Independent Sample T-test, popular among various statistical tests, which is used to develop statistical evidence for two populations average is significant or not. The Mann- Whitney U test is the non-parametric equivalent of the one-sample t-test.
The Mann- Whitney U test is the alternative non-parametric test that is used when the assumptions of the selected data are not met. The test uses the ranks of values rather than comparing means. Using ranks only requires that the data should be measured at the ordinal level. However, the main objective or the key purpose of the Mann- Whitney U test is somewhat similar to Independent Sample T-test i.e to provide statistical evidence that the sampled population are different. This test has turned into the most favoured test over that of the Independent Sample T-test. when the assumptions of the Independent Sample T-test holds, it is less likely to detect a location shift in comparison with the test. The test is performed using a two-tail test
The Mann- Whitney U test considers two assumptions for data to be used. The two main assumptions are as follows:-
- Independence
- Should have equal variance.
Note: These assumptions are sufficient for determining if the two populations are different. we take into consideration that the two populations being used should be identical. Considering all assumptions we can use the Mann- Whitney U test on our sample.
Calculating the Mann-Whitney U test
In order to determine the U test statistics, the mixed set of data is first ordered in ascending order by tied numbers getting a rank similar to the medium position of those numbers in the ordered sequence. The process involves combining all the observations from two samples into one combined sample, also note which observation comes from which samples and then rank all the combined observations from 1 to R1+R2
Let N: the sum of ranks for the first sample.
The Mann-Whitney U test statistic is calculated using:
U = r1 r2 + {r1 (r1 + 1)/2} – N ,
where r1 is first sample
and r2 is the second sample
Now, let’s take a simple example :
Sample A
Observation :22 34 45 22 12 19 14
Rank: 15.5 15.5 3.5 12.4 12.5 3.5 2.5
Sample B
Observation :20 14 41 22 12 18 12 13
Rank: 14.5 15.5 3.5 9.4 8.5 6.5 2.5 2.5
Here, T = 82.5,
R1 = 7,
R2 = 8.
Hence, U = (7 * 8) + [{7 * (7+1)}/2] – 82.5 = 8.5
We now compare the calculated U statistic value with the values present in the table. In this example, the values are provided by R1 and R2. By comparing two values we reject or accept the null hypothesis. This method provides a good approximation in the case of large samples. Now that you know everything about this alternative test to the Independent Sample T-test, use it in case of violation of assumptions in correlated-sample situations and perform the test accurately.