# Non parametric tests on two independent samples

The Mann-Whitney non-parametric test compares the ranks of two independent samples. Run it in Excel using the XLSTAT add-on statistical software.

## What are non parametric tests on two independent samples

**Non parametric tests** on two independent sample are used to compare the distribution of two independent samples.

Non-parametric tests have been put forward in order to get round the assumption that a sample is normally distributed, required for using the parametric tests (z test, Student's t test, Fisher's F test, Levene's test and Bartlett's test).

## Non-parametric Tests on two independent samples

If we designate D to be the assumed difference in position between the samples (in general we test for equality, and D is therefore 0), and P1-P2 to be the difference of position between the samples, three tests are possible depending on the alternative hypothesis chosen:

- the two-tailed test:
**H0: P1 - P2 = D and Ha: P1 - P2 ≠ D** - the left-tailed test:
**H0: P1 - P2 = D and Ha: P1 - P2 < D** - the right-tailed test:
**H0: P1 - P2 = D and Ha: P1 - P2 > D**

## Mann-Whitney's test

Use the **Mann-Whitney test** to determine if the samples come from a single population or from two different populations meaning that the two samples may be considered identical or not. This test is based on on the ranks. XLSTAT can perform a two-tailed or a one-tailed test. This test is often called the Mann-Whitney test, sometimes the Wilcoxon-Mann-Whitney test or the Wilcoxon Rank-Sum test.

Let S1 be a sample made up of n1 observations (x1, x2, …, xn1) and S2 a second sample made up of n2 observations (y1, y2, …, yn2) independent of S1. Let N be the sum of n1 and n2.

XLSTAT calculates the Wilcoxon Ws statistic which measures the difference in position between the first sample S1 and sample S2 from which D has been subtracted, we combine the values obtained for both samples, then put them in order. For XLSTAT, the Ws statistic is the sum of the ranks of the first samples.

For the expectation and variance of Ws we therefore have:

**E(Ws) = 1/2 n1(N + 1)** and **V(Ws) = 1/12 n1n2(N + 1)**

The Mann-Whitney U statistic is the sum of the number of pairs (xi, yi) where xi>yi, from among all the possible pairs. We show that:

**E(U) = n1n2/2** and **V(U) = 1/12 n1n2(N + 1)**

We may observe that the variances of Ws and U are identical. In fact, the relationship between U and Ws is:

**Ws = U + n1(n1 + 1) / 2**

The results offered by XLSTAT are those relating to Mann-Whitney's U statistic.

## Kolmogorov-Smirnov's test

Use the Kolmogorov-Smirnov's test to determine if the populations from which the samples were taken have different cumulative distribution functions. XLSTAT performs a two-tailed test.