Purpose

    This procedure compares two independent samples.

Preparations

    Run Statistics→Nonparametric Statistics →Comparing two independent samples.... command.

Results

Mann-Whitney U test or Wilcoxon Rank-Sum Test.
    This test is the nonparametric substitute for the equal-variance t-test when the assumption of normality is not valid. When in doubt about normality, play it safe and use this test. The Mann-Whitney U test assumes that the variable under consideration was measured on at least an ordinal (rank order) scale. The interpretation of the test is essentially identical to the interpretation of the result of a t-test for independent samples, except that the U test is computed based on rank sums rather than means. The U test is the most powerful (or sensitive) nonparametric alternative to the t-test for independent samples; in fact, in some instances it may offer even greater power to reject the null hypothesis than the t-test.
    With samples larger than 20, the sampling distribution of the U statistic rapidly approaches the normal distribution (see Siegel, 1956). Hence, the U statistic (adjusted for ties) will be accompanied by a z value (normal distribution variate value), and the respective p-value.

U (Mann Whitney U) - criterion statistic. U -  is defined as the total number of times a Y precedes an X in the configuration of combined samples Gibbons (1985). It is directly related to the sum of ranks. This is why this test is sometimes called the Mann-Whitney U test and other times called the Wilcoxon Rank Sum testThe Mann-Whitney U test calculates Ux and Uy.
    U = W - n_x (n_x - 1) / 2
    Mean W_x =  n_x (n_x + n_y + 1) / 2

For the Wilcoxon rank-sum test, the null and alternative hypotheses relate to the equality or non-equality of two medians..

Kolmogorov-Smirnov Test.
    This is a two-sample test for differences between two samples or distributions. If a statistical difference is found between the distributions of X and Y, the test provides no insight as to what caused the difference. For example, the difference could be due to differences in location (mean), variation (standard deviation), presence of outliers, type of skewness, type of kurtosis, number of modes, and so on.
The assumptions for this nonparametric test are: (1) there are two independent random samples; (2) the two population distributions are continuous; and (3) the data are at least ordinal in scale. This test is frequently preferred over the Wilcoxon sum-rank test when there are a lot of ties. The test statistic is the maximum distance between the empirical distribution functions (EDF) of the two samples.

Wald-Wolfowitz Runs Test.

    The Wald-Wolfowitz runs test is a nonparametric alternative to the t-test for independent samples.
    The Wald-Wolfowitz runs test works as follows: Imagine you want to compare male and female subjects on some variable. You can sort the data by that variable and look for cases when, in the sorted data, same-gender subjects are adjacent to each other. If there are no differences between male and female subjects, then the number and "lengths" of such adjacent "runs" of subjects of the same gender will be more or less random. If not, the two groups (genders in our example) are somehow different from each other. This test assumes that the variable under consideration is continuous, and that it was measured on at least an ordinal scale (i.e., rank order). The Wald-Wolfowitz runs test assesses the hypothesis that two independent samples were drawn from two populations that differ in some respect, i.e., not just with respect to the mean, but also with respect to the general shape of the distribution. The null hypothesis is that the two samples were drawn from the same population. In this respect, this test is different from the parametric t-test which strictly tests for differences in locations (means) of two samples.

Rosenbaum Test.

For the Rosenbaum test, the null and alternative hypotheses relate to the non-equality or equality of two medians..

 Criterion statistic
    Q = S₁ + S₂
where

    S₁ - number of variants from the first sample, bigger than maximum of the the second sample
    S₂ - number of variants from the second sample, bigger than maximum of the the first sample.
The criterion cannot be applied to samples with size less than 11.