close this bookStatistical Thinking for Decision Making: Revealing Facts from Figures.By.Dr. Hossein Arsham
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View the documentChapter.1:Towards Statistical Thinking for Decision Making
View the documentChapter 2:Descriptive Sampling Data Analysis
View the documentChapter.3:Probability for Statistical Inference and Modeling
View the documentChapter 4:Necessary Conditions for Statistical Decision Making
View the documentChapter 5:Estimators and Their Qualities
View the documentChapter 6:Hypothesis Testing: Rejecting a Claim
View the documentChapter 7:Hypotheses Testing for Means and Proportions
View the documentChapter 8: Applications of the Chi-square Statistic
View the documentChapter 9:Regression Modeling and Analysis
View the documentChapter 10: Unified Views of Statistical Decision Tools
View the documentChapter 11:Visualization of Statistics: Analytic-Geometry & Statistics
View the documentChapter 12: Index Numbers with Applications

Chapter.1:Towards Statistical Thinking for Decision Making

Introduction to Statistical Thinking for Decision Making

This site builds up the basic ideas of business statistics systematically and correctly. It is a combination of lectures and computer-based practical, whereby theory is firmly placed into practice. It provides an introduction to techniques for summarizing and presenting data, estimation, confidence intervals and hypothesis testing. The presentation is centered around focusing more on conceptual understanding of key concepts, and statistical thinking, and less on formulas and calculations, which can now be left to PCs using accessible and user-friendly Statistical JavaScript Applets.

Today's good decisions are driven by data. In all aspects of our lives, and importantly in the business context, an amazing diversity of data is available for inspection and enlightenment. Moreover, business managers and professionals are increasingly encouraged to justify decisions on the basis of data.

Business managers need statistical model-based decision support systems. Statistical skills enable you to intelligently collect, analyze and interpret data relevant to their decision-making. Statistical concepts and statistical thinking enable you:

  • to solve problems in a diversity of contexts
  • to add substance to decisions.
This Web site is a course in statistics appreciation; i.e., to acquire a feel for the statistical way of thinking. Appreciation of statistics is wonderful: it makes what is excellent in statistical thinking belong to you as well. It is an introductory course in statistics that is 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 making good decision and they promote one to think statistically. The cardinal objective for this Web site is to increase the extent to which statistical thinking is embedded with managerial thinking for decision making under uncertainties.

In competitive environment, business managers must design quality into products, and into the processes of making the products. They must facilitate a process of never-ending improvement at all stages of manufacturing and service. This is a strategy that employs statistical methods, particularly statistically designed experiments, and 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. This saves 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 statistical 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 processes must be based on data, not on personal opinion nor on belief.

The Devil is in the Deviations: Variation is inevitable in life! Every process, every measurement, every sample has variation. 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 it, in order to predict it, effectively.

If you have taken statistics before, and have a feeling of inability to grasp concepts, it may be 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, Professor Herman Chernoff (1996) made the following remark:

"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"

Inadequate statistical teaching during university education leads even after graduation, to one or a combination of the following scenarios:

  1. In general, people do not like statistics and therefore they try to avoid it.
  2. There is a pressure to produce scientific papers, however often confronted with "I need something quick."
  3. At many institutes in the world, there are only a few (mostly 1) statisticians, if any at all. This means that these people are extremely busy. As a result, they tend to advise simple and easy to apply techniques, or they will have to do it themselves.
  4. Communication between a statistician and decision-maker can be difficult. One speaks in statistical jargon; the other understands the monetary or utilitarian benefit of using the statistician's recommendations.

Plugging numbers into the formulas and crunching them have 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 the logical secret in any formulas not memorize them. For example, in computing the variance, consider its formula. Instead of memorizing, you should start with some why:

i. Why do we square the deviations from the mean.
Because, if we add up all deviations, we get always zero value. So, to deal with this problem, we square the deviations. Why not raise to the power of four (three will not work)? 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 is there 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 do we divide the sum of squares by n-1.
The amount of deviation should reflect also how large the sample is; so we must bring in the sample size. That is, in general, larger sample sizes have larger sum of square deviation from the mean. Why n-1 not n? The reason for n-1 is that when you divide by n-1, the sample's variance provides an estimated variance much closer to the population variance, than when you divide by n. You note that for large sample size n (say over 30), it really does not matter whether it is divided by n or n-1. The results are almost the same, and they are acceptable. The factor n-1 is what we consider as the "degrees of freedom".

This example shows how to question statistical formulas, rather than memorizing them. In fact, when you try to understand the formulas, you do not need to remember them, they are part of your brain connectivity. Clear thinking is always more important than the ability to do 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.

computer-assisted learning: The computer-assisted learning provides you a "hands-on" experience which will enhance your understanding of the concepts and techniques covered in this site.

