A client has sent us the following question:

Q.   I ran a Mann-Whitney test on two independent groups that have equal medians, the results were
significant. I thought that
the Mann-Whitney tested differences in medians. Why is the Mann-Whitney test significant
when the medians are equal?

A.   The answer is that the Mann-Whitney and the equivalent Wilcoxon test (hereafter
called the Mann-Whitney-Wilcoxon test) are rank sum tests and not median tests. Basically,
the Mann-Whitney-Wilcoxon test ranks all of the observations from both groups and then sums
the ranks from one of the groups which is compared with the expected rank sum. It is possible,
although not very common, for groups to have different rank sums and yet have equal or nearly
equal medians. An example is given below.

Consider the following example dataset of 120 observation (60 in each group) that has equal
medians and a significant Mann-Whitney-Wilcoxon test.

  +-----------------+
  |  y   grp   freq |
  |-----------------|
  | -2     1     20 |
  |  0     1     20 |
  |  5     1     20 |
  |-----------------|
  | -1     2     20 |
  |  0     2     20 |
  | 10     2     20 |
  +-----------------+

The median for each of the two groups is zero, yet the Z-approximation for the
Mann-Whitney-Wilcoxon is -2.16 with a p-value of .031. On the other hand, for the median
test the difference in medians is zero, since the two medians are equal,
the t-value is 0 and has a p-value of 1.0 (some median tests report a chi-square value which
in this case, will also be 0 with a p-value of 1.0).

The reason the Mann-Whitney-Wilcoxon is significant for the above data is the ranks
for group 1 (other than those at the median) are lower than the ranks for group 2 (again,
other than those values at the median). Here are the ranks for all the scores ignoring
the frequencies to keep it simple.

  +-------------------+
  |  y   grp     rank |
  |-------------------|
  | -2     1      1   |
  | -1     2      2   |
  |  0     2     3.5  |
  |  0     1     3.5  |
  |  5     1      5   |
  | 10     2      6   |
  +-------------------+

Note that the value of -2 has a rank of 1 and the value -1 has a rank of 2.
The values 5 and 10 have ranks of 5 and 6 respectively. The 0 scores all have a rank of 3.5.
Thus, other than the median (rank 3.5), all of the ranks for Grp1 are less than the ranks
for group 2. With sufficient sample size the difference in ranks will be large enough to be
significant even though the medians are equal.

The table below gives the sum of the ranks for each group for the full sample of 120
observations.

         Grp       obs    rank sum
          1         60        3230
          2         60        4030

Group 2 clearly has a much larger sum of ranks than group 1. The difference in the sum of ranks
is large enough to be statistically significant at the alpha equals .05 level.