Find the standard deviation for the following data:
${x_i}$
$3$
$8$
$13$
$18$
$25$
${f_i}$
$7$
$10$
$15$
$10$
$6$
Find the standard deviation for the following data:
| ${x_i}$ | $3$ | $8$ | $13$ | $18$ | $25$ |
| ${f_i}$ | $7$ | $10$ | $15$ | $10$ | $6$ |
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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