Chapter 6 Statistical Inference 251Table 6-4 Signed Ranks
Variable Values Signed Ranks
18 7.0
4 2.0
15 6.0
25 2 3.5
22 2 1.0
10 5.0
5 3.5There are seven values in this data set, so the ranks go from 1 (for lowest
in absolute value) up to 7 (for the highest in absolute value). The lowest
in absolute value is 2 2, so that observation gets the rank 1 and then is
multiplied by the sign of the observation to get the sign rank value 2 1. The
value 4 gets the sign rank value 2 and so forth. Two observations, 2 5 and
5, are tied with the same absolute value. They should get ranks 3 and 4 in
our data set, but because they’re tied, they both get an average rank of 3.5
(or 2 3.5).
Next we calculate the sum of the signed ranks. If most of the values were
positive, this would be a large positive number. If most of the values were
negative, this would be a large negative number. The sum of the signed ranks
in our example equals 7 121623 .5 211513 .5 519.
The only assumption we make with the Wilcoxon Signed Rank test is
that the distribution of the values is symmetric around the median. If under
the null hypothesis we assume that the median = 0, this would imply that
we should have as many negative ranks as positive ranks and that the sum
of the signed ranks should be 0. Using probability theory, we can then de-
termine how probable it is for a collection of 7 observations to have a total
signed rank of 19 or more if the null hypothesis is true. Without going into
the details of how to make this calculation, the p value in this particular
case is 0.133, so we would not reject the null hypothesis. In addition to
calculating p values, you can also calculate confi dence intervals using the
Wilcoxon test statistic.
One advantage in using ranks instead of the actual values is that the
hypothesis test is much less sensitive to the effect of outliers. Also, non-
parametric procedures can be applied to situations involving ordinal data,
such as surveys in which subjects rank their preferences. The downside
of nonparametric tests is that they are not as effi cient as parametric tests
when the data are normally distributed. This means that for normal data
you need a larger sample size in order to detect statistically signifi cant
effects (5% larger when the Wilcoxon Signed Rank test is used in place
of the t test). Of course, if the data are not normally distributed, you can
often detect statistically signifi cant effects with smaller sample sizes using
nonparametric procedures. The nonparametric test can be more effi cient
in those cases.