Question

Calculate the mean deviation about the mean of the set of first `n` natural numbers when `n` is odd natural number.

Answer 1

Mean of first `n` natural numbers when `n` is odd `= (n(n+1))/2`
`:.` Mean deviation `= (sum |x_i-barx|)/n`
`=1/n[(1-(n(n+1))/2)+(2-(n(n+1))/2)+...(n - (n(n+1))/2)]`
`=2/n[1+2+...(n-1)/2]`
This is a series of first `(n-1)/2` numbers,
`:. 2/n[1+2+...(n-1)/2] = 2/n((((n-1)/2)((n-1)/2+1))/2)`
`=2/n((n^2-1)/8)`
`=(n^2-1)/(4n)`,which is the required mean deviation.