Find the mean and variance for the data
${x_i}$
$6$
$10$
$14$
$18$
$24$
$28$
$30$
${f_i}$
$2$
$4$
$7$
$12$
$8$
$4$
$3$
Find the mean and variance for the data
| ${x_i}$ | $6$ | $10$ | $14$ | $18$ | $24$ | $28$ | $30$ |
| ${f_i}$ | $2$ | $4$ | $7$ | $12$ | $8$ | $4$ | $3$ |
| ${x_i}$ | ${f_i}$ | ${f_i}{x_i}$ | ${{x_i} - \bar x}$ | ${\left( {{x_i} - \bar x} \right)^2}$ | ${f_i}{\left( {{x_i} - \bar x} \right)^2}$ |
| $6$ | $2$ | $12$ | $-13$ | $169$ | $338$ |
| $10$ | $4$ | $40$ | $-9$ | $81$ | $324$ |
| $14$ | $7$ | $98$ | $-5$ | $25$ | $175$ |
| $18$ | $12$ | $216$ | $-1$ | $1$ | $12$ |
| $24$ | $8$ | $192$ | $5$ | $25$ | $200$ |
| $28$ | $4$ | $112$ | $9$ | $81$ | $324$ |
| $30$ | $3$ | $90$ | $11$ | $121$ | $363$ |
| $40$ | $760$ | $1736$ |
Here, $N = 40,\sum\limits_{i = 1}^7 {{f_1}{x_1}} = 760$
$\therefore \bar x = \frac{{\sum\limits_{i = 1}^7 {{f_1}{x_1}} }}{N} = \frac{{760}}{{40}} = 19$
Variance $ = \left( {{\sigma ^2}} \right) = \frac{1}{N}\sum\limits_{i = 1}^7 {{f_1}{{\left( {{x_1} - \bar x} \right)}^2} = } \frac{1}{{40}} \times 1736 = 43.4$
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