The standard deviation of the data given byVariate (x) 0 1 2 3 … nFrequency (f) nC0nC1 nC2nC3 …nCn

(S.D.) 2=1N∑fx2−1N∑fx21N∑fx2=12n12⋅nC1+⋯n2nCn=n2n1+2(n−1)+32(n−1)(n−2)+⋯+n=n2n1+(n−1)+(n−1)(n−2)21+⋯+1+{(n−1)+(n−1)(n−2)+⋯+(n−1)}]=n2n n−1C0+n−1C1⋯+n−1Cn−1+(n−1)=n(n+1)4 n−2C0+⋯+n−2Cn−2σ2=n(n+1)4−n24=n4.

The standard deviation of the data given byVariate (x) 0 1 2 3 … nFrequency (f) nC0nC1 nC2nC3 …nCn

(S.D.) 2=1N∑fx2−1N∑fx21N∑fx2=12n12⋅nC1+⋯n2nCn=n2n1+2(n−1)+32(n−1)(n−2)+⋯+n=n2n1+(n−1)+(n−1)(n−2)21+⋯+1+{(n−1)+(n−1)(n−2)+⋯+(n−1)}]=n2n n−1C0+n−1C1⋯+n−1Cn−1+(n−1)=n(n+1)4 n−2C0+⋯+n−2Cn−2σ2=n(n+1)4−n24=n4.

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The standard deviation of the data given by
Variate $\begin{array}{lllllll} (x) 0 1 2 3 \dots n \end{array}$
Frequency $(f)$ $^{n} {C_{0}}^{n} C_{1}^{n} {C_{2}}^{n} C_{3} \dots^{n} C_{n}$

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$2^{n} / n$

$\sqrt{(n + 1) / 2}$

none of these

$\sqrt{n / 2}$

answer is D.

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Detailed Solution

${(S.D.)}^{2} = \frac{1}{N} \sum f x^{2} - {(\frac{1}{N} \sum f x)}^{2}$

$\begin{array}{l} \frac{1}{N} \sum f x^{2} = \frac{1}{2^{n}} [1^{2} \cdot^{n} C_{1} + \dots {n^{2}}^{n} C_{n}] \\ = \frac{n}{2^{n}} [1 + 2 (n - 1) + \frac{3}{2} (n - 1) (n - 2) + \dots + n] \\ = \frac{n}{2^{n}} [\{1 + (n - 1) + \frac{(n - 1) (n - 2)}{21} + \dots + 1\} \\ + {(n - 1) + (n - 1) (n - 2) + \dots + (n - 1)}] \\ = \frac{n}{2^{n}} [^{n - 1} C_{0} +^{n - 1} C_{1} \dots +^{n - 1} C_{n - 1} + (n - 1) \\ = \frac{n (n + 1)}{4} [^{n - 2} C_{0} + \dots +^{n - 2} C_{n - 2}]] \\ σ^{2} = \frac{n (n + 1)}{4} - \frac{n^{2}}{4} = \frac{n}{4} . \end{array}$