Non parametric tests based on ranks
As previously mentioned, parametric tests are not robust because of their heavy reliance on the normality assumption. Some classical parametric tests have non parametric counterparts based on ranks.
Several non parametric tests exist including permutation test, rank (Wilcoxon) test and bootstrap. Using bootstrap method, you have additional benefit. You can even go one step further beyond the estimation of sampling distribution.
Another common application where distribution fitting procedures are useful is when we want to verify the assumption of normality before using some parametric test (see General Purpose of Nonparametric Tests).
This was true both in parametric tests (regressing the proportion of errors on both trial type [A vs B location] and condition) and in a nonparametric chi-square for the number of children searching incorrectly in each condition.
 See also: Distribution, Variance, Normal distribution, Regression, ANOVA
  
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