Does my sample have to be normally distributed for a t-test?

In small samples most statistical methods do require distributional assumptions, and the case for distribution-free rank-based tests is relatively strong. However, in the large data sets typical in public health research, most statistical methods rely on the Central Limit Theorem, which states that the average of a large number of independent random variables is approximately Normally distributed around the true population mean. It is this Normal distribution of an average that underlies the validity of the t-test.

n=5, n=10
n=30, n=50
mean and stdev are in gray

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