Write the variance of first n natural numbers. - Vidyadip

Question

Write the variance of first n natural numbers.

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Answer

Variance $=\sum\frac{\text{x}^2}{\text{n}}-\Big(\sum\frac{\text{x}}{\text{n}}\Big)^2$ But $\sum\text{x}=1+2+3+....+\text{n}=\text{n}\Big(\text{n}+\frac{1}{2}\Big)$ $\sum\text{x}^2=\frac{\text{n}(\text{n}+1)(2\text{n}+1)}{6}$ Substituting these value we get Variance = $=\frac{(\text{n}^2-1)}{12}$

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