statistics- A library of statistical types, data, and functions

Copyright(c) 2010 Neil Brown
Safe HaskellNone




Mann-Whitney U test (also know as Mann-Whitney-Wilcoxon and Wilcoxon rank sum test) is a non-parametric test for assesing whether two samples of independent observations have different mean.


Mann-Whitney U test

mannWhitneyUtest Source


:: TestType

Perform one-tailed test (see description above).

-> Double

The p-value at which to test (e.g. 0.05)

-> Sample

First sample

-> Sample

Second sample

-> Maybe TestResult

Return Nothing if the sample was too small to make a decision.

Perform Mann-Whitney U Test for two samples and required significance. For additional information check documentation of mannWhitneyU and mannWhitneyUSignificant. This is just a helper function.

One-tailed test checks whether first sample is significantly larger than second. Two-tailed whether they are significantly different.

mannWhitneyU :: Sample -> Sample -> (Double, Double) Source

The Mann-Whitney U Test.

This is sometimes known as the Mann-Whitney-Wilcoxon U test, and confusingly many sources state that the Mann-Whitney U test is the same as the Wilcoxon's rank sum test (which is provided as wilcoxonRankSums). The Mann-Whitney U is a simple transform of Wilcoxon's rank sum test.

Again confusingly, different sources state reversed definitions for U₁ and U₂, so it is worth being explicit about what this function returns. Given two samples, the first, xs₁, of size n₁ and the second, xs₂, of size n₂, this function returns (U₁, U₂) where U₁ = W₁ - (n₁(n₁+1))/2 and U₂ = W₂ - (n₂(n₂+1))/2, where (W₁, W₂) is the return value of wilcoxonRankSums xs1 xs2.

Some sources instead state that U₁ and U₂ should be the other way round, often expressing this using U₁' = n₁n₂ - U₁ (since U₁ + U₂ = n₁n₂).

All of which you probably don't care about if you just feed this into mannWhitneyUSignificant.

mannWhitneyUCriticalValue Source


:: (Int, Int)

The sample size

-> Double

The p-value (e.g. 0.05) for which you want the critical value.

-> Maybe Int

The critical value (of U).

Calculates the critical value of Mann-Whitney U for the given sample sizes and significance level.

This function returns the exact calculated value of U for all sample sizes; it does not use the normal approximation at all. Above sample size 20 it is generally recommended to use the normal approximation instead, but this function will calculate the higher critical values if you need them.

The algorithm to generate these values is a faster, memoised version of the simple unoptimised generating function given in section 2 of "The Mann Whitney Wilcoxon Distribution Using Linked Lists"

mannWhitneyUSignificant Source


:: TestType

Perform one-tailed test (see description above).

-> (Int, Int)

The samples' size from which the (U₁,U₂) values were derived.

-> Double

The p-value at which to test (e.g. 0.05)

-> (Double, Double)

The (U₁, U₂) values from mannWhitneyU.

-> Maybe TestResult

Return Nothing if the sample was too small to make a decision.

Calculates whether the Mann Whitney U test is significant.

If both sample sizes are less than or equal to 20, the exact U critical value (as calculated by mannWhitneyUCriticalValue) is used. If either sample is larger than 20, the normal approximation is used instead.

If you use a one-tailed test, the test indicates whether the first sample is significantly larger than the second. If you want the opposite, simply reverse the order in both the sample size and the (U₁, U₂) pairs.

Wilcoxon rank sum test

wilcoxonRankSums :: Sample -> Sample -> (Double, Double) Source

The Wilcoxon Rank Sums Test.

This test calculates the sum of ranks for the given two samples. The samples are ordered, and assigned ranks (ties are given their average rank), then these ranks are summed for each sample.

The return value is (W₁, W₂) where W₁ is the sum of ranks of the first sample and W₂ is the sum of ranks of the second sample. This test is trivially transformed into the Mann-Whitney U test. You will probably want to use mannWhitneyU and the related functions for testing significance, but this function is exposed for completeness.

Data types

data TestType Source

Test type. Exact meaning depends on a specific test. But generally it's tested whether some statistics is too big (small) for OneTailed or whether it too big or too small for TwoTailed



data TestResult Source

Result of hypothesis testing



Null hypothesis should be rejected


Data is compatible with hypothesis