The standard deviation of a, a+d, a+2 d, ldot | vedclass

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(C) The given series is an arithmetic progression: $a, a+d, a+2d, \ldots, a+2nd$.The number of terms $m$ is given by $a+(m-1)d = a+2nd$,which implies

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$\sqrt{\frac{n(n+1)}{3}} d$

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(C) The given series is an arithmetic progression: $a, a+d, a+2d, \ldots, a+2nd$.The number of terms $m$ is given by $a+(m-1)d = a+2nd$,which implies $m-1 = 2n$,so $m = 2n+1$.The standard deviation $\sigma$ of an arithmetic progression with $m$ terms and common difference $d$ is given by the formula $\sigma = \sqrt{\frac{m^2-1}{12}} |d|$.Substituting $m = 2n+1$:$\sigma = \sqrt{\frac{(2n+1)^2-1}{12}} |d|$$\sigma = \sqrt{\frac{4n^2+4n+1-1}{12}} |d|$$\sigma = \sqrt{\frac{4n(n+1)}{12}} |d|$$\sigma = \sqrt{\frac{n(n+1)}{3}} |d|$Thus,the correct option is $C$.

\text{Parameter}	\text{Plant } $A$ \text{ and } $B$ \text{ Data}
\text{No. of workers}	$A: 500, B: 6000$
\text{Average monthly wages}	$A: Rs. 2500, B: Rs. 2500$
\text{Variance of distribution of wages}	$A: 81, B: 100$

Classes	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$
Frequency	$11$	$29$	$18$	$4$	$5$	$3$

The standard deviation of $a, a+d, a+2 d, \ldots, a+2 n d$ is

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