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Question

From the data given below state which group is more variable A or B
Marks10-2020-3030-4040-5050-6060-7070-80
Group A917323340109
Group B1020302543157

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Solution

Class fi xiui=xiAh u2i fiuifiu2i
10-20 9 15 -3 9 -27 81
20-30 17 25 -2 4 -34 68
30-40 32 35 -1 1 -32 32
40-5033 45 0 0 0 0
50-60 4055 11 40 40
60-70 10 65 2 4 20 40
70-80 9 75 3 9 27 81
Total 150 -6 342
Here h=10,N=150
Let assumed mean A=45
Mean ¯x=A+7i=1fiuiN×h
=45+(6)×10150=450.4=44.6

Variance (σ21)=h2N2N7i=lfiu2i(7i=lfiui)2
=10022500[150×342(6)2]
=1225(51264)
=227.84
Standard deviation (σ1)=227.84=15.09

Group B:
Class fi xiui=xiAh u2i fiuifiu2i
10-20 10 15 -3 9 -30 90
20-30 20 25 -2 4 -40 80
30-40 30 35 -1 1 -30 30
40-5025 45 0 0 0 0
50-60 4355 11 43 43
60-70 15 65 2 4 30 60
70-80 7 75 3 9 2163
Total 150 -6 366
Here h=10,N=150
Let assumed mean A=45
Mean ¯x=A+7i=1fiuiN×h
=45+(6)×10150=450.4=44.6

Variance (σ22)=h2N2N7i=lfiu2i(7i=lfiui)2
=10022500[150×366(6)2]
=1225(54864)
=243.84
Standard deviation (σ2)=243.84=15.61

Since, the mean of both the groups is same the group with greater standard deviation will be more variable.
Thus group B has more variability in the marks.

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