    Source     DF        SS        MS       F      p-value
    Factor      2     79.40     39.70     4.38    0.023
    Error      27    244.90      9.07
    Total      29    324.30

    With an F = 4.38 and a p-value of .023, we reject the null at a = 0.05. This is not good news, since ANOVA, like two sample t-test, can go wrong when the equality of variances condition is not met by the Rule of 2.

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    More Than Two Dependent Populations

    SPSS program for ANOVA: More Than Two Independent Means:

    NPAR TESTS K-W Y BY GP(1,5)/



    Power of a Test

    Power of a test is the probability of correctly rejecting a false null hypothesis. This probability is inversely related to the probability of making a Type II error. Recall that we choose the probability of making a Type I error when we set a. If we decrease the probability of making a Type I error we increase the probability of making a Type II error. Therefore, there are basically two errors possible when conducting a statistical analysis; types I and II:
    • Type I error - risk (i.e. probability) of rejecting the null hypothesis when it is in fact true
    • Type II error - risk of not rejecting the null hypothesis when it is in fact false

    Power and Alpha (a)

    Thus, the probability of correctly retaining a true null has the same relationship to Type I errors as the probability of correctly rejecting an untrue null does to Type II error. Yet, as I mentioned if we decrease the odds of making one type of error we increase the odds of making the other type of error. What is the relationship between Type I and Type II errors? For a fixed sample size, decreasing one type of error increases the size of the other one.

    Power and the True Difference Between Population Means

    Anytime we test whether a sample differs from a population, or whether two samples come from 2 separate populations, there is the condition that each of the populations we are comparing has it's own mean and standard deviation (even if we do not know it). The distance between the two population means will affect the power of our test.

    Power as a Function of Sample Size and Variance s2:
    Anything that effects the extent to which the two distributions share common values will increase Beta (the likelihood of making a Type II error)

    Four factors influence power:

    • effect size (for example, the difference between the means)
    • standard error s
    • significance level a
    • number of observations, or the sample size n
    A Numerical Example: The following Figure provides an illustrative numerical example:
    Power of a Test

    Not rejecting the null hypothesis when it is false is defined as a type II error, and is denoted by the b region. In the above Figure this region lies to the left of the critical value. In the configuration shown in this Figure, b falls to the left of the critical value (and below the statistic's density under the alternative hypothesis Ha). The b is also defined as the probability of incorrectly not-rejecting a false null hypothesis, also called a miss. Related to the value of b is the power of a test. The power is defined as the probability of rejecting the null hypothesis given that a specific alternative is true, and is computed as (1- a).

    A Short Discussion: Consider testing a simple null versus simple alternative. In the Neyman-Pearson setup, an upper bound is set for the probability of type I error (a), and then it is desirable to find tests with low probability of type II error (b) given this. The usual justification for this is that "we are more concerned about a type I error, so we set an upper limit on the a we can tolerate." I have seen this sort of reasoning in elementary texts and also some advanced ones. It doesn't seem to make any sense. When the sample size is large, for most standard tests, the ratio b/a tends to 0. If we care more about type I error than type II error, why should this concern dissipate with increasing sample size?

    This is indeed a drawback of the classical theory of testing statistical hypotheses. A second drawback is that the choice lies between only two test decisions: reject the null or accept the null. It is worth considering approaches that overcome these deficiencies. This can be done, for example, by the concept of profile-tests at a 'level' a. Neither the Type I nor Type II error rates are considered separately, but they are the ratio of a correct decision. For example, we accept the alternative hypothesis Ha and reject the null H0, if an event is observed which is at least a-times greater under Ha than under H0. Conversey, we accept H0 and reject Ha, if an event is observed which is at least a-times greater under H0 than under Ha. This is a symmetric concept which is formulated within the classical approach. Furthermore, more than two decisions can also be formulated.

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    Parametric vs. Non-Parametric vs. Distribution-free Tests

    One must use a statistical technique called nonparametric if it satisfies at least one of the following five types of criteria:

    1. The data entering the analysis are enumerative - that is, count data representing the number of observations in each category or cross-category.

    2. The data are measured and/or analyzed using a nominal scale of measurement.

    3. The data are measured and/or analyzed using an ordinal scale of measurement.

    4. The inference does not concern a parameter in the population distribution - as, for example, the hypothesis that a time-ordered set of observations exhibits a random pattern.

    5. The probability distribution of the statistic upon which the analysis is based is not dependent upon specific information or assumptions about the population(s) from which the sample(s) are drawn, but only on general assumptions, such as a continuous and/or symmetric population distribution.

    By this definition, the distinction of nonparametric is accorded either because of the level of measurement used or required for the analysis, as in types 1 through 3; the type of inference, as in type 4, or the generality of the assumptions made about the population distribution, as in type 5.

    For example, one may use the Mann-Whitney Rank Test as a nonparametric alternative to Students T-test when one does not have normally distributed data.

    Mann-Whitney: To be used with two independent groups (analogous to the independent groups t-test)
    Wilcoxon: To be used with two related (i.e., matched or repeated) groups (analogous to the related samples t-test)
    Kruskall-Wallis: To be used with two or more independent groups (analogous to the single-factor between-subjects ANOVA)
    Friedman: To be used with two or more related groups (analogous to the single-factor within-subjects ANOVA)

    Non-parametric vs. Distribution-free Tests:

    Non-parametric tests are those used when some specific conditions for the ordinary tests are violated.

    Distribution-free tests are those for which the procedure is valid for all different shape of the population distribution.

    For example, the chi-square test concerning the variance of a given population is parametric since this test requires that the population distribution be normal. The chi-square test of independence does not assume normality, or even that the data are numerical. The Kolmogorov-Smirinov goodness-of-fit test is a distribution-free test which can be applied to test any distribution.

