Determine mean and standard deviation of first n terms of an $A.P.$ whose first term is a and common difference is d.
Determine mean and standard deviation of first n terms of an $A.P.$ whose first term is a and common difference is d.
$\begin{array}{|c|c|c|} \hline x_{i} & x_{i}-a & \left(x_{i}-a\right)^{2} \\ \hline a & 0 & 0 \\ \hline a+d & d & d^{2} \\ \hline a+2 d & 2 d & 4 d^{2} \\ \hline \end{array}$
$\begin{array}{|c|c|c|} \hline \ldots & \ldots & 9 d^{2} \\ \hline \ldots & \ldots & \ldots \\ \hline \ldots & \ldots & \ldots \\ \hline a+(n-1) d & (n-1) d & (n-1)^{2} d^{2} \\ \hline \Sigma x_{i}=\frac{n}{2}[2 a+(n-1) d ] & & \\ \hline \end{array}$
$\text { Mean }=\frac{\Sigma x_{i}}{n}=\frac{1}{n}\left[\frac{n}{2}(2 a+(n-1) d]=a+\frac{(n-1)}{2} d\right.$
$\Sigma\left(x_{i}-a\right)=d[1+2+3+\ldots+(n-1) d]=d \frac{(n-1) n}{2}$
and $\quad \Sigma\left(x_{i}-a\right)^{2}=d^{2} \cdot\left[1^{2}+2^{2}+3^{2}+\ldots+(n-1)^{2}\right]=\frac{d^{2} n(n-1)(2 n-1)}{6}$
$\sigma=\sqrt{\frac{\left(x_{i}-a\right)^{2}}{n}-\left(\frac{x_{i}-a}{n}\right)^{2}}$
$=\sqrt{\frac{d^{2} n(n-1)(2 n-1)}{6 n}-\left[\frac{d(n-1) n}{2 n}\right]^{2}}=\sqrt{\frac{d^{2}(n-1)(2 n-1)}{6}-\frac{d^{2}(n-1)^{2}}{4}}$
$=d \sqrt{\frac{(n-1)}{2}\left(\frac{2 n-1}{3}-\frac{n-1}{2}\right)=d \sqrt{\frac{(n-1)}{2}\left[\frac{4 n-2-3 n+3}{6}\right]}}$
${2 \sqrt{\frac{(n-1)(n+1)}{12}}=d \sqrt{\frac{n^{2}-1}{12}}}$
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