Traditional statistical tests, such as TTests and Analysis
of Variance, require the computation of means and standard
deviations.
 These computations only make sense when the data
are numeric.
In this topic we present statistical tests that
only use the ordinal information in the data.
 These tests do not involve adding, subtracting,
dividing or multiplying the numbers in the data.
 They only involve using the order of the numbers.
We discuss two tests:
 MannWhitney UTest
 This test is used when we obtain ordinal data
in the independent groups situation.
 Wilcoxon SignedRanks Test
 This test is used when we obtain ordinal data
in the paired samples situation.
MannWhitney UTest
The MannWhitney Utest evaluates the difference between
two treatments (two populations) using data from the
independent measures design.
Conceptual Basis:
 A real difference between two treatments should
make the scores in one group generally larger than
those in the other.
 If the treatment had an effect, then when we combine
the two samples together and rank order all the
combined scores, the observations for one sample
should be concentrated at one end of the scale,
and the other sample's observations should be at
the other end.
 If there is no effect, then large and small scores
should be mixed together.
Calculations
 Observations are obtained for two groups, Group
A and Group B.
 All observations from the two groups are combined
and rank ordered.
 The MannWhitney U is calculated.
 Each observation in Group A gets a point for
every observation in Group B that it is larger
than. The total number of points for Group A is
calculated.
 Each observation in Group B gets a point for
every observation in Group A that it is larger
than. The total number of points for Group B is
calculated.
 The smaller of these two sums is the MannWhitney
U.
The hypothesis test for the MannWhitney U is performed.
 Null Hypothesis:
 The two groups are identically distributed.
 Alternative Hypothesis:
 The two groups are not identically distributed.
 Note that when the treatment had such a large
effect that all scores in one group are larger
than those in the other, then U=0.
 This test is called a nonparametric test
because the hypotheses do not refer to a population
parameter.
 The value of U is looked up in a table to determine
its significance, or the computer calculates the
significance.
Example:
These data are about the manual
dexterity of threeyearold children. A sample
of 13 children was obtained, 4 boys and 8 girls.
They were asked to place a set of blocks into
a specified pattern. The time (in seconds) required
by each child to arrange the blocks was recorded.
These data are from Gravetter and Walnau,
(Ed. 4), p 607, who use them to demonstrate
MannWhitney Utest.
These data were analyzed with the Univariate
Analysis module of ViSta. The report is shown
below.
The MannWhitney suggests that the difference
between the two groups is borderline significant.



Wilcoxon SignedRanks Test
The Wilcoxon Signedranks test evaluates the difference
between two treatments, using data from a pairedsamples
(repeatedmeasures) design.
Conceptual Basis:
Since we have paired samples, we calculate the difference
scores.
 If the treatment had an effect, the scores in
one treatment would be consistently larger than
those in the other, producing difference scores
that are consistently positive or consistently
negative.
 If there is no effect, then we would expect
positive and negative differences to be intermixed
evenly.
 The Wilcoxon test uses the signs and ranks of
the differences to decide on the significance
of the differences.
Calculations
 Rank the absolute values of the difference
scores.
 Separate the ranks into those associated with
positive differences and those associated with
negative differences.
 Sum the ranks for the positive differences and
sum the ranks for the negative differences.
 The smaller sum is the Wilcoxon signed ranks
test statistic, identified as the Wilcoxon T (not
to be confused with the regular TTest T value).
 The hypothesis test for the Wilcoxon T is performed.
 Null Hypothesis:
 The two groups are identically distributed.
 Alternative Hypothesis:
 The two groups are not identically distributed.
 Note that when the treatment had such a
large effect that all scores in one group
are larger than those in the other, then T=0.
 This test is called a nonparametric
test because the hypotheses do not refer
to a population parameter.
 The value of T is looked up in a table to
determine its significance, or the computer
calculates the significance.
Example:
These data are about an intensive
campaign conducted by the Red Cross to increase
blood donations. The campaign concentrated on
10 businesses. In each company the goal was
to increase the percentage of employees who
participated. The data are the percentage participation.
These data are from Gravetter and Wallnau
(4th Ed.) p. 614, who use them to demonstrate
Wilcoxon Signed Rank test.
These data were analyzed with the Univariate
Analysis module of ViSta. The report is shown
below.
The Wilcoxon test suggests that the difference
between the two groups is significant.



