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

Class $30-40$ $40-50$ $50-60$ $60-70$ $70-80$ $80-90$ $90-100$
$f_i$ $3$ $7$ $12$ $15$ $8$ $3$ $2$

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

Similar Questions

Let ${x_1}\;,\;{x_2}\;,\;.\;.\;.\;,{x_n}$ be $n$ observations, and let $\bar x$ be their arithmaetic mean and ${\sigma ^2}$ be the variance

Statement $-1$ :Variance of $2{x_1}\;,2\;{x_2}\;,\;.\;.\;.\;,2{x_n}$ is $4{\sigma ^2}$ .

Statement $-2$: Arithmetic mean $2{x_1}\;,2\;{x_2}\;,\;.\;.\;.\;,2{x_n}$ is $4\bar x$.

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