The problem may occur when we try to use a normal distribution-based test to analyze data from variables that are themselves not normally distributed. In such cases, we have two general choices. First, we can use some alternative "nonparametric" test (or so-called "distribution-free test", but this is often inconvenient because such tests are typically less powerful and less flexible in terms of types of conclusions that they can provide.

Alternatively, in many cases we can still use the normal distribution-based test if we only make sure that the size of our samples is large enough. The latter option is based on an extremely important principle that is largely responsible for the popularity of tests that are based on the normal function. Namely, as the sample size increases, the shape of the sampling distribution (i.e., distribution of a statistic from the sample; this term was first used by Fisher, 1928a) approaches normal shape, even if the distribution of the variable in question is not normal. There is a very important corollary to this principle too, called The Central Limit Theorem explained in the next panel.