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Question

Let r be the range and S2=1n1ni=1(xi¯x)2 be the S.D. of a set of observations x1,x2,.....xn, then

A
Srnn1
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B
S=rnn1
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C
Srnn1
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D
none of these
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Solution

The correct option is A Srnn1
Here range =r= largest value smallest value =Max(xixj)(ij) and
S=1n1ni=1(xi¯x)2
now (xi¯x)2=[xix1+x2+.....+xnn]2
=1n2[(xix1)+(xix2)+...+(xixn)]2 =1n2[(xix1)+(xix2)+...+(xixi1)+(xixi+1)+...+(xixn)]2
.(xi¯x)21n2[(n1)r]2(xixjr)
ni=1(xi¯x)2n11n2(n1)[(n1)r]2 (summing up and dividing (n-1) both sides)
1n21n1n(n1)2r2 =n1nr2 <nn1.r2 (n>1n>1n)
S2<nn1.r2orS<rnn1

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