Calculate the standard deviation for the given frequency distribution :
$C . I .$ $f$
$1 - 5$
$6 - 10$
$11 - 15$
$16 - 20$ $1$
$2$
$3$
$4$
$N = 10$

Open in App

Solution

Given $N = 10$

Class Frequency $(f)$ Midpoint $(M)$ $M^{2}$ $f \times M^{2}$
$1 - 5$ $1$ $3$ $9$ $9$
$6 - 10$ $2$ $8$ $64$ $128$
$11 - 15$ $3$ $13$ $169$ $507$
$16 - 20$ $4$ $18$ $324$ $1296$
Total $= 1940$
$\sum f \times M = 3 + 16 + 39 + 72 = 130$
Mean $(μ) = \frac{\sum f \times M}{N} = \frac{130}{10} = 13$

$S^{2} = \frac{(\sum f \times M^{2}) - N (μ)^{2}}{N - 1}$
$= \frac{1940 - 10 \times 169}{9}$
$= \frac{1940 - 1690}{9}$
$= \frac{250}{9}$
$= 27.7777$
$∴$ $S^{2} = 27.7777$
$∴$ Standard Deviation $(S) = 5.270$ .

Suggest Corrections

X	$10$	$11$	$12$	$13$	$14$	$15$	$16$	$17$	$18$
f	$2$	$7$	$10$	$12$	$15$	$11$	$10$	$6$	$3$

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

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

$C . I .$	$f$
$1 - 5$ $6 - 10$ $11 - 15$ $16 - 20$	$1$ $2$ $3$ $4$