Find the standard deviation for the following data:

${x_i}$ $3$ $8$ $13$ $18$ $25$
${f_i}$ $7$ $10$ $15$ $10$ $6$

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Let us form the following Table :

${x_i}$ ${f_i}$ ${f_i}{x_i}$ ${x_i}^2$ ${f_i}{x_i}^2$
$3$ $7$ $21$ $9$ $63$
$8$ $10$ $80$ $64$ $640$
$13$ $15$ $195$ $169$ $2535$
$18$ $10$ $180$ $324$ $3240$
$23$ $6$ $138$ $529$ $3174$
  $48$ $614$   $9652$

Now, by formula $(3),$ we have

$\sigma  = \frac{1}{N}\sqrt {N\sum {{f_i}x_i^2 - {{\left( {\sum {{f_i}{x_i}} } \right)}^2}} } $

$=\frac{1}{48} \sqrt{48 \times 9652-(614)^{2}}$

$=\frac{1}{48} \sqrt{463296-376996}$

$=\frac{1}{48} \times 293.77=6.12$

Therefore, Standard deviation $(c)=6.12$

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