Chapter 5: Analysing the Data Part II : Inferential Statistics

# Normality and Equality of Variance

To test hypotheses about population parameters, we must assume that the population distribution of the variable being measured is normal in form. The nonparametric tests that we will meet later have been developed at least partly to deal with data in which the normality condition seems not to be met. How important is the normality assumption? What happens if we violate the assumption and use the t test for data that are not normally distributed?

Because so many tests assume the normal distribution and equal variances in multiple condition designs, those assumptions have received a great deal of attention from statisticians in the past several years. We won't try to survey this expanding area of statistical literature in detail, but will summarise what seems to be an emerging consensus of opinion and offer guidelines for researchers.

Much current opinion, but by no means all of it, is that, in general, you shouldn't worry a lot about normality and equal variance. Research seems to indicate that most of the parametric (that is, normal-curve-based) inference procedures are fairly well-behaved in the face of moderate departures from both normality and equality of variance. Tests and estimates that are relatively uninfluenced by violations in their assumptions are known as robust procedures, and a substantial literature has developed in the field of robust statistics.

A well-equipped statistician has a number of tools to draw upon at different stages of the analysis. In the initial stages, when you're trying to get a general "feel for" the data, all of the descriptive procedures are useful. When it comes to actually making inferences, though, parametric procedures are often the best because they are frequently more powerful.