Basic Statistics

194 NONPARAMETRIC STATISTICS

and the sum of the negative values T- is 94. The sum of all the ranks is 105, Note
that n(n+1)/2=105. So the sum of all the ranks is n(n + 1)/2 and the mean of T+
and T- is one half of the sum of the ranks, or n(n + 1)/4
For large samples, the normal approximation (see van Belle et al. [2004]) can be
used with

E(T) = n(n + 1)/4 = 52.5

and the variance

Var (T) = n(n + 1)(2n + 1)/24

and

z = (T - E(T))/(dGT)

A correction should be made for the two 3.5’s and the two 5.5’s. Let ti equal the
number of ties. The numerator in the variance formula is reduced by

or

and if there are c ties,

or Var (T) = (14(15)(29) - 6)/24 = 6084/24 = 253.5 and the square root of the
variance is 15.9, so

2 = (94 - 52.5)/15.9 = 2.61

Since the z value of 2.61 is greater than 1.96, the null hypothesis that the differences

between each pair of observations comes from a distribution that is symmetric with

a zero mean or median is rejected for cy = .05.

13.2.2 Wilcoxon Signed Ranks Test for Small Samples

When the sample size is less than 15 it is recommended that the significance of the
results be obtained fromTable 12 in Conover [ 19991, Table C in Gibbons [ 19931, Table
19 in Dixon and Massey [1983], Table 3 in Daniel [1978], or the use of a software
package that gives exact results. The use of an exact methods is especially important
if the significance level is borderline. StatXact, SPSSX, and SAS can perform exact
calculations. Stata and Minitab also perform the Wilcoxon signed ranks test. In some
of the software packages, the results are given in terms of medians, as mentioned for
the sign test.