Find the variance of the following data: $6,8,10,12,14,16,18,20,22,24$

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From the given data we can form the following Table The mean is calculated by step-deviation method taking $14$ as assumed mean. The number of observations is $n=10$

${x_i}$ ${d_i} = \frac{{{x_i} - 14}}{2}$

Deviations orom mean 

$\left( {{x_i} - \bar x} \right)$

$\left( {{x_i} - \bar x} \right)$
$6$ $-4$ $-9$ $81$
$8$ $-3$ $-7$ $49$
$10$ $-2$ $-5$ $25$
$12$ $-1$ $-3$ $9$
$14$ $0$ $-1$ $1$
$16$ $1$ $1$ $1$
$18$ $2$ $3$ $9$
$20$ $3$ $5$ $25$
$22$ $4$ $7$ $49$
$24$ $5$ $9$ $81$
  $5$   $330$

Therefore    $Mean\,\,\bar x = $ assumed mean $ + \frac{{\sum\limits_{i = 1}^n {{d_i}} }}{n} \times h$

$ = 14 + \frac{5}{{10}} \times 2 = 15$

and    Veriance $\left( {{\sigma ^2}} \right) = \frac{1}{n}\sum\limits_{i = 1}^{10} {{{\left( {{x_i} - \bar x} \right)}^2} = \frac{1}{{10}} \times 330 = 33} $

Thus Standard deviation $\left( \sigma  \right) = \sqrt {33}  = 5.74$

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