Question

Find S.D. of the following frequency distribution
$\begin{tabular}{|l|c|c|c|c|c|} x_{i} & 1 & 2 & 3 & 4 & 5 \\ f_{i} & a & 2a & 3a & 4a & 5a \end{tabular}$

Answer 1

Solution:

$\begin{table}[] \begin{tabular}{|l|l|l|l|l|} \cline{1-5} x_{i} & \begin{tabular}[c]{@{}l@{}}Frequency\\ f_{i}\end{tabular} & (x_{i} f_{i}) & (x_{i})^{2} & f_{i}(x_{i}^{2}) \\ \cline{1-5} 1 & a & a & 1 & a \\ \cline{1-5} 2 & 2a & 4a & 4 & 8a \\ \cline{1-5} 3 & 3a & 9a & 9 & 27a \\\cline{1-5} 4 & 4a & 16a & 16 & 64a \\ \cline{1-5} 5 & 5a & 25a & 25 & 125a \\ \cline{1-5} Total & 15a & 55a & & 225a \\ \cline{1-5} \end{tabular} \end{table}$

Since N = 15 a

$\Sigma f_{i}x_{i} = 55a \\ \\\Sigma f_{i}(x_{i})^{2}= 225a$

Standard Deviation

$\sigma =\frac{1}{15a} \sqrt{N\Sigma f_{i}(x_{i})^{2} - (\Sigma f_{i}x_{i}})^{2} } \\ \\ \sigma=\frac{1}{15a} \sqrt{15a \times 225a - ( {55a)}^{2} } \\ \\ = \frac{1}{15} \sqrt{3375 - 3025} \\ \\ = \frac{1}{50} \sqrt{350} \\ \\ = \frac{18.70}{15} \\ \\SD = 1.24$
Hope it helps you