    Pearson's and Spearman's Correlations

    There are measures that describe the degree to which two variables are linearly related. For the majority of these measures, the correlation is expressed as a coefficient that ranges from 1.00, indicating a perfect linear relationship such that knowing the value of one variable will allow perfect prediction of the value of the related value, to 0.00, indicating no predictability by a linear model, with negative values indicating that when the value of one variable is high, the other is low (and vice versa), and positive values indicating that when the value of one variable is high, so is the other (and vice versa). Correlation has similar interpretation compared with the derivative you have learned in you calculus (a deterministic course).

    The Pearson's product correlation is an index of the linear relationship between two variables.


    • = S xi / n
      This is just the mean of the x values.
    • = S yi / n
      This is just the mean of the y values.
    • Sxx = S(xi - )2 = Sxi2 - [ Sxi) ] 2 / n
    • Syy = S(yi - )2 = Syi2 - [ Syi ] 2 / n
    • Sxx = S(xi - )(yi - ) = Sxi. yi - [ Sx(i) . Syi ] / n
    The Pearson's correlation is

    r = Sxy / (Sxx . Syy)0.5

    If there is a positive relationship an individual has a score on variable x that is above the mean of variable x, this individual is likely to have a score on variable y that is above the mean of variable y, and vice versa. A negative relationship would be an x score above the mean of x and a y score below the mean of y. It is a measure of the relationship between variables and an index of the proportion of individual differences in one variable that can be associated with the individual differences in another variable. In essence, the product-moment correlation coefficient is the mean of the cross-products of scores. If you have three values for r of .40, .60, and .80. you cannot say that the difference between r = .40 and r = .60 is the same as the difference between r =.60 and r = .80, or that r = .80 is twice as large as r = .40 because the scale of values for the correlation coefficient is not interval or ratio, but ordinal. Therefore, all you can say is that, for example, a correlation coefficient of +.80 indicates a high positive linear relationship and a correlation coefficient of +.40 indicates a some what lower positive linear relationship. It can tell us how much of the total variance of one variable can be associated with the variance of another variable. The square of the correlation coefficient equals the proportion of the total variance in Y that can be associated with the variance in x.

    However, in engineering/manufacturing/development, an r of 0.7 is often considered weak, and +0.9 is desirable. When the correlation coefficient is around +0.9, it is time to make a prediction and confirmation trial(s). Note that a correlation coefficient is usually done on linear correlations. If the data forms a symmetric quadratic hump, a linear correlation of x and y will produce an r of 0!. So one must be careful and look at data.

    Spearman rank-order correlation coefficient is used as a non-parametric version of Pearson's. It is expressed as:

    r = 1 - (6 S d2) / [n(n2 - 1)],

    where d is the difference rank between each X and Y pair.

    Spearman correlation coefficient can be algebraically derived from the Pearson correlation formula by making use of sums of series. Pearson contains expressions for S x(i), S y(i), S x(i)2 and S y(i)2.

    In the Spearman case, the x(i)'s and y(i)' are ranks, and so the sums of the ranks, and the sums of the ranks squared, are entirely determined by the number of cases (without any ties).

    S i = (N+1)N/2, S i2 = N(N+1)(2N+1)/6

    The Spearman formula then is equal to:

    [12P - 3N(N+1)2] / [N(N2 - 1)],

    where P is the sum of the product of each pair of ranks x(i)y(i). This reduces to:

    r = 1 - (6 S d2) / [n(n2 - 1)],

    where d is the difference rank between each x(i) and y(i) pair.

    An important consequence of this is that if you enter ranks into a Pearson formula, you get precisely the same numerical value as that obtained by entering the ranks into the Spearman formula. This comes as a bit of a shock to those who like to adopt simplistic slogans such as "Pearson is for interval data, Spearman is for ranked data". Spearman doesn't work too well if there are lots of tied ranks. That's because the formula for calculating the sums of squared ranks no longer holds true. If one has lots of tied ranks, use the Pearson formula.

    Visit also the Web sites: Correlation Pearsons r, Spearman's Rank Correlation

    Independence vs. Correlated

    In the sense that it is used in statistics, i.e., as an assumption in applying a statistical test, a random sample from the entire population provides a set of random variables X1,...., Xn that are identically distributed and and mutually independent (mutually independent is stronger than pairwise independence). The random variables are mutually independent if their joint distribution is equal to the product of their marginal distributions.

    In the case of joint normality, independence is equivalent to zero correlation but not in general. Independence will imply zero correlation (if the random variables have second moments) but not conversely. Not that not all random variables have a first moment let alone a second moment and hence there may not be a correlation coefficient.

    However if the correlation coefficient of two random variables (theoretical) is not zero then the random variables are not independent.

    Correlation, and Level of Significance

    It is intuitive that with very few data points, a high correlation may not be statistically significant. You may see statements such as, "correlation is significant between x and y at the a = .005 level" and "correlation is significant at the a = .05 level." The question is that how to determine these numbers?
    For simple correlation, you can look at the test as a test on r2. Looking at a simple correlation, the formula for F, where F is the square of the t-statistic, becomes

    F= (n-2) r2 / (1-r2), n 2.

    As you may see, this is monotonic in r2and in n. If the degrees of freedom (n-2) is large, then the F-test is very closely approximated by the chisquared - so that a value of 3.84 is what is needed for reaching a = 5% level. The cutoff value of F changes little enough that the same value, 3.84, gives a pretty good estimate even when the n is small. You can look up an F-table or chisquared table to see the cutoff values needed for other a levels.

