Mann-Whitney U-Test

An alternative to the Independent Measures t-test

• The Mann-Whitney test is designed to use the data from two separate samples to evaluate the difference between two treatments.

• Individual scores in the two samples should be rank-ordered.

• The Mann-Whitney test compares two distributions rather than two means.

• If the 2 samples are combined and all the scores placed in rank order, the scores from one sample should be concentrated at one end of the line and the scores from the other sample should be concentrated at the other end.

• If there is no treatment difference, large and small scores will be mixed evenly in the 2 samples because there is no reason for one set of scores to be systematically larger or smaller than the other.

ExamplenA = 6nB = 6

Sample A 27 2 9 48 6 15

Sample B 71 63 18 68 94 8

STEP 1. State your hypothesis.Ho: There is no difference between the two

treatments. Therefore, there is no tendency for the ranks in one treatment condition to be systematically higher (or lower) than the ranks in the other treatment condition.

H1: There is a difference between the two treatments. Therefore, the ranks in one treatment condition are systematically higher (or lower) than the ranks in the other treatment condition.

STEP 2. Select alpha level & look up critical U value.

a. Select alpha level and type of test.α = .05,

b. Look at Mann-Whitney U table. For nA = 6, nB = 6, the critical U value is 5.

STEP 3. Compute RA and RB

a. Compute ΣRA (sum of the ranks of scores in sample A).

ΣRA = 1 + 2 + 4 + 5 + 7 + 8 = 27

b. Compute ΣRB (sum of the ranks of scores in sample B).

ΣRB = 3 + 6 + 9 + 10 + 11 + 12 = 51

STEP 4. Compute UA and UB

a. UA = nAnB + nA (nA + 1) - ΣRA

2 = 6(6) + 6(7) – 27

2 = 36 + 21 – 27

UA = 30

b. UB = nAnB + nB (nB + 1) - ΣRB

2= 6(6) + 6(7) – 51

2 = 36 + 21 – 51

UB = 6

The Mann-Whitney U value is the smaller of these two, U = 6.

Each score in sample A is assigned 1 point for every score in sample B that has a higher rank.

UA = 6 + 6 + 5 + 5 + 4 + 4 = 30UB = 6

UA + UB = nAnB

30 + 6 = 6(6)U = 36

RANK SCORE SAMPLE POINTS FOR SAMPLE A

1 2 (A) 6 points

2 6 (A) 6 points

3 8 (B)

4 9 (A) 5 points

5 15 (A) 5 points

6 18 (B)

7 27 (A) 4 points

8 48 (A) 4 points

9 63 (B)

10 68 (B)

11 71 (B)

12 94 (B)

STEP 5. State your conclusion

• U = 6, p > .05• Thus, we retain Ho.

• The data do not provide enough evidence to conclude that there is a significant difference between the two treatments.