Calculate mean, variance and standard deviation for following distribution: Class 30-40 40-50 50-60

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13.Statistics

hard

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

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

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

A

$14.18$

B

$14.18$

C

$14.18$

D

$14.18$

Solution

From the given data, we construct the following Table

Class	Freq $\left( {{f_i}} \right)$	Mid-point $\left( {{x_i}} \right)$	${f_i}{x_i}$	${\left( {{x_i} – \bar x} \right)^2}$	${f_i}{\left( {{x_i} – \bar x} \right)^2}$
$30-40$	$3$	$35$	$105$	$729$	$2187$
$40-50$	$7$	$45$	$315$	$289$	$2023$
$50-60$	$12$	$55$	$660$	$49$	$588$
$60-70$	$15$	$6$	$975$	$9$	$135$
$70-80$	$8$	$75$	$600$	$169$	$1352$
$80-90$	$3$	$85$	$255$	$529$	$1587$
$90-100$	$2$	$95$	$190$	$1089$	$2178$
	$50$		$3100$		$10050$

Thus Mean $\bar x = \frac{1}{N}\sum\limits_{i = 1}^7 {{f_i}{x_i}} = \frac{{3100}}{{50}} = 62$

Variance $\left( {{\sigma ^2}} \right) = \frac{1}{N}\sum\limits_{i = 1}^7 {{f_i}{{\left( {{x_i} – \bar x} \right)}^2}} $

$ = \frac{1}{{50}} \times 10050 = 201$

and Standerd deviation $\left( \sigma \right) = \sqrt {201} = 14.18$

Standard 11

Mathematics

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