    Resampling Techniques: Jackknifing, and Bootstrapping

    Statistical inference techniques that do not require distributional assumptions about the statistics involved. These modern non-parametric methods use large amounts of computation to explore the empirical variability of a statistic, rather than making a priori assumptions about this variability, as is done in the traditional parametric t- and z- tests. Monte Carlo simulation allows for the evaluation of the behavior of a statistic when its mathematical analysis is intractable. Bootstrapping and jackknifing allow inferences to be made from a sample when traditional parametric inference fails. These techniques are especially useful to deal with statistical problems such as small sample size, statistics with no well-developed distributional theory, and parametric inference conditions violations. Both are comouter intensive. Bootstrapping involves taking repeated samples from a popular with the operating rule that you delete n from the sample each time. Jackknifing involves systematically doing n steps, of omitting 1 case from a sample at a time, or, more generally, n/k steps of omitting k cases; computations that compare "included" vs. "omitted" can be used (especially) to reduce the bias of estimation.

    Bootstrapping means you take repeated samples from a sample and then make statements about a population. Bootstrapping entails sampling-with-replacement from a sample. Both have applications in reducing biase in estimations.

    Resampling -- including the bootstrap, permutation, and other non-parametric tests -- is a method for hypothesis tests, confidence limits, and other applied problems in statistics and probability. It involves no formulas or tables. Resampling procedure-free for all tests.

    Following the first publication of the general technique (and the bootstrap) in 1969 by Julian Simon and subsequent independent development by Bradley Efron, resampling has become an alternative approach for test of hypotheses.

    There are other findings, "The bootstrap started out as a good notion in that it presented, in theory, an elegant statistical procedure that was free of distributional conditions. Unfortunately, it doesn't work very well, and the attempts to modify it make it more complicated and more confusing than the parametric procedures that it was meant to replace."

    For the pros and cons of the bootstrap, read
    Young G., Bootstrap: More than a Stab in the Dark?, Statistical Science, l9, 382-395, 1994.
    visit also, the Web sites
    Resampling, and
    Bootstrapping with SAS.

    Sampling Methods

    From the food you eat to the TV you watch, from political elections to school board actions, much of your life is regulated by the results of sample surveys. In the information age of today and tomorrow, it is increasingly important that sample survey design and analysis be understood by many so as to produce good data for decision making and to recognize questionable data when it arises. Relevant topics are: Simple Random Sampling, Stratified Random Sampling, Cluster Sampling, Systematic Sampling, Ratio and Regression Estimation, Estimating a Population Size, Sampling a Continuum of Time, Area or Volume, Questionnaire Design, Errors in Surveys.

    A sample is a group of units selected from a larger group (the population). By studying the sample it is hoped to draw valid conclusions about the larger group.

    A sample is generally selected for study because the population is too large to study in its entirety. The sample should be representative of the general population. This is often best achieved by random sampling. Also, before collecting the sample, it is important that the researcher carefully and completely defines the population, including a description of the members to be included.

    Random sampling of size n from a population size N. Unbiased estimate for variance of is Var() = S2(1-n/N)/n, where n/N is the sampling fraction. For sampling fraction less than 10% the finite population correction factor (N-n)/(N-1) is almost 1.

    The total T is estimated by N. , its variance is N2Var().

    For 0, 1, (binary) type variables, variation in Pbar is

    S2 = Pbar.(1-Pbar).(1-n/N)/(n-1).

    For ratio r = Sxi/Syi= / , the variation for r is

    [(N-n)(r2S2x + S2y -2 r Cov(x, y)]/[n(N-1).2].

    Stratified Sampling: s = S Wt. Bxart, over t=1, 2, ..L (strata), and t is SXit/nt.

    Its variance is:

    SW2t /(Nt-nt)S2t/[nt(Nt-1)]

    Population total T is estimated by N. s, its variance is


    Since the survey usually measures several attributes for each population member, it is impossible to find an allocation that is simultaneously optimal for each of those variables. Therefore, in such a case we use the popular method of allocation which use the same sampling fraction in each stratum. This yield optimal allocation given the variation of the strata are all the same.

    Determination of sample sizes (n) with regard to binary data: Smallest integer greater than or equal to:

    [t2 N p(1-p)] / [t2 p(1-p) + a2 (N-1)]

    with N being the size of the total number of cases, n being the sample size, a the expected error, t being the value taken from the t distribution corresponding to a certain confidence interval, and p being the probability of an event.

    Cross-Sectional Sampling:: Cross-Sectional Study the observation of a defined population at a single point in time or time interval. Exposure and outcome are determined simultaneously.

    For more information on sampling methods, visit the Web sites :
    Sampling In Research
    Sampling, Questionnaire Distribution and Interviewing
    SRMSNET: An Electronic Bulletin Board for Survey Researchers
    Sampling and Surveying Handbook

    Warranties: Statistical Planning and Analysis

    In today market place, warranty has become an increasingly important component of a product package and most consumer and industrial products are sold with a warranty. The warranty serves many purposes. It provides protection for both buyer and manufacturer. For a manufacturer, a warranty also serves to communicate information about product quality, and, as such, may be used as a very effective marketing tool.

    Warranty decisions involve both technical and commercial considerations. Because of the possible financial consequences of these decisions, effective warranty management is critical for the financial success of a manufacturing firm. This requires that management at all levels be aware of the concept, role, uses and cost and design implications of warranty.

    The aim is to understand:

    the concept of warranty and its uses; warranty policy alternatives; the consumer/manufacturer perspectives with regards warranties; the commercial/technical aspects of warranty and their interaction; strategic warranty management; methods for warranty cost prediction; warranty administration

    References and Further Readings:
    Brennan J., Warranties: Planning, Analysis, and Implementation, McGraw Hill, New York, 1994.

    Factor Analysis

    Factor Analysis is a technique for data reduction that is, explaining the variation in a collection of continuous variables by a smaller number of underlying dimensions (called factors). Common factor analysis can also be used to form index numbers or factor scores by using correlation or covariance matrix. The main problem with factor analysis concept is that it is very subjective in its interpretation of the results.

    Delphi Analysis

    Delphi Analysis is used in decision making process, in particular in forecasting. Several "experts" sit together and try to compromise on something they cannot agree on.

