Counting for Something

Frontmatter

Chapter 1. Statistics—The Word

Abstract

Two views about quantification are expressed by a scientist and a poet. First, Lord Kelvin:

When you can measure what you arc speaking about, and express it in numbers, then you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind.

William S. Peters

Chapter 2. Distributions

Abstract

The two examples of chest measurements and math scores were presented in the first chapter without much preparation. We hoped it would he more or less obvious how the tables were obtained. We pause now to be more systematic about the concepts of variables and frequency distributions, and the main measures used to describe them.

William S. Peters

Chapter 3. Special Averages

Abstract

One reason the mean is such a widely used average is that it lends itself to calculations of the following kind.

William S. Peters

Chapter 4. Making Comparisons

Abstract

In 1954 Branch Rickey wrote an article for Life Magazine entitled “Goodbye to Some Old Baseball Ideas.” [1] Rickey criticized some traditional baseball statistics and proposed some of his own, which included formulas for team offense and pitching efficiency. For individual hitting Rickey recommended the sum of on-base average (OBA) plus extra base power (EBP). These were defined as

$$\begin{array}{l} OBA = \frac{{hits + bases\;on\;balls + hit\;by\;pitcher}}{{at bats + bases\;on\;balls + hit\;by\;pitcher}},\\ EBP = \frac{{total\;bases - hits}}{{at bats}} \end{array}$$

.

William S. Peters

Chapter 5. Probability

Abstract

Modern probability theory had its start in 1654 when a French nobleman known as Chevalier de Mere wrote a letter to Blaise Pascal (1623–1662), a celebrated mathematician. The letter concerned some problems de Mere had encountered at the gaming tables. Pascal initiated a correspondence with another mathematician. Pierre de Fermat (1601–1655) about these problems. Neither Pascal nor Fermat ever published any of his work on probability, but most of the correspondence has survived.

William S. Peters

Chapter 6. Craps and Binomial

Abstract

One of our illustrations of combinations in the last chapter was the number of ways that the baseball World Series could last four games, five games, six games, and seven games. Our interest there was to show how counting numbers of ways is part and parcel of probability calculations. That example led to a set of probabilities for the possible lengths of the series. When the events of concern in an uncertain situation are numerical and we have obtained the probabilities for all possible events, we have what is called a probability distribution.

William S. Peters

Chapter 7. Horsekicks, Deadly Quarrels, and Flying Bombs

Abstract

In his 1951 book, Facts from Figures, M. J. Moroney has a chapter entitled “Goals, Floods, and Horsekicks.” [1] One can only conclude that violence has increased in the world when we start with horsekicks and go on to deadly quarrels and flying bombs. All of these are examples of a distribution carrying the name of a French mathematician, Simeon D. Poisson (1781–1840).

William S. Peters

Chapter 8. Normal Distribution

Abstract

Abraham De Moivre (1667–1754), who was born in France but lived most of his adult life in England, has the largest claim to the discovery of the normal distribution. The Frenchman Laplace and the German Gauss used the normal distribution especially in astronomic and geodesic measurements, and the Belgian Quetelet first applied it extensively to social statistics. [1]

William S. Peters

Chapter 9. Political Arithmetic

Abstract

The word statistics, or state arithmetic, was coined by Gottfried Achenwall (1719–1772), in 1749. Achenwall was a professor at the universities of Gottingen and Marborough in Germany. By state arithmetic is meant the carrying on of those counting and calculating activities necessary to the operation of a modern nation-state. M.J. Moroney explains to us that rulers

… must know just how far they may go in picking the pockets of their subjects. A king going to war wishes to know what reserves of manpower and money he can call on. How many men need be put in the field to defeat the enemy? How many guns and shirts, how much food, will they need? How much will all this cost? Have the citizens the necessary money to pay for the king’s war? Taxation and military service were the earliest fields for the use of statistics. [1]

William S. Peters

Chapter 10. Regression and Correlation

Abstract

Most people are acquainted with linear relationships. If a salesperson is paid $2000 a month plus a commission of 5 percent of sales, monthly income can he expressed as

$$Y = \$ 2000 + 0.05X$$

, where X is the sales for the month. If no sales occur, X = 0 and Y = $2000; if sales are $20,000, then Y = $2000 + 0.05($20,000) = $3000. What we have here is a mathematical relationship expressing an agreement about compensation.

William S. Peters

Chapter 11. Karl Pearson

Abstract

Karl Pearson (1857–1936) has been called “the founder of the science of statistics.” [1] Helen Walker identifies the start of the first great wave of modern statistics with the publication of Galton’s Natural Inheritance in 1889 and with Pearson’s series of lectures on The Scope and Concepts of Modern Science in 1891 and his series in 1893 on Mathematical Contributions to the Theory of Evolution. [2] Prior to this time statistical theory had been primarily the work of astronomers and mathematicians concerned with errors of measurement.

