Find the variance and standard deviation of the following frequency distribution:
$x_{i}$ $4$ $8$ $11$ $17$ $20$ $24$ $32$
$f_{i}$ $3$ $5$ $9$ $5$ $4$ $3$ $1$

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Solution

Mean $= ¯ x = \frac{\sum f_{i} x_{i}}{\sum f_{i}}$
$= \frac{4 \times 3 + 8 \times 5 + 11 \times 9 + 17 \times 5 + 20 \times 4 + 24 \times 3 + 32 \times 1}{3 + 5 + 9 + 5 + 4 + 3 + 1}$
$= \frac{12 + 40 + 99 + 85 + 80 + 72 + 32}{30}$
$= \frac{420}{30} = 14$
Variance $= \frac{\sum f_{i} (x_{i} - ¯ x)^{2}}{\sum f_{i}}$
$= \frac{3 (4 - 14)^{2} + 5 (8 - 14)^{2} + 9 (11 - 14)^{2} + 5 (17 - 14)^{2} + 4 (20 - 14)^{2} + 3 (24 - 14)^{2} + 1 (32 - 14)^{2}}{30}$
$= \frac{300 + 180 + 81 + 45 + 144 + 300 + 324}{30}$
$= \frac{1374}{30}$
$= 45.8$
Standard deviation equals $\sqrt{45.8} = 6.767$

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Class:	1–10	10–20	20–30	30–40	40–50	50–60
Frequency:	11	29	18	4	5	3

$x_{i}$	$3$	$5$	$7$	$9$	$11$	$13$
$f_{i}$	$6$	$8$	$15$	$3$	$8$	$4$

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$x_{i}$	$1$	$2$	$3$	$4$	$5$
$f_{i}$	$5$	$4$	$f$	$2$	$3$

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Class	0-4	4-8	8-12	12-16	16-20	20-24	24-28	28-32	32-36
Frequency	6	8	17	23	16	15	-	4	3

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$x_{i}$	$4$	$8$	$11$	$17$	$20$	$24$	$32$
$f_{i}$	$3$	$5$	$9$	$5$	$4$	$3$	$1$