    Delbecq, A., Group Techniques for Program Planning, Scott Foresman, 1975.

    Binomial Distribution

    Application: Gives probability of exact number of successes in n independent trials, when probability of success p on single trial is a constant. Used frequently in quality control, reliability, survey sampling, and other industrial problems.

    Example: What is the probability of 7 or more "heads" in 10 tosses of a fair coin?

    Know that the binomial distribution is to satisfy the five following requirements: each trial can have only two outcomes or its outcomes can be reduced to two categories which is called pass and fail, there must be a fixed number of trials, the outcome of each trail must be independent, the probabilities must remain constant, and the outcome of interest is the number of successes.

    Comments: Can sometimes be approximated by normal or by Poisson distribution.


    Application: Gives probability of exactly x independent occurrences during a given period of time if events take place independently and at a constant rate. May also represent number of occurrences over constant areas or volumes. Used frequently in quality control, reliability, queuing theory, and so on.

    Example: Used to represent distribution of number of defects in a piece of material, customer arrivals, insurance claims, incoming telephone calls, alpha particles emitted, and so on.

    Comments: Frequently used as approximation to binomial distribution.

    Exponential Distribution

    Application: Gives distribution of time between independent events occurring at a constant rate. Equivalently, probability distribution of life, presuming constant conditional failure (or hazard) rate. Consequently, applicable in many, but not all reliability situations.

    Example: Distribution of time between arrival of particles at a counter. Also life distribution of complex non redundant systems, and usage life of some components - in particular, when these are exposed to initial burn-in, and preventive maintenance eliminates parts before wear-out.

    Comments: Special case of both Weibull and gamma distributions.

    Uniform Distribution

    Application: Gives probability that observation will occur within a particular interval when probability of occurrence within that interval is directly proportional to interval length.

    Example: Used to generate random valued.

    Comments: Special case of beta distribution.

    The density of geometric mean of n independent uniforms(0,1) is:

    P(X = x) = n x(n - 1) (Log[1/xn])(n -1) / (n - 1)!.

    zL = [UL-(1-U)L] / L is said to have Tukey's symmetrical lambda distribution.

    Student's t-Distributions

    The t distributions were discovered in 1908 by William Gosset who was a chemist and a statistician employed by the Guinness brewing company. He considered himself a student still learning statistics, so that is how he signed his papers as pseudonym "Student". Or perhaps he used a pseudonym due to "trade secrets" restrictions by Guinness.

    Note that there are different t distributions, it is a class of distributions. When we speak of a specific t distribution, we have to specify the degrees of freedom. The t density curves are symmetric and bell-shaped like the normal distribution and have their peak at 0. However, the spread is more than that of the standard normal distribution. The larger the degrees of freedom, the closer the t-density is to the normal density.

    Critical Values for the t-Distribution

    Annotated Review of Statistical Tools on the Internet

    Visit also the Web site Computational Tools and Demos on the Internet

    Introduction: Modern, web-based learning and computing provides the means for fundamentally changing the way in which statistical instruction is delivered to students. Multimedia learning resources combined with CD-ROMs and workbooks attempt to explore the essential concepts of a course by using the full pedagogical power of multimedia. Many Web sites have nice features such as interactive examples, animation, video, narrative, and written text. These web sites are designed to provide students with a "self-help" learning resource to complement the traditional textbook.

    In a few pilot studies, [Mann, B. (1997) Evaluation of Presentation modalities in a hypermedia system, Computers & Education, 28, 133-143. Ward M. and D. Newlands (1998) Use of the Web in undergraduate teaching, Computers & Education, 31, 171-184.] compared the relative effectiveness of three versions of hypermedia systems, namely, Text, Sound/Text, and Sound. The results indicate that those working with Sound could focus their attention on the critical information. Those working with the Text and Sound/Text version however, did not learn as much and stated their displeasure with reading so much text from the screen. Based on this study, it is clear at least at this time that such web-based innovations cannot serve as an adequate substitute for face-to-face live instruction [See also Mcintyre D., and F. Wolff, An experiment with WWW interactive learning in university education, Computers & Education, 31, 255-264, 1998].

    Online learning education does for knowledge what just-in-time delivery does for manufacturing: It delivers the right tools and parts when you need them.

    The Java applets are probably the most phenomenal way of simplifying various concepts by way of interactive processes. These applets help bring into life every concept from central limit theorem to interactive random games and multimedia applications.

    The Flashlight Project develops survey items, interview plans, cost analysis methods, and other procedures that institutions can use to monitor the success of educational strategies that use technology.

    Read also, Critical notice: we are blessed with the emergence of the WWW? Edited by B. Khan, and R. Goodfellow, Computers and Education, 30(1-2), 131-136, 1998.

    The following compilation summarizes currently available public domain web sites offering statistical instructional material. While some sites may have been missed, I feel that this listing is fully representative. I would welcome information regarding any further sites for inclusion, E-mail.

    Academic Assistance Access It is a free tutoring service designed to offer assistance to your statistics questions.

    Basic Definitions, by V. Easton and J. McColl, Contains glossary of basic terms and concepts.

    Basic principles of statistical analysis, by Bob Baker, Basics concepts of statistical models, Mixed model, Choosing between fixed and random effects, Estimating variances and covariance, Estimating fixed effects, Predicting random effects, Inference space, Conclusions, Some references.

