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

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

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

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