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 and your predictor variable is also nominal scale . 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 , 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 , interval or ratio scale variable, and our predictor is a binary “grouping” variable . 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.