    Briefbook of Data Analysis, has many contributors. The most comprehensive dictionary of statistics. Includes ANOVA, Analysis of Variance, Attenuation, Average, Bayes Theorem, Bayesian Statistics, Beta Distribution, Bias, Binomial Distribution, Bivariate Normal Distribution, Bootstrap, Cauchy Distribution, Central Limit Theorem, Bootstrap, Chi-square Distribution, Composite Hypothesis, Confidence Level, Correlation Coefficient, Covariance, Cramer-Rao Inequality, Cramer-Smirnov-Von Mises Test, Degrees of Freedom, Discriminant Analysis, Estimator, Exponential Distribution, F-Distribution, F-test, Factor Analysis, Fitting, Geometric Mean, Goodness-of-fit Test, Histogram, Importance Sampling, Jackknife, Kolmogorov Test, Kurtosis, Least Squares, Likelihood, Linear Regression, Maximum Likelihood Method, Mean, Median, Mode, Moment, Monte Carlo Methods, Multinomial Distribution, Multivariate Normal, Distribution Normal Distribution, Outlier, Poisson Distribution, Principal Component Analysis, Probability, Probability Calculus, Random Numbers, Random Variable, Regression Analysis, Residuals, Runs Test, Sample Mean, Sample Variance, Sampling from a Probability Density Function, Scatter Diagram, Significance of Test, Skewness, Standard Deviation, Stratified Sampling, Student's t Distribution, Student's test, Training Sample, Transformation of Random Variables, Trimming, Truly Random Numbers, Uniform Distribution, Validation Sample Variance, Weighted Mean, etc., References, and Index.

    Calculus Applied to Probability and Statistics for Liberal Arts and Business Majors, by Stefan Waner and Steven Costenoble, contains: Continuous Random Variables and Histograms; Probability Density Functions; Mean, Median, Variance and Standard Deviation.

    Computing Studio, by John Behrens, Each page is a data entry form that will allow you to type data in and will write a page that walks you through the steps of computing your statistic: Mean, Median, Quartiles, Variance of a population, Sample variance for estimating a population variance, Standard-deviation of a population, Sample standard-deviation used to estimate a population standard-deviation, Covariance for a sample, Pearson Product-Moment Correlation Coefficient (r), Slope of a regression line, Sums-of-squares for simple regression.

    CTI Statistics, by Stuart Young, CTI Statistics is a statistical resource center. Here you will find software reviews and articles, a searchable guide to software for teaching, a diary of forthcoming statistical events worldwide, a CBL software developers' forum, mailing list information, contact addresses, and links to a wealth of statistical resources worldwide.

    Data and Story Library, It is an online library of datafiles and stories that illustrate the use of basic statistics methods.

    DAU Stat Refresher, has many contributors. Tutorial, Tests, Probability, Random Variables, Expectations, Distributions, Data Analysis, Linear Regression, Multiple Regression, Moving Averages, Exponential Smoothing, Clustering Algorithms, etc.

    Descriptive Statistics Computation, Enter a column of your data so that the mean, standard deviation, etc. will be calculated.

    Elementary Statistics Interactive, by Wlodzimierz Bryc, Interactive exercises, including links to further reading materials, includes on-line tests.

    Elementary Statistics, by J. McDowell. Frequency distributions, Statistical moments, Standard scores and the standard normal distribution, Correlation and regression, Probability, Sampling Theory, Inference: One Sample, Two Samples.

    Evaluation of Intelligent Systems, by Paul Cohen (Editor-in-Chief), covers: Exploratory data analysis, Hypothesis testing, Modeling, and Statistical terminology. It also serves as community-building function.

    First Bayes, by Tony O'Hagan, First Bayes is a teaching package for elementary Bayesian Statistics.

    Fisher's Exact Test, by Řyvind Langsrud, To categorical variables with two levels.

    Gallery of Statistics Jokes, by Gary Ramseyer, Collection of Statistical Joks.

    Glossary of Statistical Terms, by D. Hoffman, Glossary of major keywords and phrases in suggested learning order is provided.

    Graphing Studio, Data entry forms to produce plots for two-dimensional scatterplot, and three-dimensional scatterplot.

    HyperStat Online, by David Lane. It is an introductory-level statistics book.

    Interactive Statistics, Contains some nice Java applets: guessing correlations, scatterplots, Data Applet, etc.

    Interactive Statistics Page, by John Pezzullo, Web pages that perform mostly needed statistical calculations. A complete collection on: Calculators, Tables, Descriptives, Comparisons, Cross-Tabs, Regression, Other Tests, Power&Size, Specialized, Textbooks, Other Stats Pages.

    Internet Glossary of Statistical Terms, by By H. Hoffman, The contents are arranged in suggested learning order and alphabetical order, from Alpha to Z score.

    Internet Project, by Neil Weiss, Helps students understand statistics by analyzing real data and interacting with graphical demonstrations of statistical concepts.

    Introduction to Descriptive Statistics, by Jay Hill, Provides everyday's applications of Mode, Median, Mean, Central Tendency, Variation, Range, Variance, and Standard Deviation.

    Introduction to Quantitative Methods, by Gene Glass. A basic statistics course in the College of Education at Arizona State University.

    Introductory Statistics Demonstrations, Topics such as Variance and Standard Deviation, Z-Scores, Z-Scores and Probability, Sampling Distributions, Standard Error, Standard Error and Z-score Hypothesis Testing, Confidence Intervals, and Power.

    Introductory Statistics: Concepts, Models, and Applications, by David Stockburger. It represents over twenty years of experience in teaching the material contained therein by the author. The high price of textbooks and a desire to customize course material for his own needs caused him to write this material. It contains projects, interactive exercises, animated examples of the use of statistical packages, and inclusion of statistical packages.

    The Introductory Statistics Course: A New Approach, by D. Macnaughton. Students frequently view statistics as the worst course taken in college. To address that problem, this paper proposes five concepts for discussion at the beginning of an introductory course: (1) entities, (2) properties of entities, (3) a goal of science: to predict and control the values of properties of entities, (4) relationships between properties of entities as a key to prediction and control, and (5) statistical techniques for studying relationships between properties of entities as a means to prediction and control. It is argued that the proposed approach gives students a lasting appreciation of the vital role of the field of statistics in scientific research. Successful testing of the approach in three courses is summarized.

