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