The Wilcoxon Signed Rank
Test is a nonparametric equivalent of the
one sample t-test.
It tests the hypothesis:
H_{0} the mean equals zero
H_{1} the mean [is less than/greater
than/not equal to] zero
The test involves:
- sort the values into order of their
absolute magnitude (ignoring the signs)
and allocate ranks
- calculate the sum the ranks of the
data values are positive (S_{+})
An example would be:
Rank |
1 |
2 |
3 |
4 |
5 |
6 |
Data Value |
-1.0 |
-2.0 |
2.5 |
3.0 |
3.0 |
3.5 |
If the mean is non-zero the positive values
will be clustered at one end or other of
the ordered data set and S_{+} and
S_{-} will be very different.
For the example S_{+} = 3 + 4 +
5 + 6 = 18
For small data sets values the critical
value is found from published tables. Because
the sum of the ranks must be an integer,
it is not usually possible to find the exact
critical value:
For larger data sets the z-statistic can
be found from:
This gives the upper tail
test (the mean is less than zero). For the
lower tail test, use the sum S+ and for
the two tailed test use both, remembering
to use the table value twice.
The p-value
can be calculated using normal distribution
tables. |