    Java Applets, by many contributors. Distributions (Histograms, Normal Approximation to Binomial, Normal Density, The T distribution, Area Under Normal Curves, Z Scores & the Normal Distribution. Probability & Stochastic Processes (Binomial Probabilities, Brownian Motion, Central Limit Theorem, A Gamma Process, Let's Make a Deal Game. Statistics (Guide to basic stats labs, ANOVA, Confidence Intervals, Regression, Spearman's rank correlation, T-test, Simple Least-Squares Regression, and Discriminant Analysis.

    The Knowledge Base, by Bill Trochim, The Knowledge Base is an online textbook for an introductory course in research methods.

    Lies, Damn Lies, and Psychology, by David Howell, This is the homepage for a course modeled after the Chance course.

    Math Titles: Full List of Math Lesson Titles, by University of Illinois, Lessons on Statistics and Probability topics among others.

    Nonparametric Statistical Methods, by Anthony Rossini, almost all widely used nonparametric tests are presented.

    On-Line Statistics, by Ronny Richardson, contains the contents of his lecture notes on: Descriptive Statistics, Probability, Random Variables, The Normal Distribution, Create Your Own Normal Table, Sampling and Sampling Distributions, Confidence Intervals, Hypothesis Testing, Linear Regression Correlation Using Excel.

    Online Statistical Textbooks, by Haiko Lüpsen.

    Power Analysis for ANOVA Designs, by Michael Friendly, It runs a SAS program that calculates power or sample size needed to attain a given power for one effect in a factorial ANOVA design. The program is based on specifying Effect Size in terms of the range of treatment means, and calculating the minimum power, or maximum required sample size.

    Practice Questions for Business Statistics, by Brian Schott, Over 800 statistics quiz questions for introduction to business statistics.

    Prentice Hall Statistics, This site contains full description of the materials covers in the following books coauthored by Prof. McClave: A First Course In Statistics, Statistics, Statistics For Business And Economics, A First Course In Business Statistics.

    Probability Lessons, Interactive probability lessons for problem-solving and actively.

    Probability Theory: The logic of Science, by E. Jaynes. Plausible Reasoning, The Cox Theorems, Elementary Sampling Theory, Elementary Hypothesis Testing, Queer Uses for Probability Theory, Elementary Parameter Estimation, The Central Gaussian, or Normal, Distribution, Sufficiency, Capillarity, and All That, Repetitive Experiments: Probability and Frequency, Physics of ``Random Experiments'', The Entropy Principle, Ignorance Priors -- Transformation Groups, Decision Theory: Historical Survey, Simple Applications of Decision Theory, Paradoxes of Probability Theory, Orthodox Statistics: Historical Background, Principles and Pathology of Orthodox Statistics, The A --Distribution and Rule of Succession. Physical Measurements, Regression and Linear Models, Estimation with Cauchy and t--Distributions, Time Series Analysis and Auto regressive Models, Spectrum / Shape Analysis, Model Comparison and Robustness, Image Reconstruction, Nationalization Theory, Communication Theory, Optimal Antenna and Filter Design, Statistical Mechanics, Conclusions Other Approaches to Probability Theory, Formalities and Mathematical Style, Convolutions and Cumulates, Circlet Integrals and Generating Functions, The Binomial -- Gaussian Hierarchy of Distributions, Courier Analysis, Infinite Series, Matrix Analysis and Computation, Computer Programs.

    Probability and Statistics, by Beth Chance. Covers the introductory materials supporting Moo re and McCabe, Introduction to the practice of statistics, ND edition, WHO Freeman, 1999. text book.

    Rice Virtual Lab in Statistics, by David Lane et al., An introductory statistics course which uses Java script Monte Carlo.

    Sampling distribution demo, by David Lane, Applet estimates and plots the sampling distribution of various statistics given population distribution, sample size, and statistic.

    Selecting Statistics, Cornell University. Answer the questions therein correctly, then Selecting Statistics leads you to an appropriate statistical test for your data.

    Simple Regression, Enter pairs of data so that a line can be fit to the data.

    Scatterplot, by John Behrens, Provides a two-dimensional scatterplot.

    Selecting Statistics, by Bill Trochim, An expert system for statistical procedures selection.

    Some experimental pages for teaching statistics, by Juha Puranen, contains some - different methods for visualizing statistical phenomena, such as Power and Box-Cox transformations.

    Statlets: Download Academic Version (Free), Contains Java Applets for Plots, Summarize, One and two-Sample Analysis, Analysis of Variance, Regression Analysis, Time Series Analysis, Rates and Proportions, and Quality Control.

    Statistical Analysis Tools, Part of Computation Tools of Hyperstat.

    Statistical Demos and Monte Carlo, Provides demos for Sampling Distribution Simulation, Normal Approximation to the Binomial Distribution, and A "Small" Effect Size Can Make a Large Difference.

    Statistical Education Resource Kit, by Laura Simon, This web page contains a collection of resources used by faculty in Penn State's Department of Statistics in teaching a broad range of statistics courses.

    Statistical Instruction Internet Palette, For teaching and learning statistics, with extensive computational capability.

    Statistical Terms, by The Animated Software Company, Definitions for terms via a standard alphabetical listing.

    Statiscope, by Mikael Bonnier, Interactive environment (Java applet) for summarizing data and descriptive statistical charts.

    Statistical Calculators, Presided at UCLA, Material here includes: Power Calculator, Statistical Tables, Regression and GLM Calculator, Two Sample Test Calculator, Correlation and Regression Calculator, and CDF/PDF Calculators.

    Statistical Home Page, by David C. Howell, This is a Home Page containing statistical material covered in the author's textbooks (Statistical Methods for Psychology and Fundamental Statistics for the Behavioral Sciences), but it will be useful to others not using those book. It is always under construction.

    Statistics Page, by Berrie, Movies to illustrate some statistical concepts.

