Question

what is the value of n for which the numbers 1,2.,3,.......,n have variance 2?​

Answer 1

Variance=

(Σ(observations^2)/n) - (Mean)^2

Mean of first n natural number

=( 1+2+3+……+n)/n

(since mean is the sum of observations upon number of observations)

Mean=n(n+1)/2n=(n+1)/2 , {1+2+3…..+n=n(n+1)/2}

Variance= (1^2 + 2^2 +……n^2)/n - ((n+1)/2)^2

=n(n+1)(2n+1)/6n - ((n+1)/2)^2

= (n^2 - 1)/2 {1^2 + 2^2 +……n^2

=n(n+1)(2n+1)/6 }

So Variance

= (n^2 - 1)/2