Question

Find the mean, standard deviation and variance of first 10 multiples of 3.

Answer 1

First 10 multiples of 3 are 3, 6, 9, 12, . . . 30.

This is an AP with a = 3, n = 10 and l = 30.

$\therefore$ sum $\frac{n}{2}(a+1)=\frac{10}{2}.(3+30)=165$

$\therefore$ mean, $\overline{x}=\frac{165}{10}=16.5$

Variance, ${\sigma }^2=\frac{\sum x_i^2}{n}-(\overline{x}{)}^2$

$=\{\frac{3^2+6^2+9^2+\cdots +{30}^2}{10}-(16.5{)}^2\}$

$=\frac{3^2\times \{1^2+2^2+3^2+\cdots +{10}^2\}}{10}-(16.5{)}^2$

$=\frac{9\times 10\times (10+1)(2\times 10+1)}{6\times 10}-(16.5{)}^2$

$[\because \sum n^2=\frac{n(n+1)(2n+1)}{6}]$

$=(\frac{9\times 10\times 11\times 21}{6\times 10}-272.25)=(\frac{693}{2}-272.25)$

=$(346.5-272.325)=74.25$

$\therefore$ variance, ${\sigma }^2=74.25$

Standard deviation, $\sigma =\sqrt{74.25}=8.61$