Statistical Analysis for Education and Psychology Researchers

Many statistical textbooks give special tables for the test statistic SR with upper and lower
critical values for different combinations of sample sizes. In some of these tables the p-
values associated with sample sizes of 20 or more are actually large sample
approximations. With very small sample sizes of n≤5 the exact sampling should be used,
for explanation of the use of these tables and illustrated worked examples, see Conover
(1980) and Siegel and Castallan (1988). The normal approximation of SR with continuity
correction is adequate for most occasions.

Interpretation

The obtained Z-value of 0.0330 has an associated p-value of 0.4880, the nearest value in
the Table of Z-scores in Appendix A4. This is the probability beyond a Z-value of 0.0330,
for a two-tailed test this is doubled. The p-value is therefore 0.976. This is clearly not
significant and the null hypothesis is therefore not rejected at the 5 per cent level. We
conclude that teachers’ estimates of active learning in maths lessons do not differ among
lower and upper school maths teachers.
Rejection of the null hypothesis is usually interpreted as a difference in the central
tendency of the two distributions. However, caution is required with this interpretation
because the null hypothesis may be rejected when the means of the two samples are very
similar (not so in this example). This is because the Wilcoxon M-W test is sensitive to
differences in variance in the two samples (the Behrens Fisher problem). Under the null
hypothesis, the Wilcoxon M-W test assumes that the two samples come from a single
population with an underlying continuous distribution even though measurement is only
at the ordinal level. If we assume that the two distributions are the same we are also
assuming that the variances are the same. If they are not, the means may still be similar
but clearly the distributions are not identical.

Computer Analysis

The SAS procedure PROC NPAR1WAY with the option WILCOXON performs a
Wilcoxon M-W test on the ranks of scores when there are two independent groups (data
is classified by a variable into two levels). The following SAS code illustrates use of the
procedure PROC NPAR1WAY:

data a;

input group response @@;

cards;

1 30 1 51 1 48 1 28 1 26 1 42 1 44 1 66 1 68 1 21

2 30 2 31 2 54 2 32 2 34 2 38 2 40 2 52 2 65 2 30 2 40

2 39

;

proc nparlway data=a wilcoxon;

class group;

var response;

run;

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