
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 MannWhitney 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  MannWhitney (testing medians)
>2 independent samples
Parametric  ANOVA (testing means)
NonParametric  Mood'sMedian or KruskalWallis 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
Six Sigma Modules
Green Belt Program (1,000+ Slides)
Basic Statistics
SPC
Process Mapping
Capability Studies
MSA
Cause & Effect Matrix
FMEA
Multivariate Analysis
Central Limit Theorem
Confidence Intervals
Hypothesis Testing
T Tests
1Way Anova Test
ChiSquare Test
Correlation and Regression
Control Plan
Kaizen
Error Proofing