Question

The marks obtained by $60$ students in a test are given as follows:

Marks	$5-15$	$15-25$	$25-35$	$35-45$	$45-55$	$55-65$
No. of students	$8$	$12$	$20$	$10$	$7$	$3$

Calculate the mean and standard deviation of the distribution. Also interpret the results.

Answer 1

C-I	F	x	Fx	Step deviation $d=\frac{x-A}{c}$ $A=30$	$d^2$	fd	$f{d}^2$
5-15	8	10	80	-2	4	-16	32
15-25	12	20	240	-1	1	-12	12
25-35	20	30	600	0	0	0	0
35-45	10	40	400	+1	1	+10	10
45-55	7	50	350	+2	4	+14	28
55-65	3	60	180	+3	9	+9	27
	n=60		$\Sigma =1850$			$\Sigma fd=+5$	$\Sigma f{d}^2=109$

(i) $Mean=\frac{\sum fx}{n}=\frac{1850}{60}=30.8$

The mid-point of class intervals have $C=10$ as a common factor and the mean is not an integer.
$\therefore$ Let us follow step deviation method.
(ii) $S.D=\sqrt{\frac{\sum f{d}^2}{n}-{\left(\frac{\sum fd}{n}\right)}^2}\times C=\sqrt{\frac{109}{60}-{\left(\frac{5}{60}\right)}^2}\times 10=\sqrt{1.81}\times 10=1.34\times 10,\sigma =13.4$.
This means, on an average each score deviates from the mean value $30.8$ by $13.4$.