A nonparametric test is a hypothesis test that does not require the population's distribution to be characterized by certain parameters. For example, many hypothesis tests rely on the assumption that the population follows a normal distribution with parameters μ and σ. Nonparametric tests do not have this assumption, so they are useful when your data are strongly nonnormal and resistant to transformation.

In parametric statistics, we assume that samples are drawn from fully specified distributions characterized by one or more unknown parameters we want to make inference about. In a nonparametric method, we assume that the parent distribution of the sample is unspecified and we are often interested in making inference about the center of the distribution. For examples, many tests in parametric statics such as the 1-sample t-test are derived under the assumption that the data come from normal population with unknown mean. In a nonparametric study the normality assumption is removed.

Nonparametric methods are useful when the normality assumption does not hold and your sample size is small. However, nonparametric tests are not completely free of assumptions about your data. For instance, it is crucial to assume that the observations in the samples are independent and come from the same distribution. Also, in two-sample designs the assumption of equal shape and spread is required.

For example, salary data are heavily skewed to the right, with many people earning modest salaries and fewer people earning larger salaries. You can use nonparametric tests on this data to answer questions such as the following:

- Is the median salary at your company equal to a certain value? Use the 1-sample sign test.
- Is the median salary at a bank's urban branch greater than the median salary of the bank's rural branch? Use the Mann-Whitney test or the Kruskal-Wallis test.
- Are median salaries different in rural, urban, and suburban bank branches? Use Mood's median test.
- How does education level affect salaries at the rural and urban branch? Use Friedman test.

Nonparametric tests have the following limitations:

- Nonparametric tests are usually less powerful than corresponding parametric test when the normality assumption holds. Thus, you are less likely to reject the null hypothesis when it is false if the data comes from the normal distribution.
- Nonparametric tests often require you to modify the hypotheses. For example, most nonparametric tests about the population center are tests about the median instead of the mean. The test does not answer the same question as the corresponding parametric procedure if the population is not symmetric.

When a choice exists between using a parametric or a nonparametric procedure, and you are relatively certain that the assumptions for the parametric procedure are satisfied, then use the parametric procedure. You may also be able to use the parametric procedure when the population is not normally distributed if the sample size is adequately large.

The following is a list of the nonparametric tests, and their parametric alternatives.

Nonparametric test | Alternative parametric test |
---|---|

1-sample sign test | 1-sample Z-test, 1-sample t-test |

1-sample Wilcoxon test | 1-sample Z-test, 1-sample t-test |

Mann-Whitney test | 2-sample t-test |

Kruskal-Wallis test | One-way ANOVA |

Mood's Median test | One-way ANOVA |

Friedman test | Two-way ANOVA |