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1
Wilcoxon RankSum Test, also known as the MannWhitney test
•
Rank the data. That is replace, the data values by their ranks, from
smallest to largest. For example, the pH samples are:
Group 1:
8.53 8.52 8.01 7.99 7.93
7.89 7.85 7.82 7.80
Group 2:
7.85 7.73 7.58 7.40 7.35
7.30 7.27 7.27 7.23
are replaced by the ranks
Group 1: 18 17 16 15 14 13 11.5 10
9
Group 2: 11.5 8
7
6
5
4
3
2
1
The tied values (7.85, 7.85) would have had ranks 11 and 12 were they
slightly diﬀerent. In the case of ties, assign the “midrank” [here (11+12)/2]
to both values.
•
Calculate
W
, which is the sum of the ranks in the ﬁrst group. In this case,
W
= 123
.
5.
•
Let
n
1
be the number of observations in the ﬁrst group,
n
2
the number in
the second group, and
N
=
n
1
+
n
2
. Here
n
1
=
n
2
= 9 and
N
= 18.
•
Under the null hypothesis that the two disbributions are the same, the
mean and variance of
W
are
μ
W
=
n
1
(
N
+1)
/
2 and
σ
2
W
=
n
1
n
2
(
N
+1)
/
12,
•
the observed value of the test statistic is
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 Spring '12
 daniel

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