Why would the non-parametric test be used?

Question

Why would the non-parametric test be used?

Answer 1

Non-parametric tests are a class of tests that cover two distinct groups of problems, distribution free problems and parameter free problems.

Certain tests, like the t-test or the z-test, depend upon the Central Limit Theorem for validity.  If the underlying assumptions of the Theorem are violated then the t-test or z-test is invalid.  There are two basic reasons for such a violation.  The first is that the distribution is not one of "finite variance," an example of which is a Cauchy distribution.  The second is that either the way the data was collected or they way the data appears in nature results in a violation of either "independence," of the data collected, or of being "identically distributed."

In such a case, you may not be able to know in advance what distribution you are facing.  The z-test depends upon the Gaussian  or Normal distribution.  The t-test depends upon the Student's distribution.  If those distributions do not apply, then any results from those tests have no reliability at all.  In such a case, you want to use non-parametric tests which do not have an assumption as to which distribution to use.  These tests are far less powerful.

In the case of parameter free tests, you would use them when you know either in fact or for a theoretical reason that the parameters you are using are biased.  For example, if you were comparing wages in two communities, but Bill Gates happened to live in one of those communities, then you should know that the mean wage is strongly skewed by Bill Gates alone and so any t-test of differences in the mean should result in a clear difference between the two communities, even though if Gates were removed there would be no difference.  Parameter free tests do not consider the mean at all, but rather the structure of the data.