Question

Formula for standard deviation of first n natural numbers

Answer 1

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$\underline{Formula }$

CBSE XI >> Math >> Statistics

Next similar questionQ)

Find the standard deviation of the first n natural numbers.

(A)n2−112−−−−−√(B)(n+1)(C)0(D)n−112

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A)

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The formula to calculate SD is SD=∑x2n+(∑xn)2−−−−−−−−−−−−−−√

The first n natural numbers are 1,2,3........n

As we know sum of n natural numbers are ∑x=n(n+1)2

∑x2=∑(12+22+32+.........+n2)

Sum of square numbers ∑x2=n(n+1)(2n+1)6

SD=∑x2n−(∑xn)2−−−−−−−−−−−−−−√

=n(n+1)(2n+1)6n−(n(n+1)2n)2−−−−−−−−−−−−−−−−−−−−−−−√

=(n+1)(2n+1)6−((n+1)2)2−−−−−−−−−−−−−−−−−−−−−√

=(n+1)[2n+16−(n+1)4]−−−−−−−−−−−−−−−−−−−−√

=(n+1)(n−1)12−−−−−−−−−√

=n2−112−−−−−√

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