MedCalc

# Mann-Whitney test (independent samples)

## Description

The Mann-Whitney test is the non-parametric equivalent of the independent samples t-test (it is sometimes - wrongly - called a 'non-parametric t-test').

This test should be used when the sample data are not Normally distributed, and they cannot be transformed to a Normal distribution by means of a logarithmic transformation.

## Required input

Select the variables for sample 1 and sample 2. You can use the button to select variables and filters in the variables list.

Caveat: if the two variables are the same, then the two filters must define distinct groups so that the same case is not included in the two samples.

## Results

### Summary statistics

The results windows for the Mann-Whitney test (independent samples) displays summary statistics of the two samples.

The statistics include the Hodges-Lehmann median difference (the Hodges-Lehmann estimate of location shift) and its 95% confidence interval (Conover, 1999). For two independent samples with sample size m and n, the Hodges-Lehmann median difference is the median of all m × n paired differences between the observations in the two samples. Differences are calculated as sample 2 − sample 1. The confidence interval is derived according to Conover (1999, p. 281).

Note that the Hodges-Lehmann median difference is not necessarily the same as the difference between the two medians.

### Mann-Whitney test results

The Mann-Whitney test (independent samples) combines and ranks the data from sample 1 and sample 2 and calculates a statistic on the difference between the sum of the ranks of sample 1 and sample 2.

• In the presence of ties, or when either or both sample sizes are larger than 25, MedCalc uses the Normal approximation (Siegel & Castellan, 1988; Hollander et al., 2014) to calculate the P-value.
• For smaller sample sizes (both N≤25) MedCalc calculates the exact probability (Mann & Whitney, 1947; Dinneen & Blakesley, 1973).

If the resulting P-value is small (P<0.05) then a statistically significant difference between the two samples can be accepted.

Note that in MedCalc P-values are always two-sided.

## Literature

• Conover WJ (1999) Practical non-parametric statistics, 3rd edition. New York: John Wiley & Sons.
• Dinneen LC, Blakesley BC (1973) Algorithm AS 62: A generator for the sampling distribution of the Mann-Whitney U statistic. Journal of the Royal Statistical Society. Series C (Applied Statistics) 22:269-273
• Hollander M, Wolfe DA, Chicken E (2014). Non-parametric Statistical Methods. 3rd ed. Hoboken NJ: John Wiley & Sons.
• Lentner C (ed) (1982) Geigy Scientific Tables, 8th edition, Volume 2. Basle: Ciba-Geigy Limited.
• Mann HB, Whitney DR (1947) On a test of whether one of two random variables is stochastically larger than the other. The Annals of Mathematical Statistics 18:50-60.
• Siegel S, Castellan NJ Jr (1988) Non-parametric statistics for the behavioral sciences. 2nd ed. Singapore: McGraw-Hill Book Company.