Explorations in statistics: the assumption of normality

Abstract

Learning about statistics is a lot like learning about science: the learning is more meaningful if you can actively explore. This twelfth installment of Explorations in Statistics explores the assumption of normality, an assumption essential to the meaningful interpretation of a t test. Although the data themselves can be consistent with a normal distribution, they need not be. Instead, it is the theoretical distribution of the sample mean or the theoretical distribution of the difference between sample means that must be roughly normal. The most versatile approach to assess normality is to bootstrap the sample mean, the difference between sample means, or t itself. We can then assess whether the distributions of these bootstrap statistics are consistent with a normal distribution by studying their normal quantile plots. If we suspect that an inference we make from a t test may not be justified-if we suspect that the theoretical distribution of the sample mean or the theoretical distribution of the difference between sample means is not normal-then we can use a permutation method to analyze our data.

Keywords: Central Limit Theorem; bootstrap; normal quantile plot; permutation methods.