This first chapter begins by discussing what statistics are and why the study of statistics is important. Subsequent sections cover a variety of topics all basic to the study of statistics. One theme common to all of these sections is that they cover concepts and ideas important for other chapters in the book.

Graphing data is the first and often most important step in data analysis. In this day of computers, researchers all too often see only the results of complex computer analyses without ever taking a close look at the data themselves. This is all the more unfortunate because computers can create many types of graphs quickly and easily.

The of this chapter gives an example in which a well-construed graph makes it clear that there was a bias in the draft lottery of 1969. The two following sections discuss common graphs for qualitative and quantitative variables.

Descriptive statistics often involves using a few numbers to summarize a distribution. One important aspect of a distribution is where its center is located. Measures of central tendency are discussed first. A second aspect of a distribution is how spread out it is. In other words, how much the numbers in the distribution vary from one another. The second section describes measures of variability. Distributions can differ in shape. Some distributions are symmetric whereas others have long tails in just one direction. The third section describes measures of the shape of distributions. The final two sections concern (1) how transformations affect measures summarizing distributions and (2) the variance sum law, an important relationship involving a measure of variability. 

A dataset with two variables contains what is called bivariate data. This chapter discusses ways to describe the relationship between two variables. For example, you may wish to describe the relationship between the heights and weights of people to determine the extent to which taller people weigh more. 

The introductory section gives more examples of bivariate relationships and presents the most common way of portraying these relationships graphically. The next five sections discuss Pearson’s correlation, the most common index of the relationship between two variables. The final section, “Variance Sum Law II” makes use of Pearson’s correlation to generalize this law to bivariate data.

Probability is an important and complex field of study. Fortunately, only a few basic issues in probability theory are essential for understanding statistics at the level covered in this book. These basic issues are covered in this chapter. 

The introductory section discusses the definitions of probability. This is not as simple as it may seem. The section on basic concepts covers how to compute probabilities in a variety of simple situations. The Gambler’s Fallacy Simulation provides an opportunity to explore this fallacy by simulation. The Birthday Demonstration illustrates the probability of finding two or more people with the same birthday. The Binomial Demonstration shows the binomial distribution for different parameters. The section on base rates discusses an important but often-ignored factor in determining probabilities. It also presents Bays’ Theorem. The Bays’ Theorem Demonstration shows how a tree diagram and Bays’ Theorem result in the same answer. Finally, the Monty Hall Demonstration lets you play a game with a very counterintuitive result.nbsp;

Most of the statistical analyses presented in this book are based on the bell-shaped or normal distribution. The introductory section defines what it means for a distribution to be normal and presents some important properties of normal distributions. The interesting history of the discovery of the normal distribution is described in the second section. Methods for calculating probabilities based on the normal distribution are described in Areas of Normal Distributions. The Varieties of Normal Distribution Demo allows you to enter values for the mean and standard deviation of a normal distribution and see a graph of the resulting distribution. A frequently used normal distribution is called the Standard Normal distribution and is described in the section with that name. The binomial distribution can be approximated by a normal distribution. The section Normal Approximation to the Binomial shows this approximation. The Normal Approximation Demo allows you to explore the accuracy of this approximation. 

The concept of a sampling distribution is perhaps the most basic concept in inferential statistics. It is also a difficult concept to teach because a sampling distribution is a theoretical distribution rather than an empirical distribution. 

The introductory section defines the concept and gives an example for both a discrete and a continuous distribution. It also discusses how sampling distributions are used in inferential statistics.

The Basic Demo is an interactive demonstration of sampling distributions. It is designed to make the abstract concept of sampling distributions more concrete. The Sample Size Demo allows you to investigate the effect of sample size on the sampling distribution of the mean. The Central Limit Theorem (CLT) Demo is an interactive illustration of a very important and counter-intuitive characteristic of the sampling distribution of the mean.

The remaining sections of the chapter concern the sampling distributions of important statistics: the Sampling Distribution of the Mean, the Sampling Distribution of the Difference Between Means, the Sampling Distribution of r, and the Sampling Distribution of a Proportion.

Many if not most experiments are designed to compare means. The experiment may involve only one sample mean that is to be compared to a specific value. Or the experiment could be testing differences among many different experimental conditions, and the experimenter could be interested in comparing each mean with each other mean. This chapter covers methods of comparing means in many different experimental situations. 

The topics covered here in sections D, E, G, and H are typically covered in other texts in a chapter on Analysis of Variance. We prefer to cover them here since they bear no necessary relationship to analysis of variance. As has been pointed out elsewhere, it is not logical to consider the procedures in this chapter tests to be performed subsequent to an analysis of variance. Nor is it logical to call them post-hoc tests as some computer programs do.

Statisticians are often called upon to develop methods to predict one variable from other variables. For example, one might want to predict college grade point average from high school grade point average. Or, one might want to predict income from the number of years of education.

Chi Square is a distribution that has proven to be particularly useful in statistics. The first section describes the basics of this distribution. The following two sections cover the most common statistical tests that make use of the Chi Square distribution. The section “One-Way Tables” shows how to use the Chi Square Distribution to test the difference between theoretically expected and observed frequencies. The section “Contingency Tables” shows how to use Chi Square to test the association between two nominal variables. This use of Chi Square is so common that it is often referred to as the “Chi Square Test.”