Java, once an esoteric programming language for animating Web pages, is now a full-fledged platform for building JavaScript E-labs' learning objects with useful applications. As you used to do experiments in physics labs to learn physics, computer-assisted learning enables you to use any online interactive tool available on the Internet to perform experiments. The purpose is the same; i.e., to understand statistical concepts by using statistical applets which are entertaining and educating.

The appearance of computer software, JavaScript Applets, Statistical Demonstration Applets, and Online Computation are the most important events in the process of teaching and learning concepts in model-based, statistical decision making courses. These e-lab tools allow you to construct numerical examples to understand the concepts, and to find their significance for yourself.

Unfortunately, most classroom courses are not learning systems. The way the instructors attempt to help their students acquire skills and knowledge has absolutely nothing to do with the way students actually learn. Many instructors rely on lectures and tests, and memorization. All too often, they rely on "telling." No one remembers much that's taught by telling, and what's told doesn't translate into usable skills. Certainly, we learn by doing, failing, and practicing until we do it right. The computer assisted learning serves this purpose.

A course in appreciation of statistical thinking gives business professionals an edge. Professionals with strong quantitative skills are in demand. This phenomenon will grow as the impetus for data-based decisions strengthens and the amount and availability of data increases. The statistical toolkit can be developed and enhanced at all stages of a career. Decision making process under uncertainty is largely based on application of statistics for probability assessment of uncontrollable events (or factors), as well as risk assessment of your decision.


Further Readings:
Chernoff H., A Conversation With Herman Chernoff, Statistical Science, Vol. 11, No. 4, 335-350, 1996.
Churchman C., The Design of Inquiring Systems, Basic Books, New York, 1971. Early in the book he stated that knowledge could be considered as a collection of information, or as an activity, or as a potential. He also noted that knowledge resides in the user and not in the collection.
Rustagi M., et al. (eds.), Recent Advances in Statistics: Papers in Honor of Herman Chernoff on His Sixtieth Birthday, Academic Press, 1983.

The Birth of Probability and Statistics



Probability has much longer history. Probability is derived from the verb to probe meaning to "find out" what is not too easily accessible or understandable. The word "proof" has the same origin that provides necessary details to understand what is claimed to be true.

Probability originated from the study of games of chance and gambling during the 16th century. Probability theory was a branch of mathematics studied by Blaise Pascal and Pierre de Fermat in the seventeenth century. Currently in 21st century, probabilistic modeling is 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.


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.

Further Readings:
Daston L., Classical Probability in the Enlightenment, Princeton University Press, 1988.
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.

Statistical Modeling for Decision-Making under Uncertainties:
From Data to the Instrumental Knowledge

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 departures and patterns.

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.




As an example of statistical modeling with managerial implications, such as "what-if" analysis, consider regression analysis. Regression analysis is a powerful technique for studying relationship between dependent variables (i.e., output, performance measure) and independent variables (i.e., inputs, factors, decision variables). Summarizing relationships among the variables by the most appropriate equation (i.e., modeling) allows us to predict or identify the most influential factors and study their impacts on the output for any changes in their current values.

Frequently, for example the marketing managers are faced with the question, What Sample Size Do I Need? This is an important and common statistical decision, which should be given due consideration, since an inadequate sample size invariably leads to wasted resources. The sample size determination section provides a practical solution to this risky decision.



Knowledge is what we know. Information is the communication of knowledge. In every knowledge exchange, there is a sender and a receiver. The sender makes common what is private, does the informing, the communicating. Information can be classified as explicit and tacit forms. The explicit information can be explained in structured form, while tacit information is inconsistent and fuzzy to explain.


Fact becomes knowledge, when it is used in the successful completion of a decision process. Knowledge needs wisdom. Wisdom is the power to put our time and our knowledge to the proper use. Once you have a massive amount of facts integrated as knowledge, then your mind will be superhuman in the same sense that mankind with writing is superhuman compared to mankind before writing. The following figure illustrates the statistical thinking process based on data in constructing statistical models for decision making under uncertainties.

From Data to Knowledge

The above figure depicts the fact that as the exactness of a statistical model increases, the level of improvements in decision-making increases. 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 purpose of statistical thinking is to get acquainted with the statistical techniques, to be able to execute procedures using available JavaScript Applets, and to be conscious of the conditions and limitations of various techniques.

