# Chapter 3. How to Test for Differences between Groups

Statistical methods are used to summarize data and test hypotheses with those data. Chapter 2 discussed how to use the mean, standard deviation, median, and percentiles to summarize data and how to use the standard error of the mean to estimate the precision with which a sample mean estimates the population mean. Now we turn our attention to how to use data to test scientific hypotheses. The statistical techniques used to perform such tests are called *tests of significance*; they yield the highly prized *P value.* We now develop procedures to test the hypothesis that, on the average, different treatments all affect some variable identically. Specifically, we will develop a procedure to test the hypothesis that diet has no effect on the mean cardiac output of people living in a small town. Statisticians call this hypothesis of no effect the *null hypothesis.*

The resulting test can be generalized to analyze data obtained in experiments involving any number of treatments. In addition, it is the archetype for a whole class of related procedures known as *analysis of variance.*

To begin our experiment, we randomly select four groups of seven people each from a small town with 200 healthy adult inhabitants. All participants give informed consent. People in the control group continue eating normally; people in the second group eat only spaghetti; people in the third group eat only steak; and people in the fourth group eat only fruit and nuts. After 1 month, each person has a cardiac catheter inserted and his or her cardiac output is measured.

As with most tests of significance, we begin with the hypothesis that all treatments (diets) have the same effect (on cardiac output). Since the study includes a control group (as experiments generally should), this hypothesis is equivalent to the hypothesis that diet has no effect on cardiac output. Figure 3-1 shows the distribution of cardiac outputs for the entire population, with each individual's cardiac output represented by a circle. The specific individuals who were randomly selected for each diet are indicated by shaded circles, with different shading for different diets. Figure 3-1 shows that the null hypothesis is, in fact, true. Unfortunately, as investigators we cannot observe the entire population and are left with the problem of deciding whether or not to reject the null hypothesis from the limited data shown in Figure 3-2. There are obviously differences between the samples; the question is: *Are these differences due to the fact that the different groups of people ate differently or are these differences simply a reflection of the random variation in cardiac output between individuals?*

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