The raw data x_1, x_2, ldots, x_{n} is an A.P | vedclass

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(D) The terms are $x_i = (i-1)d$ for $i=1, 2, \ldots, n$.$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} (i-1)d = \frac{d}{n} \frac{(n-1)n}{2} = \frac{(n-1)d}{2

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$\frac{(n^2-1) d^2}{12}$

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(D) The terms are $x_i = (i-1)d$ for $i=1, 2, \ldots, n$.$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} (i-1)d = \frac{d}{n} \frac{(n-1)n}{2} = \frac{(n-1)d}{2}$.$\sum x_i^2 = \sum_{i=1}^{n} (i-1)^2 d^2 = d^2 \sum_{k=0}^{n-1} k^2 = d^2 \frac{(n-1)n(2n-1)}{6}$.$\sigma^2 = \frac{1}{n} \sum x_i^2 - (\bar{x})^2 = \frac{d^2(n-1)(2n-1)}{6} - \frac{(n-1)^2 d^2}{4}$.$\sigma^2 = \frac{d^2(n-1)}{2} \left[ \frac{2n-1}{3} - \frac{n-1}{2} \right] = \frac{d^2(n-1)}{2} \left[ \frac{4n-2-3n+3}{6} \right] = \frac{d^2(n-1)(n+1)}{12} = \frac{(n^2-1)d^2}{12}$.

The raw data $x_1, x_2, \ldots, x_{n}$ is an $A$.$P$. with common difference $d$ and first term $0$. If $\bar{x}$ and $\sigma^2$ are the mean and variance of $x_{i}, i=1, 2, \ldots, n$,then $\sigma^2$ is:

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