Section author: Danielle J. Navarro and David R. Foxcroft

Comparing two means

In chapter Categorical data analysis we covered the situation when your outcome variable is nominal scale nominal and your predictor variable is also nominal scale nominal. Lots of real world situations have that character, and so you’ll find that χ²-tests in particular are quite widely used. However, you’re much more likely to find yourself in a situation where your outcome variable is interval scale or higher continuous, and what you’re interested in is whether the average value of the outcome variable is higher in one group or another. For instance, a psychologist might want to know if anxiety levels are higher among parents than non-parents, or if working memory capacity is reduced by listening to music (relative to not listening to music). In a medical context we might want to know if a new drug increases or decreases blood pressure. An agricultural scientist might want to know whether adding phosphorus to Australian native plants will kill them.[1] In all these situations our outcome variable is a fairly continuous continuous, interval or ratio scale variable, and our predictor is a binary “grouping” variable nominal. In other words, we want to compare the means of the two groups.

The standard answer to the problem of comparing means is to use a t-test, of which there are several varieties depending on exactly what question you want to solve. As a consequence, the majority of this chapter focuses on different types of t-test: one sample *t*-tests are discussed first, followed by two different flavours of the independent samples t-test: The Student test assumes that the groups have the same standard deviation, the Welch test does not. Afterwards, paired samples *t*-tests are discussed. We’ll then talk about one-sided tests and, after that, we’ll talk a bit about Cohen’s d, which is the standard measure of effect size for a t-test. The later sections of the chapter focus on the assumptions of the t-tests, especially normality and possible remedies if they are violated. However, before discussing any of these useful things, we’ll start with a discussion of the z-test.

[1]Informal experimentation in my garden suggests that yes, it does. Australian natives are adapted to low phosphorus levels relative to everywhere else on Earth, so if you’ve bought a house with a bunch of exotics and you want to plant natives, keep them separate; nutrients to European plants are poison to Australian ones.