Data Analysis with Microsoft Excel: Updated for Office 2007

Chapter 6 Statistical Inference 265

Applying a Nonparametric Test to

Two-Sample Data

The two-sample nonparametric test is the Mann-Whitney test. In the Mann-

Whitney test we rank all of the values from smallest to largest and then sum

the ranks in each sample. Unlike the Wilcoxon test, we do not rank the abso-

lute data values or multiply the ranks by the sign of the original data. Table 6-7

shows an example of two sample data along with the calculated ranks.

Table 6-7 Two-Sample data

Sample 1 Values Ranks Sample 2 Values Ranks

22 12.0 23 3.0

16 11.0 21 4.0

1 5.0 2 6.0

24 1.5 8 9.0

7 8.0 24 1.5

3 7.0

9 10.0

Note that we don’t need to have equal sample sizes. Our null hypothesis

is that both samples have the same median value. In this example, the sum

of the Sample 1 ranks is 54.5, and the sum of the Sample 2 ranks is 23.5. We

can use probability theory to determine the probability of the fi rst sample

having a rank sum of 54.5 or greater if the null hypothesis were true. In this

case, that p value would be 0.176, which would not support rejecting the

null hypothesis.

When using the Mann-Whitney test, we also need to calculate the median

difference between the two samples. This is done by calculating the differ-

ence for each pair of observations taken from Sample 1 and Sample 2 and

then determining the median of those differences. For the data in Table 6-6,

there are 35 pairs, starting with the difference between 22 and 2 3 (the fi rst

observations in the samples) and going down to the difference between 9

and 2 4 (the last observations). The median of these 35 differences is 7. By

comparison, the difference of the sample averages is 7.31, so the median dif-

ference is pretty close. When the sample sizes get large, these calculations

cannot be easily done by hand.

The Mann-Whitney test makes only four assumptions.

1. Both samples are random samples taken from their respective probability
   distributions.


2. The samples are independent of each other.


3. The measurement scale is at least ordinal.


4. The two distributions have the same shape.