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Statistical Tests & Distributions

October 29, 2010

Are all test statistics normally distributed?

Not all, but most of them are either based on the normal distribution directly or on distributions that are related to and can be derived from normal, such as t, F, or Chi-square.

Typically, these tests require that the variables analyzed are themselves normally distributed in the population, that is, they meet the so-called “normality assumption.” Many observed variables actually are normally distributed, which is another reason why the normal distribution represents a “general feature” of empirical reality.

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 ; 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.

via Elementary Concepts in Statistics.

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