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 $\left( \sigma \right) = \sqrt {33} = 5.74$
If the variance of the frequency distribution is $160$ , then the value of $\mathrm{c} \in \mathrm{N}$ is
$X$ | $c$ | $2c$ | $3c$ | $4c$ | $5c$ | $6c$ |
$f$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ |
The mean of the numbers $a, b, 8,5,10$ is $6$ and their variance is $6.8$. If $M$ is the mean deviation of the numbers about the mean, then $25\; M$ is equal to
Find the mean and variance for the data
${x_i}$ | $6$ | $10$ | $14$ | $18$ | $24$ | $28$ | $30$ |
${f_i}$ | $2$ | $4$ | $7$ | $12$ | $8$ | $4$ | $3$ |
The mean and standard deviation of $100$ observations were calculated as $40$ and $5.1$ , respectively by a student who took by mistake $50$ instead of $40$ for one observation. What are the correct mean and standard deviation?