First slide

Introduction to statistics

Question

The variance of the first n natural numbers is

Easy

A

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

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

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

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

Solution

$\begin{array}{l} Variance = (SD)^{2} = \frac{1}{n} Σ x^{2} - {(\frac{Σ x}{n})}^{2}, (∵ \bar{x} = \frac{Σ x}{n}) \\ = \frac{n (n + 1) (2 n + 1)}{6 n} - {(\frac{n (n + 1)}{2 n})}^{2} = \frac{n^{2} - 1}{12} \end{array}$

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