Page 152 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 7: Measures of Dispersion

X A Notes
Let u i i , X = A + hu .... (2)
h i i
Also f X = f (A + hu ) = AN + h f u
i i i i i i
Dividing both sides by N, we have
f X i f u
i i
i
N A h N or X A hu .... (3)
Substituting the values of X and X in equation (1), we have
i
2 1 2 2 1 2 2 2
i
X f A hu i A hu h f u i u h u .... (4)
i
N N
The result shows that variance is independent of change of origin but not of change of scale.
Using this we can write down a short-cut formula for variance of X.
2 2
f u f u
2 2 i i i i
X h .... (5)
N N
Further, when only change of origin is made
f d 2 f d 2
2 i i i i
X , where d = X – A
i
i
N N
Example: Calculate standard deviation of the following series:
Weekly wages No. of workers Weekly wages No. of workers
100 -105 200 130 -135 410
105 -110 210 135 -140 320
110 -115 230 140 -145 280
115 -120 320 145 -150 210
120 -125 350 150 -155 160
125 -130 520 155 -160 90
Solution:
Calculation of S.D. by using d (= X – A)
i i
Class Intervals No. of Workers ( f ) Mid values (X) d=X - 127.5 fd fd 2
100-105 200 102.5 -25 -5000 125000
105-110 210 107.5 -20 -4200 84000
110-115 230 112.5 -15 -3450 51750
115-120 320 117.5 -10 -3200 32000
120-125 350 122.5 -5 -1750 8750
125-130 520 127.5 0 0 0
130-135 410 132.5 5 2050 10250
135-140 320 137.5 10 3200 32000
140-145 280 142.5 15 4200 63000
145-150 210 147.5 20 4200 84000
150-155 160 152.5 25 4000 100000
155-160 90 157.5 30 2700 81000
Total 3300 2750 671750

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