1.

Find the standard deviation of first n natural numbers.

Answer»


Solution :Now`Sigmax_(i)=1+2+3+4+.....+n=(n(n+1))/2`
`Sigmax_(i)^(2)=1^(2)+2^(2)+3^(3)+…..+n^(2)=(n(n+1)(2N+1))/(6)`
`sigma=sqrt((Sigmax_(i)^(2))/N-(Sigma_(x_(i))/N)^(2)`
`sqrt((n(n+1)(2n+1))/(6N)-(n^(2)(n+1)^(2))/(4N^(2))`
`sqrt(((n+1)(2n+1))/(6)-((n+1)^(2))/(4)`
`sqrt((2(2n^(2)+3n+1)-3(n^(2)+2n+1))/(12)`
`sqrt((4n^(2)+6n+2-3n^(2)-6n-3)/(12)`
`sqrt((n^(2)-1)/12)`


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