Solved – Paired sample t test normality violation

Question

I would like to conduct a paired sample t test and thus, I'm checking for the assumptions of normality. Upon conducting normality testing, each group scores were found to be normally distributed however the difference between pairs of scores are found to be not normally distributed. So my question here is, what went wrong and what can I do about this?

Answer 1

Nothing has gone wrong, and you don't need to do anything about it. The paired sample t-test relies on the average of the differences of the paired samples being approximately normally distributed, not the differences themselves. The two statements are not incompatible; even if the differences are significantly skewed, the distribution of the sample mean will become "more normal" as the sample size gets larger, subject to a few conditions that are almost always met in practical applications.

For example, consider the sample mean of twelve observations from a Uniform$(0,1)$ distribution. Clearly the U$(0,1)$ distribution is a long way from Normal, but the sample mean, even of only 12 observations, is not:

> x <- rep(0,1000)
> for  (i in 1:1000) x[i] <- mean(runif(12))
> hist(x)

which produces:

Not too bad, and with 10,000 draws, it would be better still...