9.1.11, 2.2.12, D.1.8, D.1.13, D.2.2 - Mann–Whitney U decisions
Learn how to compare two independent groups using ranked scores, calculate Mann–Whitney U and make a justified significance decision from the supplied Pearson tables.
Choosing Mann–Whitney for independent groups
Mann–Whitney U asks whether the pattern of scores differs between two independent groups.
Imagine comparing recall errors after learning words in quiet or with background speech. In an independent-groups design, different people provide the scores in each condition: one participant's quiet score is not paired with their own speech score. A test of difference is appropriate because the question compares conditions, rather than asking whether two co-variables rise together.
Mann–Whitney U
A non-parametric test of difference between two independent groups, using scores that can be ranked.
Non-parametric means this test does not require the scores themselves to follow a normal distribution. Data must be at least ordinal: ordered ratings are suitable, and numerical error counts can also be ranked. Nominal category frequencies require a different test. For paired scores, such as the same people tested twice, use an appropriate related-samples test such as Wilcoxon instead. Independent observations are still necessary; repeatedly recording one person does not create several independent participants.
The test compares rank distributions. When the groups have similarly shaped distributions, a difference can be interpreted as a shift in their typical scores. If their shapes or spreads differ substantially, do not automatically describe a significant U as proof of a difference in medians alone. Ranking also discards the original distances between scores: the gap between two adjacent ranks is always one even if their raw scores differ greatly.
Pooling scores and giving tied ranks
Ranking both groups together shows where each group's scores fall within the complete dataset.
The following illustrative data were created for this worked example; they are not a report of a completed practical. Five people in each condition made these numbers of recall errors:
| Quiet, group A | Background speech, group B |
|---|---|
| 2 | 4 |
| 3 | 6 |
| 3 | 7 |
| 4 | 8 |
| 5 | 9 |
Put all ten scores in ascending order and assign rank 1 to the lowest. Keep each score's group label. The two scores of 3 occupy positions 2 and 3, so each receives their mean rank, . The two scores of 4 occupy positions 4 and 5, so each receives rank 4.5, even though they came from different groups. After a tie, continue with the next unused position: 5 receives rank 6, not rank 5.
| Score | Group | Pooled rank |
|---|---|---|
| 2 | A | 1 |
| 3 | A | 2.5 |
| 3 | A | 2.5 |
| 4 | A | 4.5 |
| 4 | B | 4.5 |
| 5 | A | 6 |
| 6 | B | 7 |
| 7 | B | 8 |
| 8 | B | 9 |
| 9 | B | 10 |
Add the ranks within each group, not the raw scores. Group A has ; group B has . Their combined total is 55, which matches , the sum of ranks 1 to 10. Mean ranks preserve this total. Ranking each group separately would give both groups the same rank total and remove the comparison the test needs.
Calculating and checking the smaller U
The two U values use each group's size and pooled rank total, and the test statistic is the smaller value.
The current Pearson examination aid supplies these formulae, so practise substitution rather than treating memorisation as the learning goal:
Here and are the numbers of scores in each group, and and are their pooled rank sums.
For the recall-error data, . Therefore:
Continue this lesson.
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