Let r be the range and S2=1n−1∑ni=1(xi−¯x)2 be the S.D. of a set of observations x1,x2,.....xn, then
A
S≤r√nn−1
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B
S=r√nn−1
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C
S≥r√nn−1
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D
none of these
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Solution
The correct option is AS≤r√nn−1 Here range =r= largest value − smallest value =Max∣∣(xi−xj)∣∣(i≠j) and S=1n−1n∑i=1(xi−¯x)2 now (xi−¯x)2=[xi−x1+x2+.....+xnn]2 =1n2[(xi−x1)+(xi−x2)+...+(xi−xn)]2=1n2[(xi−x1)+(xi−x2)+...+(xi−xi−1)+(xi−xi+1)+...+(xi−xn)]2 ⇒.(xi−¯x)2≤1n2[(n−1)r]2(∵∣∣xixj∣∣≤r) ⇒∑ni=1(xi−¯x)2n−1≤1n2(n−1)∑[(n−1)r]2 (summing up and dividing (n-1) both sides) ⇒1n21n−1n(n−1)2r2=n−1nr2<nn−1.r2(∵∀n>1n>1n) ∴S2<nn−1.r2orS<r√nn−1