William S. Peters

Chapter 12. Pearson to Gosset to Fisher

Abstract

A double play in baseball is a beautiful thing to behold, and an infield that can turn the double play is much to be valued. The most common double play is short-stop to second baseman to first baseman. In the early 1900s in baseball the most famous combination was Tinker to Evers to Chance for the Chicago Cubs. Our next episode in statistical methods has the heroic quality of the double play. Our team has one putout on the opposition up to now (1908). A spectacular double play is about to happen which gets us out of the inning and ready to go on the offensive. We call the play Pearson to Gosset to Fisher.

William S. Peters

Chapter 13. More Regression

Abstract

We ended our introduction to regression with Galton’s statement that he had found a “great subject to write upon.” The torch was then passed to Karl Pearson, who developed most of the mathematics of regression analysis. Then Gosset. Pearson’s student, posed the problem which he ultimately solved himself by developing the t-distribution. The t-distribution allowed the error of a sample mean to be correctly stated using only the information of the sample.

William S. Peters

Chapter 14. R. A. Fisher

Abstract

In our Pearson to Gosset to Fisher chapter we left our historical account at Gossett’s 1908 paper on what later became, at R. A. Fisher’s suggestion, the Student t-distribution. We illustrated the distribution in its final form, as clarified by Fisher in 1925.

William S. Peters

Chapter 15. Sampling: Polls and Surveys

Abstract

In an article called “Samplingin a Nutshell,” Morris Slonim has us deal with a population of mixed nuts. [l] The mixture has one-half peanuts, one-third filberts, and one-sixth walnuts. The mean weight of the nuts in the mixture is 23 mg and the standard deviation is 17.2 mg.

William S. Peters

Chapter 16. Quality Control

Abstract

In our chapter on the normal distribution we illustrated the important idea of the probability distribution of the sample mean with an example of a process control chart. While the probability distribution of the sample mean based on a known population standard deviation was known since the time of Gauss, the application to manufacturing control, as suggested in the example, did not take place until the third decade of the twentieth century. It appears to have been an outgrowth of the emphasis on small samples and exact sampling distribution theory begun by Gosset in 1908 and extended greatly by R. A. Fisher by the early 1920s. [1]

William S. Peters

Chapter 17. Principal Components and Factor Analysis

Abstract

In Chapter 10 on correlation and regression we used an example of the numbers of runs scored in the 1975 and 1976 seasons by American League baseball clubs. The data were shown in Table 10–1 and the correlation was 0.63. Figure 17-1 shows the data plotted as a scatter diagram in standardized units.

William S. Peters

Chapter 18. Jerzy Neyman

Abstract

Jerzy Neyman (1894–1981) was born in Bendery near the border between Russia and Rumania. His father was a lawyer. Soon after his father’s death in 1906, Neyman’s family moved to Kharkov, in the Ukraine. He studied mathematics at the University of Kharkov, where the mathematician S. N. Bernstein introduced him to Karl Pearson’s Grammar of Science. Having twice been rejected for military service, he graduated in 1917 and continued at the University to prepare for an academic career. He was married in 1919, imprisoned twice during the Russian revolution, received his master’s degree in 1920, and subsequently lectured at the University in Kharkov. He went to Poland in 1921 in an exchange of nationals agreed to by Poland and Russia.

William S. Peters

Chapter 19. The Bayesian Bandwagon

Abstract

We ended our discussion of Laplace’s principles of probability without exploring the tenth principle, which Laplace termed moral hope. We know this concept as utility.

William S. Peters

Chapter 20. Observational Studies and Program Evaluation

Abstract

The principles of experimental design were taught by R. A. Fisher and succeeding generations of statisticians and researchers. First among these principles is the random assignment of experimental material to treatments. This ensures that variables not controlled in the experiment do not introduce spurious effects and permits a measure of error separate from the effects of the treatments. This error is used as the basis for tests and estimates concerning treatment effects.

William S. Peters

Chapter 21. Nonparametrics and Robust Methods

Abstract

The chapters in this book have traced the origin and development of some of the major ideas and applications of statistics. A large part of this history has to do with inference about the mean of a distribution. In stating a confidence interval or testing a hypothesis about a mean based on sample data, the usual classical technique is to use the Student t-distribution. Indeed, Gosset’s discovery of this distribution in 1908 was a major step forward in the history of statistics.

William S. Peters

Chapter 22. Statistics—A Summing Up

Abstract

Our journey that began with William Petty’s political arithmetic and Blaise Pascal’s correspondence on gaming is at an end. The tour has provided an introduction to the principles and practice of economic and social statistics. The journey has taken us back in time and we have viewed the principal developments in the mirror of history.

William S. Peters

Backmatter