In Chapter 7 we studied statistical inference for means in one- and two-sample procedures. The more general case of comparing several means, one-way ANOVA, was the subject of Chapter 12. The methods in those chapters are valid when the sample means are approximately Normal. In this chapter we discuss alternative procedures that do not require this condition. The methods detailed in this chapter will help us answer questions such as

• Do baseball games in the American League have more hits than games in the National League?
• Do illustrations improve how well children retell a story?
• Does the type of background music influence performance on a mathematics task?

The condition that the sample means are Normal is satisfied when the population distributions are Normal. In practice, of course, no distribution is exactly Normal. Fortunately, our usual methods for inference about population means (the one-sample and two-sample t procedures and analysis of variance) are quite robust. That is, the results of inference are not very sensitive to moderate lack of Normality, especially when the samples are reasonably large. Methods that are not sensitive to conditions such as Normality are called robust.

What can we do if plots suggest that the population distribution is clearly not Normal, especially when we have only a few observations? This is not a simple question. Here are the basic options:

1. Outliers. If lack of Normality is due to outliers, it may be legitimate to remove the outliers. An outlier is an observation that may not come from the same population as the other observations. Equipment failure that produced a bad measurement, for example, entitles you to remove the outlier and analyze the remaining data.

2. Transformations. Sometimes we can transform our data so that their distribution is more nearly Normal. Transformations such as the logarithm that pull in the long tail of right-skewed distributions are particularly helpful. Example 7.9 (page 399) illustrates use of the logarithm.

3. Other standard distributions. In some settings, other standard distributions replace the Normal distributions as models for the overall pattern in the population. We mentioned in Chapter 5 (page 293) that the Weibull and exponential distributions are common models for the lifetimes in service of equipment in statistical studies of reliability. Also, we studied the exponential distributions (page 288) and the Poisson distributions (page 315) in Chapter 5. There are inference procedures for the parameters of these distributions that replace the t procedures when we use specific non-Normal models.

4. Bootstrap methods and permutation tests. Modern bootstrap methods and permutation tests do not require Normality conditions or any other specific form of sampling distribution. Moreover, you can base inference on resistant statistics such as the trimmed mean. We recommend these methods unless the sample is so small that it may not represent the population well. Chapter 16 includes a full discussion.

5. Nonparametric procedures. Finally, there are other nonparametric procedures that do not require any specific form for the distribution of the population. Unlike bootstrap and permutation methods, common nonparametric procedures do not make use of the actual values of the observations. We have already discussed the sign test (page 401), which works with counts of observations. This chapter presents rank tests based on the rank (place in order) of each observation in the set of all the data.

The methods of this chapter are designed to replace the t tests and one-way analysis of variance when the Normality conditions for those tests are not met. Figure 15.1 presents an outline of the standard tests (based on Normal distributions) and the rank tests that compete with them.

The rank tests we will study concern the center of a population or populations. When a population has at least a roughly Normal distribution, we describe its center by the mean. The “Normal tests” in Figure 15.1 test hypotheses about population means. When distributions are strongly skewed, we often prefer the median to the mean as a measure of center. In simplest form, the hypotheses for rank tests just replace the mean by the median.

We devote a section of this chapter to each of the rank procedures. Section 15.1, which discusses the most common of these tests, also contains general information about rank tests. The big idea of using ranks, the kind of assumptions required, the nature of the hypotheses tested, and using exact distributions for small samples and approximate distributions for larger samples are common to all rank tests. Sections 15.2 and 15.3 more briefly describe other rank tests.