Data Analysis - Quantitative Analysis - Inferential Statistics

This test is the non-parametric equivalent of the paired samples t-test. The null and alternative hypotheses are the same as those for the paired samples t-test although they are often stated in terms of the median, thus:

H₀: In the population the median difference is zero
H₁: In the population the median difference is not zero

However, an assumption behind this test is that the population distributions are symmetrical, in which case the mean and medians are identical.

Worked Example

Blood pressures (mmHg)
Subject	Standing	Supine	Difference
1	132	136	4
2	146	145	1
3	135	140	5
4	141	147	6
5	139	142	3
6	162	160	-2
7	128	137	9
8	137	136	-1
9	145	149	4
10	151	158	7
11	131	120	-11
12	143	150	7
Mean	140.83	143.33	2.50
SD	9.49	10.83	5.50

The frugal Minitab output simply shows the estimated median difference and the 94.5% confidence interval (with non-parametric tests it is not always possible to achieve exact probability levels). Since this confidence interval contains zero, we fail to reject the null hypothesis that the population median difference is zero.

The SPSS output provides, among other things, the p-value for a two-tailed test (0.099).

The conclusion is that there is insufficient evidence to suggest a difference in standing and supine systolic blood pressures

Computer output from Minitab and SPSS

The Wilcoxon Test using Minitab

To perform this test use
Stat > Nonparametrics > 1-sample Wilcoxon... after creating a column of differences between the pairs in the sample.

Wilcoxon Signed Rank Confidence Interval

		Estimated	Achieved
	N	Median	Confidence	Confidence Interval
Difference	12	3.00	94.5	(-1.00, 6.00)

The Wilcoxon Test using SPSS

Use
Analyze > Nonparametric Tests > 2 Related Samples... to obtain the following output

Wilcoxon Signed Ranks Test

table of ranks

table of test stats