Statistical Decision-Making Process

Unlike the deterministic decision-making process, in decision making under pure uncertainty, the variables are often more numerous and more difficult to measure and control. However, the steps are the same. They are:
  1. Simplification
  2. Building a decision model
  3. Testing the model
  4. Using the model to find the solution:
    • It is a simplified representation of the actual situation
    • It need not be complete or exact in all respects
    • It concentrates on the most essential relationships and ignores the less essential ones.
    • It is more easily understood than the empirical situation, and hence permits the problem to be solved more readily with minimum time and effort.
  5. It can be used again and again for similar problems or can be modified.


  • An auditor can use random sampling techniques to audit the accounts receivable for clients.
  • A sales manager may use statistical techniques to forecast sales for the coming year.

Questions Concerning Statistical the Decision-Making Process:


Further Readings:
Corfield D., and J. Williamson, Foundations of Bayesianism, Kluwer Academic Publishers, 2001. Contains Logic, Mathematics, Decision Theory, and Criticisms of Bayesianism.
Lapin L., Statistics for Modern Business Decisions, Harcourt Brace Jovanovich, 1987.
Pratt J., H. Raiffa, and R. Schlaifer, Introduction to Statistical Decision Theory, The MIT Press, 1994.

What is Business Statistics?

The main objective of Business Statistics is to make inferences (e.g., prediction, making decisions) about certain characteristics of a population based on information contained in a random sample from the entire population. The condition for randomness is essential to make sure the sample is representative of the population.

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, 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.


At the planning stage of a statistical investigation, the question of sample size (n) is critical. For example, sample size for sampling from a finite population of size N, is set at: N½+1, rounded up to the nearest integer. Clearly, a larger sample provides more relevant information, and as a result a more accurate estimation and better statistical judgement regarding test of hypotheses.

What is statistics?

Activities Associated with the General Statistical Thinking
Click on the image to enlarge it

The above figure illustrates the idea of statistical inference from a random sample about the population. It also provides estimation for the population's parameters; namely the expected value µx, the standard deviation, and the cumulative distribution function (cdf) Fx, s and their corresponding sample statistics, mean , sample standard deviation Sx, and empirical cumulative distribution function (cdf), respectively.

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 too 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, Greek letters represent parameters and Latin letters represent statistics (as shown in the above Figure).





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 sell or that your customer buys? 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 sell or that you customer buys. 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 sell or you customer buy, how does the product/service compare with other existing similar types? That is, comparing several m 's, and several s 's .

Common Statistical Terminology with Applications

Like all profession, also statisticians have their own keywords and phrases to ease a precise communication. However, one must interpret the results of any decision making in a language that is easy for the decision-maker to understand. Otherwise, he/she does not believe in what you recommend, and therefore does not go into the implementation phase. This lack of communication between statisticians and the managers is the major roadblock for using statistics.

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

Qualitative and Quantitative Variables: Any object or event, which can vary in successive observations either in quantity or quality is called a "variable." Variables are classified accordingly as quantitative or qualitative. A qualitative variable, unlike a quantitative variable does not vary in magnitude in successive observations. The values of quantitative and qualitative variables are called "Variates" and "Attributes", respectively.



Sample: A subset of a population or universe.


Statistical Experiment: An experiment in general is an operation in which one chooses the values of some variables and measures the values of other variables, as in physics. A statistical experiment, in contrast is an operation in which one take a random sample from a population and infers the values of some variables. For example, in a survey, we "survey" i.e. "look at" the situation without aiming to change it, such as in a survey of political opinions. A random sample from the relevant population provides information about the voting intentions.



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 of May in any of the years.




Primary data and Secondary data sets: If the data are from a planned experiment relevant to the objective(s) of the statistical investigation, collected by the analyst, it is called a Primary Data set. However, if some condensed records are given to the analyst, it is called a Secondary Data set.

Random Variable: A random variable is a real function (yes, it is called " variable", but in reality it is a function) that assigns a numerical value to each simple event. For example, in sampling for quality control an item could be defective or non-defective, therefore, one may assign X=1, and X = 0 for a defective and non-defective item, respectively. You may assign any other two distinct real numbers, as you wish; however, non-negative integer random variables are easy to work with. Random variables are needed since one cannot do arithmetic operations on words; the random variable enables us to compute statistics, such as average and variance. Any random variable has a distribution of probabilities associated with it.

Probability: Probability (i.e., probing for the unknown) 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. 

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 executed. For example, a person; a sample of soil; a pot of seedlings; a zip code area; a doctor's practice.






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.


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 were found to be $25, then $25 would be her estimate.




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.

Inferential Statistics: 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. Any inferred conclusion from a sample data to the population from which the sample is drawn must be expressed in a probabilistic term. Probability is the language and a measuring tool for uncertainty in our statistical conclusions.






Notice that to be able to estimate the population parameters, the sample size n must 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.

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