In general, the power of parametric tests are greater than the power of the alternative nonparametric test. As the sample size increases and becomes larger, the power of the nonparametric test approaches it parametric alternative.
Nonparametric test also assume that the underlying distributions are symmetric by not necessarily normal. When the choice exist on whether to use the parametric or nonparametric test, if the distribution is fairly symmetric, the standard parametric test are better choices than the nonparametric alternatives.
For example, if you are not sure if two data sets are normally distributed it may be safer to substitute the Mann-Whitney test to reduce the risk of drawing a wrong conclusion when testing two means.
A sample and a target (or given value)
Parametric - One Sample t test (testing means)
Nonparametric - One Sample Wilcoxon or One Sample Sign (testing medians)
Two independent samples
Parametric - Two Sample t test (testing means)
NonParametric - Mann-Whitney (testing medians)
>2 independent samples
Parametric - ANOVA (testing means)
NonParametric - Mood's-Median or Kruskal-Wallis Test (testing medians)
Use the Runs Test to examine the randomness of the data
Follow these steps when you believe the data does not meet normality assumptions:
Six Sigma Modules
Green Belt Program (1,000+ Slides)
Cause & Effect Matrix
Central Limit Theorem
1-Way Anova Test
Correlation and Regression