Calculate mean, variance and standard deviation for following distribution. Classes 30-40 40-50 50-6

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13.Statistics

hard

Calculate mean, variance and standard deviation for the following distribution.

Classes $30-40$ $40-50$ $50-60$ $60-70$ $70-80$ $80-90$ $90-100$

${f_i}$ $3$ $7$ $12$ $15$ $8$ $3$ $2$

A

$14.18$

B

$14.18$

C

$14.18$

D

$14.18$

Solution

Let the assumed mean $A =65 .$ Here $h=10$

We obtain the following Table from the given data :

Class	Frequency ${f_i}$	Mid-point ${x_i}$	${y_i} = \frac{{{x_i} – 65}}{{10}}$	${y_i}^2$	${f_i}{y_i}$	${f_i}{y_i}^2$
$30-40$	$3$	$35$	$-3$	$9$	$-9$	$27$
$40-50$	$7$	$45$	$-2$	$4$	$-14$	$28$
$50-60$	$12$	$55$	$-1$	$1$	$-12$	$12$
$60-70$	$15$	$65$	$0$	$0$	$0$	$0$
$70-80$	$8$	$75$	$1$	$1$	$8$	$8$
$80-90$	$3$	$85$	$2$	$4$	$6$	$12$
$90-100$	$2$	$95$	$3$	$9$	$6$	$18$
	$N=50$				$-15$	$105$

Therefore $\bar x = A + \frac{{\sum {{f_i}{y_i}} }}{{50}} \times h = 65 – \frac{{15}}{{50}} \times 10 = 62$

Variance ${\sigma ^2} = \frac{{{h^2}}}{{{N^2}}}\left[ {N{{\sum {{f_i}{y_i}} }^2} – {{\left( {\sum {{f_i}{y_i}} } \right)}^2}} \right]$

$=\frac{(10)^{2}}{(50)^{2}}\left[50 \times 105-(-15)^{2}\right]$

$=\frac{1}{25}[5250-225]=201$

and standard deviation $(\sigma)=\sqrt{201}=14.18$

Standard 11

Mathematics

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