Statistical Analysis for Education and Psychology Researchers

differences is either greater or less than zero. The test statistic, T, is the rank sum totals
for selected pairs of observations. The absolute values of difference scores are first
ranked then the sign of the difference score is assigned (positive and negative). The
signed ranked differences are then summed separately (positive differences and negative
differences), the smaller of the two sums of the absolute values of the signed rank
differences (the +/− signs are again ignored) is the test statistic T. A sufficiently small
value of T provides evidence for rejection of the null hypothesis. The exact sampling
distribution for T with sample sizes ≤15 is determined and tabulated, for larger sample
sizes there is a normal approximation.

Test Assumptions

These are:

• Data consists of pairs of observations which have been selected at random and each pair
  is independent of other pairs.


• The differences between pairs of observations are also independent.


• The original measures in the two samples are rankable (in practice may be ratio, interval
  or ordinal).


• The differences between the two measures are rankable.


• Pairs of observations may be two measurements taken on the same subject or two
  subjects that are matched (paired) with respect to one or more important variables.

Example from the Literature

Teachers are frequently cited as the reason pupils dislike school. In a study involving 101
pupils from three schools, Boser and Poppen (1978) examined the qualities of teachers
that are generally liked and disliked by pupils. Pupils were asked to describe, using seven
descriptive categories, a teacher with whom they had poor relationships, ‘difficult getting
on with’ and a teacher with whom they had good relationships, ‘get along with well’.
One quality identified in the research literature that is associated with good teacher-
pupil relationships is the teacher role of ‘sharing’ that is when a teacher shares personal
ideas, opinions, or feelings about things. The investigators wanted to determine whether
there was a difference in sharing behaviour between best relationships and poorest
relationship situations. Sharing behaviour was scored on an ordinal scale 1–5 where 1
means always behaves like this and 5 means never behaves like this.
The null hypothesis tested was that teachers sharing behaviour is the same in poorest
and best relationship situations. The alternative hypothesis was nondirectional (two-
tailed). In this situation the Wilcoxon signed ranks test was used rather than a related t-
test because measures were ordinal. The investigators did not specify in advance an alpha
level but reported that sharing behaviour was significantly different at the 0.0001 level on
a two-tailed test. The mean score for sharing in best relationship situation was 2.198
compared with 3.830 in poorest relationship situation, a low mean denotes a high
frequency of behaviour.

Inferences involving rank data 227