First slide

Introduction to statistics

Question

The variance of first n natural numbers, is

Moderate

A

$\frac{n + 1}{2}$
B

$\frac{n^{2} + 1}{12}$
C

$\frac{n^{2} - 1}{6}$
D

$\frac{n^{2} - 1}{12}$

Solution

Here $x_{i} = i; i = 1, 2, \dots, n$
$\begin{array}{l} ∴ σ^{2} = \frac{1}{n} (\sum_{i = 1}^{n} x_{i}^{2}) - {(\frac{1}{n} \sum_{i = 1}^{n} x_{i})}^{2} \\ \Rightarrow σ_{2} = \frac{1}{n} (1^{2} + 2^{2} + \dots + n^{2}) - {(\frac{n + 1}{2})}^{2} \\ \Rightarrow σ^{2} = \frac{n (n + 1) (2 n + 1)}{6 n} - {(\frac{n + 1}{2})}^{2} \\ \Rightarrow σ^{2} = \frac{(n + 1) (2 n + 1)}{6} - \frac{(n + 1)^{2}}{4} = \frac{n^{2} - 1}{12} \end{array}$

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