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# 2.First n natural numbers

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Solution

## It is given that the data contains first n natural numbers. The formula to calculate the mean is the ratio of sum of observations to the number of observations. x ¯ = 1 n ∑ i=1 n x i (1) The formula to calculate the sum of n natural numbers is, ∑ n = n( n+1 ) 2 Since the total number of observation is n, thus substitute n( n+1 ) 2 for ∑ i=1 n x i in equation (1). x ¯ = 1 n ( n( n+1 ) 2 ) = ( n+1 ) 2 Therefore, the mean of the given data is ( n+1 ) 2 . The formula to calculate the variance is, σ 2 = 1 n ∑ i=1 n ( x i − x ¯ ) 2 (2) Substitute ( n+1 ) 2 for x ¯ in equation (2). σ 2 = 1 n ∑ i=1 n ( x i − ( n+1 ) 2 ) 2 Simplify the brackets by using square formula and apply summation on each part separately, σ 2 = 1 n ∑ i=1 n ( x i − ( n+1 ) 2 ) 2 = 1 n ∑ i=1 n ( x i ) 2 + 1 n ∑ i=1 n ( n+1 2 ) 2 − 1 n ∑ i=1 n 2 x i ( n+1 2 ) = 1 n ∑ i=1 n ( x i ) 2 + 1 n ∑ i=1 n ( n+1 ) 2 4 − 1 n [ ∑ i=1 n x i × ∑ i=1 n ( n+1 ) ] The formula to calculate the square of n natural numbers is, ∑ n 2 = n( n+1 )( 2n+1 ) 6 Substitute n( n+1 )( 2n+1 ) 6 for ∑ i=1 n ( x i ) 2 and n( n+1 ) 2 for ∑ i=1 n x i in the above equation. σ 2 = 1 n × n( n+1 )( 2n+1 ) 6 + 1 4n ∑ i=1 n ( n+1 ) 2 − 1 n [ n( n+1 ) 2 × ∑ i=1 n ( n+1 ) ] = ( n+1 )( 2n+1 ) 6 + 1 4n ∑ i=1 n ( n+1 ) 2 − ( n+1 ) 2 × 1 n ∑ i=1 n ( n+1 ) Since the sum is done up to n natural numbers, therefore the summation of any part is multiplied by n, σ 2 = ( n+1 )( 2n+1 ) 6 + 1 4n ×n ( n+1 ) 2 − ( n+1 ) 2 × 1 n ×n( n+1 ) = ( n+1 )( 2n+1 ) 6 + ( n+1 ) 2 4 − ( n+1 ) 2 2 = ( n+1 )( 2n+1 ) 6 − ( n+1 ) 2 4 Take ( n+1 ) common and simplify the above equation by taking L.C.M. σ 2 =( n+1 )[ 2( 2n+1 )−3( n+1 ) 12 ] =( n+1 )[ 4n+2−3n−3 12 ] =( n+1 )[ n−1 12 ] = n 2 −1 12 Thus, the variance of the given data is n 2 −1 12 .  Suggest Corrections  1      Similar questions  Explore more