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where
n
1
and
n
2
are the sizes of samples 1 and 2, respectively, and
R
1
and
R
2
are the sums of the ranks of samples 1 and 2, respectively. h e required test
statistic
U
is the smaller of the two variables
U
1
and
U
2
, which we compare
with a critical
U
value that depends on the sample sizes
n
1
and
n
2
, and on the
signii cance level ʱ. Alternatively, we can use the
U
value to calculate
if
n
1
≥8 and
n
2
≥8 (Mann and Whitney 1978, Hedderich and Sachs 2012, page
486). h e null hypothesis can be rejected if the absolute measured
z
-value is
higher than the absolute critical
z
-value, which depends on the signii cance
level ʱ (Section 3.4).
In practice, data sets ot en contain tied values, i.e., some of the values in
the sample 1 and/or sample 2 are identical. In this case, the average ranks
of the tied values are used instead of the true ranks. h is means that the
equation for the
z
-value must be corrected for tied values
where
S
=
n
1
+
n
2
,
r
is the number of tied values and
t
i
is the number of
occurrences of the
i
-th tied value. Again, the null hypothesis can be rejected
if the absolute measured
z
-value is higher than the absolute critical
z
-value,
which depends on the signii cance level ʱ.
h e MATLAB code presented here has been tested with an example
contained in the topic by Hedderich and Sachs (2012, page 489). h eexample
uses the Mann-Whitney test to test whether two samples (
data1
and
data2
),
each consisting of eight measurements with some tied values, come from
the same population (
null hypothesis
) or from two dif erent populations
(
alternative hypothesis
). We clear the workspace and dei ne two samples,
