Question

2.First n natural numbers

Answer 1

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 .