    Statistical Procedures, by Phillip Ingram, Descriptions of various statistical procedures applicable to the Earth Sciences: Data Manipulation, One and Two Variable Measures, Time Series Analysis, Analysis of Variance, Measures of Similarity, Multi-variate Procedures, Multiple regression, and Geostatistical Analysis.

    Statistical Tests, Contains Probability Distributions (Binomial, Gaussian, Student-t, Chi-Square), One-Sample and Matched-Pairs tests, Two-Sample tests, Regression and correlation, and Test for categorical data.

    Statistical Tools, Pointers for demos on Binomial and Normal distributions, Normal approximation, Sample distribution, Sample mean, Confidence intervals, Correlation, Regression, Leverage points and Chisquare.

    Statistics, This server will perform some elementary statistical tests on your data. Test included are Sign Test, McNemar's Test, Wilcoxon Matched-Pairs Signed-Ranks Test, Student-t test for one sample, Two-Sample tests, Median Test, Binomial proportions, Wilcoxon Test, Student-t test for two samples, Multiple-Sample tests, Friedman Test, Correlations, Rank Correlation coefficient, Correlation coefficient, Comparing Correlation coefficients, Categorical data (Chi-square tests), Chi-square test for known distributions, Chi-square test for equality of distributions.

    Statistics Homepage, by StatSoft Co., Complete coverage of almost all topics

    Statistics: The Study of Stability in Variation, Editor: Jan de Leeuw. It has components which can be used on all levels of statistics teaching. It is disguised as an introductory textbook, perhaps, but many parts are completely unsuitable for introductory teaching. Its contents are Introduction, Analysis of a Single Variable, Analysis of a Pair of Variables, and Analysis of Multi-variables.

    Statistics Every Writer Should Know, by Robert Niles and Laurie Niles. Treatment of elementary concepts.

    Statistics Glossary, by V. Easton and J. McColl, Alphabetical index of all major keywords and phrases

    Statistics Network A Web-based resource for almost all statistical kinds of information.

    Statistics Online A good collection of links on: Statistics to Use, Confidence Intervals, Hypothesis Testing, Probability Distributions, One-Sample and Matched-Pairs Tests, Two-Sample Tests, Correlations, Categorical Data, and Statistical Tables.

    Statistics on the Web, by Clay Helberg, Just as the Web itself seems to have unlimited resources, Statistics on the web must have hundreds of sites listing such statistical areas as: Professional Organizations, Institutes and Consulting Groups, Educational Resources, Web courses, and others too numerous to mention. One could literally shop all day finding the joys and treasures of Statistics!

    Statistics To Use, by T. Kirkman, Among others it contains computations on: Mean, Standard Deviation, etc., Student's t-Tests, chi-square distribution test, contingency tables, Fisher Exact Test, ANOVA, Ordinary Least Squares, Ordinary Least Squares with Plot option, Beyond Ordinary Least Squares, and Fit to data with errors in both coordinates.

    Stat Refresher, This module is an interactive tutorial which gives a comprehensive view of Probability and Statistics. This interactive module covers basic probability, random variables, moments, distributions, data analysis including regression, moving averages, exponential smoothing, and clustering.

    Tables, by William Knight, Tables for: Confidence Intervals for the Median, Binomial Coefficients, Normal, T, Chi-Square, F, and other distributions.

    Two-Population T-test

    SURFSTAT Australia, by Keith Dear. Summarizing and Presenting Data, Producing Data, Variation and Probability, Statistical Inference, Control Charts.

    UCLA Statistics, by Jan de Leeuw, On-line introductory textbook with datasets, Lispstat archive, datasets, and live on-line calculators for most distributions and equations.

    VassarStats, by Richard Lowry, On-line elementary statistical computation.

    Web Interface for Statistics Education, by Dale Berger, Sampling Distribution of the Means, Central Limit Theorem, Introduction to Hypothesis Testing, t-test tutorial. Collection of links for Online Tutorials, Glossaries, Statistics Links, On-line Journals, Online Discussions, Statistics Applets.

    WebStat, by Webster West. Offers many interactive test procedures, graphics, such as Summary Statistics, Z tests (one and two sample) for population means, T tests (one and two sample) for population means, a chi-square test for population variance, a F test for comparing population variances, Regression, Histograms, Stem and Leaf plots, Box plots, Dot plots, Parallel Coordinate plots, Means plots, Scatter plots, QQ plots, and Time Series Plots.

    WWW Resources for Teaching Statistics, by Robin Lock.

    Interesting and Useful Sites

    Selected Reciprocal Web Sites
    | ABCentral | Bulletin Board Libraries |Business Problem Solving |Business Math |Casebook |Chance |CTI Statistics |Cursos de estadística |Demos for Learning Statistics |Electronic texts and statistical tables |Epidemiology and Biostatistics |Financial and Economic Links | Hyperstat |Intro. to Stat. |Java Applets |Lecturesonline |Lecture summaries | Maths & Stats Links|

    | Online Statistical Textbooks and Courses |Probability Tutorial | Research Methods & Statistics Resources | Statistical Demos and Calculations |Statistical Education Resource Kit|

    |Statistical Resources |Statistical Resources on the Web |Statistical tests |Statistical Training on the Web |Statistics Education-I |Statistics Education-II |

    | Statistics Network |Statistics on the Web |Statistics, Statistical Computing, and Mathematics |Statoo |Stats Links |st@tserv |StatSoft | StatsNet |

    | StudyWeb | SurfStat |Using Excel |Virtual Library |WebEc | Web Tutorial Links |Yahoo:Statistics|

    More reciprocal sites may be found by clicking on the following search engines:

