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Find the mean and variance of the frequency distribution given below:
$\begin{array}{|l|l|l|l|l|} \hline x & 1 \leq x<3 & 3 \leq x<5 & 5 \leq x<7 & 7 \leq x<10 \\ \hline f & 6 & 4 & 5 & 1 \\ \hline \end{array}$
Solution
$\begin{array}{|c|c|c|c|c|} \hline x & f _{ i } & x _{ i } & f x _{ i } & f x _{ i }^{ 2 } \\ \hline 1-3 & 6 & 2 & 12 & 24 \\ \hline 3-5 & 4 & 4 & 16 & 64 \\ \hline 5-7 & 5 & 6 & 30 & 180 \\ \hline 7-10 & 1 & 8.5 & 8.5 & 72.25 \\ \hline \text { Total } & n=16 & & \Sigma f_{i} x_{i}=66.5 & \Sigma f_{i}{ }_{i}^{2}=340.25 \\ \hline \end{array}$
$\therefore \quad$ Mean $=\frac{\Sigma f_{i} x_{i}}{\Sigma f_{i}}=\frac{66.5}{16}=4.13$
And variance $=\sigma^{2}=\frac{\Sigma f_{i} x_{i}^{2}}{\Sigma f_{i}}-\left(\frac{\Sigma f_{i} x_{i}}{\Sigma f_{i}}\right)^{2}=\frac{340.25}{16}-(4.13)^{2}$
$\quad=21.2656-17.0569=4.21$
