Statistical Analysis for Education and Psychology Researchers

conclude that the two distributions are not significantly different and that the average of
teachers’ estimates of active learning in maths for lower and upper school maths teachers
are not significantly different. As this is a two-tailed test, a Z-value of ≥1.96 would be
required for the results to be statistically significant at the 5 per cent level.

7.5 Wilcoxon Signed Ranks test (for related data)

When to Use

This is a test of the difference between pairs of related observations or matched pairs of
subjects, and is the nonparametric equivalent of the related t-test. The Wilcoxon signed
ranks test should be considered when researchers are interested in comparing two related
samples on some rankable measure and when the shape of the population is unknown or
assumptions underlying use of the related t-test are not met, typically distribution of the
measures in the population are non normal or measurements are not on an interval or
ratio scale (see Chapter 8 for details of t-test assumptions).
The Wilcoxon Signed Ranks test is a more powerful version of the sign test because
the procedure uses information about both the direction and magnitude of differences
within pairs. When we can determine the magnitude of a difference, observations can be
ranked. Parametric tests are often held to be more powerful than nonparametric
counterparts but this is only true when underlying normal theory assumptions are met. It
is seldom acknowledged that nonparametric tests may be as powerful or more powerful
than parametric counterparts under certain circumstances, for example, ‘heavy-tailed’
distributions, log-normal distributions and exponential distributions, in these situations
the Wilcoxon Signed Ranks test is more powerful than the related t-test, with a truncated
normal distribution the Wilcoxon Signed Ranks test and related t-test are equal in terms
of statistical power (Blair and Higgins, 1985).
The logic underpinning this test is elegantly simple. The aim of the test is to find out
about the distribution of the difference scores, that is the difference for each pair of
observations. We can think of, for example, a pre-post-test study design where each
individual has a before (pre) and after (post) intervention score. The distribution of
difference scores, pre-post-, would be asymmetrical about zero, that is predominantly
negative if the majority of subjects showed an improvement in scores after the
intervention. If there were an equal number of positive and negative differences, such as
only chance differences, and these differences were roughly equal magnitude, this would
suggest no significant difference between pre-and post-intervention samples of scores.

Statistical Inference and Null Hypothesis

The null hypothesis tested is that the median of the population differences is zero and that
the distribution of differences is symmetrical about zero. It is based on the assumption
that the amount of positive and negative difference which occurs by chance should be
approximately equal in each direction. A non-directional alternative hypothesis (two-
sided) would be that the median of the population of differences is non-zero and a
directional alternative hypothesis would be that the median of the population of

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