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Let r be the range and S^(2)=(1)/(n-1) sum_(i=1)^(n) (x_(i)-overline(x))^(2) be the SD of a set of observations x_(1),x_(2),.., x_(n), then |
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Answer» `S le r sqrt((N)/(n-1))` and `S^(2)=(1)/(n-1) underset(i=1)OVERSET(n)(sum)(x_(i)-overline(x))^(2)` Now, `(x_(i)-overline(x))^(2)=(x_(i)-(x_(1)+x_(2)+..+x_(n))/(n))^(2)` `=(1)/(n^(2))[(x_(i)-x_(1))+(x_(i)-x_(2))+..+(x_(i)-x_(i)-1)+(x_(i)-x_(1)+1)+..+(x_(i)-x_(n))] le (1)/(n^(2))[(n-1)r]^(2) "" [because]x_(i)-x_(j)|le r|` `(x_(i)-overline(x))^(2) le r^(2)` `implies underset(i=1)overset(n)(sum)(x_(i)overline(x))^(2) le NR^(2)` `implies (1)/(n-1) underset(i=1)overset(n)(sum)(x_(i)-overline(x))^(2) le(nr^(2))/((n-1))` `implies S^(2) le (nr^(2))/((n-1))` `implies S le r sqrt((n)/(n-1))` |
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