    GoTo| HotBot| InfoSeek| LookSmart| Lycos|

    General References
    | The MBA Page | What is OPRE? | Desk Reference| Another Desk Reference | Spreadsheets | All Topics on the Web | Contacts to Statisticians | Statistics Departments (by country)|

    | ABCentral | Syllabits | World Lecture Hall | Others Selected Links | Virtual Library

    | Argus Clearinghouse | TILE.NET | CataList | Maths and Computing Lists

    Statistics References
    | Careers in Statistics | Conferences | | Statistical List Subscription | Statistics Mailing Lists | Edstat-L | Mailbase Lists | Stat-L | Stats-Discuss | Stat Discussion Group | StatsNet | List Servers|

    | Math Forum Search|

    | Statistics Journals | Books and Journal | Main Journals | Journal Web Sites|

    Statistical Societies & Organizations

    Statistics Resources
    | Statistics Main Resources | Statistics and OPRE Resources | Statistics Links | STATS | StatsNet | Resources | UK Statistical Resources|

    | Mathematics Internet Resources | Mathematical and Quantitative Methods |Stat Index | StatServ | Study WEB | Ordination Methods for Ecologists|

    WWW Resources | StatLib: Statistics Library | Guide for Statisticians|

    | Stat Links | Use and Abuse of Statistics | Statistics Links| | Statistical Links | Statistics Handouts | Statistics Related Links | Statistics Resources | OnLine Text Books|

    Probability Resources
    |Probability Tutorial |Probability | Probability & Statistics |Theory of Probability | Virtual Laboratories in Probability and Statistics

    |Let's Make a Deal Game |Central Limit Theorem | The Probability Web | Probability Abstracts

    | Coin Flipping |Java Applets on Probability | Uncertainty in AI |Normal Curve Area | Topics in Probability | PQRS Probability Plot | The Birthday Problem|

    Data and Data Analysis
    |Histograms | Statistical Data Analysis | Exploring Data | Data Mining |Books on Statistical Data Analysis|

    | Evaluation of Intelligent Systems | AI and Statistics|

    Statistical Software
    | Statistical Software Providers | S-PLUS | WebStat | QDStat | Statistical Calculators on Web | MODSTAT | The AssiStat|

    | Statistical Software | Mathematical and Statistical Software | NCSS Statistical Software|

    | Choosing a Statistical Analysis Package | Statistical Software Review| Descriptive Statistics by Spreadsheet | Statistics with Microsoft Excel|

    Learning Statistics
    | How to Study Statistics | Statistics Education | Web and Statistical Education | Statistics & Decision Sciences | Statistics | Statistical Education through Problem Solving|

    | Exam, tests samples | INFORMS Education and Students Affairs | CHANCE Magazin | Chance Web Index|

    | Statistics Education Bibliography | Teacher Network | Computers in Teaching Statistics|

    Glossary Collections

    The following sites provide a wide range of keywords & phrases. Visit them frequently to learn the language of statisticians.

    |Data Analysis Briefbook | Glossary of Statistical Terms |Glossary of Terms |Glossary of Statistics |Internet Glossary of Statistical Terms |Lexicon |Selecting Statistics Glossary |Statistics Glossary | SurfStat glossary|

    Selected Topics
    |ANOVA |Confidence Intervals |Regression

    | Kolmogorov-Smirnov Test | Topics in Statistics-I | Topics in Statistics-I | Statistical Topics | Resampling | Pattern Recognition | Statistical Sites by Applications | Statistics and Computing|

    | Biostatistics | Biomathematics and Statistics | Introduction to Biostatistics Bartlett Corrections|

    | Statistical Planning | Regression Analysis | AI-Geostats | Total-Qality | Analysis of Variance and Covariance|

    | Significance Testing | Hypothesis Testing | Two-Tailed Hypothesis Testing | Commentaries on Significance Testing | Bayesian | Philosophy of Testing|

    Questionnaire Design, Surveys Sampling and Analysis
    |Questionnaire Design and Statistical Data Analysis |Summary of Survey Analysis Software |Sample Size in Surveys Sampling |Survey Samplings|

    |Multilevel Statistical Models | Write more effective survey questions|

    | Sampling In Research | Sampling, Questionnaire Distribution and Interviewing | SRMSNET: An Electronic Bulletin Board for Survey|

    | Sampling and Surveying Handbook |Surveys Sampling Routines |Survey Software |Multilevel Models Project|

    Econometric and Forecasting
    | Time Series Analysis for Official Statisticians | Time Series and Forecasting | Business Forecasting | International Association of Business Forecasting | Institute of Business Forecasting |Principles of Forecasting|

    | Financial Statistics | Econometric-Research | Econometric Links | Economists | RFE: Resources for Economists | Business & Economics Scout Reports|

    | A Business Forecasting Course | A Forecasting Course | Time Series Data Library | Journal of Forecasting|

    | Economics and Teaching |Box-Jenkins Methodology |

    Statistical Tables
    The following Web sites provide critical values useful in statistical testing and construction of confidence intervals. The results are identical to those given in almost all textbook. However, in most cases they are more extensive (therefore more accurate).

    |Normal Curve Area |Normal Calculator |Normal Probability Calculation |Critical Values for the t-Distribution | Critical Values for the F-Distribution | Critical Values for the Chi- square Distribution|

    A selection of:

    Academic Info: Business, AOL: Science and Technology, Biz/ed: Business and Economics, BUBL Catalogue, Business & Economics: Scout Report, Business & Finance, Business & Industrial,

    Business Nation, Dogpile: Statistics, HotBot Directory: Statistics, IFORS, LookSmart: Statistics, LookSmart: Data & Statistics, MathForum: Business, McGraw-Hill: Business Statistics, NEEDS: The National Engineering Education Delivery System, Netscape: Statistics, NetFirst,

    SavvySearch Guide: Statistics, Small Business, Social Science Information Gateway, WebEc, and the Yahoo

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    Kindly e-mail me your comments, suggestions, and concerns. Thank you.

    Professor Hossein Arsham