Calculation of Wilcoxon test statistic for QB choice problem
Below is a quick illustration of the calculation of the
Wilcoxon statistic for the QB choice problem.
Calculation
of Wilcoxon Test Statistic for Difference in
First
QB and Second QB Career Touchdowns


c1

c2

c3

c4

c5

c6

c7

First
Choice QB

Second Choice QB

D

Abs(D)

DUM=1 if D is positive

Rank ABS D

DUM*Rank

65

79

14

14

0

2

0

128

363

235

235

0

30

0

102

0

102

102

1

13

13

64

234

170

170

0

26

0

491

14

477

477

1

33

33

1

161

160

160

0

22

0

77

11

66

66

1

9

9

174

208

34

34

0

7

0

15

113

98

98

0

12

0

251

50

201

201

1

29

29

16

48

32

32

0

6

0

2

8

6

6

0

1

0

154

5

149

149

1

17

17

165

0

165

165

1

23

23

170

0

170

170

1

26

26

275

7

268

268

1

31

31

203

19

184

184

1

28

28

207

40

167

167

1

25

25

247

94

153

153

1

18

18

300

29

271

271

1

32

32

3

100

97

97

0

11

0

3

136

133

133

0

16

0

86

60

26

26

1

4

4

33

199

166

166

0

24

0

100

28

72

72

1

10

10

3

159

156

156

0

20

0

124

0

124

124

1

15

15

156

182

26

26

0

4

0

155

0

155

155

1

19

19

124

16

108

108

1

14

14

3

17

14

14

0

2

0

164

125

39

39

1

8

8

212

55

157

157

1

21

21

The Wilcoxon W is obtained from the SUMPRODUCT command for
c5 and c6 or just add up c7.
The expected value of W is
EW = n x (n+1) /4
Where n is the sample size minus ties.
The STDW is
STDW = ((n x (n+1)
x (2n+1))/24)^{0.5}
The pvalue for the Wilcoxon test was obtained from z
Z=(WEW)/STDW
The function NORM.DIST in excel can be used to calculate the
p value for this Z.
Footnote: The D_{i} used to calculate the
Wilcoxon are not normally distributed.
The upper tail is very large because of Manning and Elway two very
successful firstchoice picks and the lower tail is very large because of Simms
and Testaverde, two great secondchoice picks.
The tests may be more affected by the size of the tails than by the
location of the means or medians.
The hypothesis testing problem:
http://dailymathproblem.blogspot.com/2014/04/hypothesistestsforqbperformance.html
The hypothesis testing problem:
http://dailymathproblem.blogspot.com/2014/04/hypothesistestsforqbperformance.html
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