You have several options when you want to perform a hypothesis test with nonnormal data.
Although many hypothesis tests are formally based on the assumption of normality, you can still obtain good results with nonnormal data if your sample is large enough. The amount of data you need depends on how nonnormal your data are but a sample size of 20 is often adequate. The relationship between robustness to normality and sample size is based on the central limit theorem. This theorem proves that the distribution of the mean of data from any distribution approaches the normal distribution as the sample size increases. Therefore, if you're interested in making an inference about a population mean the normality assumption is not critical so long as your sample is large enough.
Nonparametric tests do not assume a specific distribution for the population. Minitab provides several nonparametric tests that you can use instead of tests that assume normality. These tests can be especially useful when you have a small sample that is skewed or a sample that contains several outliers.
|Test that assumes normality||Nonparametric test equivalents|
|1-Sample Z, 1-sample-t||1-Sample Sign, 1-Sample Wilcoxon|
|ANOVA||Kruskal-Wallis, Mood's median, Friedman|
Nonparametric tests are not completely free of assumptions about your data: for example, they still require the data to be an independent random sample.
Sometimes you can transform your data by applying a function to make your data fit a normal distribution, so that you can finish your analysis.