Normal sampling theory: test for difference between several sample means, analysis of variance, design of experiments

Abstract

In Chapter 6 the t test was used to decide whether the difference between two sample means was too great for them to come from the same population. We now describe the test for the case of more than two samples (Note: applying the t test to the difference between largest and smallest mean will give the wrong answer). This test is based on the fact that the mean of a sample of size n from a normal population with given σ, is itself normally distributed with variance σ²/n (see Section 5.7). If, then, we have m samples, and their means have a greater variance than would be expected from m individuals from a population whose variance is σ²/n, we conclude that, very probably, there are some real physical differences between the means. Unfortunately, σ² is not usually known and so has to be estimated from the variance within each sample (it is assumed that all population variances are the same).