The variance of first n natural numbers isnone of these

The correct option is A We have σ^2 = 1/n∑_i^n = _1 x _i^2 [ 1/n∑ _i^n= _1 x_i ]^2 =1/n(1^2 + 2^2 + ......+ n^2) [ 1/n (1+2+....+n) ]^2 =1/nn(n+1)(2n+1)/6 ...

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Standard XII

Mathematics

Standard Deviation

The variance ...

Question

The variance of first n natural numbers is

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none of these

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Solution

The correct option is B
$W e h a v e σ^{2} = \frac{1}{n} \sum_{i}^{n} =_{1} x_{i}^{2} - {[\frac{1}{n} \sum_{i}^{n} =_{1} x_{i}]}_{1}^{2} = \frac{1}{n} (1^{2} + 2^{2} + . . . . . . + n^{2}) - {[\frac{1}{n} (1 + 2 + . . . . + n)]}^{2} = \frac{1}{n} \frac{n (n + 1) (2 n + 1)}{6} - {[\frac{n + 1}{2}]}^{2} = \frac{n^{2} - 1}{12}$

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The variance of first n natural numbers is

The correct option is B We have σ2=1n∑ni=1x2i−[1n∑ni=1xi]2=1n(12+22+......+n2)−[1n(1+2+....+n)]2=1nn(n+1)(2n+1)6−[n+12]2=n2−112