﻿ Mann-Whitney U-test for Independent-Samples

# Mann-Whitney U-test for Independent-Samples

### Statistics > Nonparametric Tests > Mann-Whitney

Performs Mann-Whitney U-test for Independent-Samples.

This procedure performs a 2-sample rank test (also called the Mann-Whitney U-test for Independent-Samples, or the two-sample Wilcoxon rank sum test) of the equality of two population medians, and calculates the corresponding point estimate and confidence interval. This procedure is a nonparametric alternative to the Independent-Samples t-Test.

The Mann-Whitney U test is used to test whether two samples are drawn from the same population. It is most appropriate when the likely alternative is that the two populations are shifted with respect to each other. The test is performed by ranking the combined data set, dividing the ranks into two sets according the group membership of the original observations, and calculating a two sample z statistic, using the pooled variance estimate. For large samples, the statistic is compared to percentiles of the standard normal distribution. For small samples, the statistic is compared to what would result if the data were combined into a single data set and assigned at random to two groups having the same number of observations as the original samples.

Assumptions of the Mann-Whitney U-test:

the two samples are independent (unpaired)

the two populations have the same shape and same variance

the measurement scale is at least ordinal

When samples are large, a normal approximation is used for the hypothesis test and for the confidence interval.

### Dialog box items

Samples in one column:
Choose if the sample data are in a single column, differentiated by factor levels in a second column.

oSamples: Enter the columns containing the sample data.

oFactor and Levels: Enter the columns containing the sample factor, and select the levels.

Samples in different columns:
Choose if the data of the two samples are in separate columns.

oFirst Sample: Enter the column containing one sample.

oSecond Sample: Enter the column containing the other sample

Report:
The display of outputs of VisualStat.

### Data

Data can be entered in one of two ways:

Both samples in a single numeric column with another grouping column (called factor) to identify the population. The grouping column may be categorical, numeric or text.

Each sample in a separate numeric column.

The sample sizes do not need to be equal. Missing values are ignored.

### Options

Summary Statistics:
Check to compute summaries for each sample.

Adjusted for ties:
Check if the statistic is computed with average ranks used in the case of ties.

Continuity Correction:
Choose to use a continuity correction in the normal approximation to the distribution of the test statistics. This correction is valid only for dichotomous categories.

Alternative Hypothesis:
Enter Two-sided, Upper One-sided, or Lower One-sided. If you choose an One-sided hypothesis test, an upper or lower confidence bound will be constructed, respectively, rather than a confidence interval.

Confidence Level:
Enter the level of confidence desired. Enter any number between 0 and 100. Entering 90 will result in a 90% confidence interval. The default is 95%.

### Example

Two processing systems were used to clean wafers. The data represent the (coded) particle counts. The null hypothesis is that there is no difference between the medians of the particle counts; the alternative hypothesis is that there is a difference.

1.Open the DataBook nonparam.vstz

2.Select the sheet mw

3.Choose the tab Statistics, the group Nonparametric Tests and the command Mann-Whitney

4.Choose Samples in different columns

5.In First Sample, enter Group A

6.In Second Sample, enter Group B

7.Click Statistics. Check Summary Statistics

8.Click OK

Report window output

Mann-Whitney U-test for Independent-Samples

Tow-sample rank test of the equality of two population medians

alternative hypothesis: true ETA1 is not equal to ETA2

With continuity correction

Summary Statistics

N    Mean  Median  Std Dev  Mean Rank  Rank Sum        U

Group A  11  0.5745  0.5500   0.1041     9.6364  106.0000  40.0000

Group B  11  0.6373  0.6500   0.0912    13.3636  147.0000  81.0000

Mann-Whitney U-test for Independent-Samples (adjusted for ties)

Group A ; Group B

Distribution    Normal Approximation

Mann-Whitney U  40.0000

Wilcoxon W      106.0000

Z Statistic     -1.3152

p-value         0.1885

alpha-level     0.05

Critical Value  1.9600

Conclusion      Reject the Alternative Hypothesis

### Interpreting the results

VisualStat calculates the sample medians of the ordered data as 0.55 and 0.65 . The test statistic U = 60 has a p-value of 0.1885 . Since the p-value is not less than the chosen a level of 0.05, you conclude that there is insufficient evidence to reject the Null Hypothesis. That means, the two populations have the same median.

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