$x_{i}$ $10$ $15$ $18$ $20$ $25$
$f_{i}$ $3$ $2$ $5$ $8$ $2$
Find the Variance of the given data.

14

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15

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16

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17

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Solution

The correct option is D $17$
$f_{i}$ $x_{i}$ $x_{i} - E (x)$ $(x_{i} - E (x))^{2}$ $f_{i} (x_{i} E (x))^{2}$
$3$ $10$ $- 8$ $64$ $192$
$2$ $15$ $- 3$ $9$ $18$
$5$ $18$ $0$ $0$ $0$
$8$ $20$ $2$ $4$ $32$
$2$ $25$ $7$ $49$ $98$
Mean, $E (x) = \frac{f_{1} x_{1} + f_{2} x_{2} + f_{3} x_{3} + . . . . . . . + f_{n} x_{n}}{f_{1} + f_{2} + f_{3} + . . . . . . + f_{n}}$

$E (x) = \frac{3 \times 10 + 2 \times 15 + 5 \times 18 + 8 \times 20 + 2 \times 25}{3 + 2 + 5 + 8 + 2} = 18$

Variance = $\frac{f_{1} (x_{1} - E (x))^{2} + f_{2} (x_{2} - E (x))^{2} + f_{3} (x_{3} - E (x))^{2} + . . . . . . + f_{n} (x_{n} - E (x))^{2}}{f_{1} + f_{2} + f_{3} + . . . . . + f_{n}}$

Varience = $\frac{3 \times 192 + 2 \times 18 + 5 \times 0 + 8 \times 32 + 2 \times 98}{3 + 2 + 5 + 8 + 2}$

Variance = $340 / 20 = 17$

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