Question

The Mean of n observation x₁, x₂,..., x_n is `bar"X"`. If (a - b) is added to each of the observation, show that the mean of the new set of observation is `bar"X"` + (a - b).

Answer 1

We have
`bar"X" = (x_1 + x_2 + ... + x_"n")/"n"`      ...(i)
Let `bar"X"` be the mean of x₁ + (a - b), x₂ + (a - b),...,x_n + (a - b). Then
`bar"X" = ([x_1 + (a - b)] + [x_2 + (a - b)] + ... + [x_2 + (a - b)])/"n"`
= `(x_1 + x_2 + ... + x_"n" + "n"(a - b))/"n"`
= `(x_1 + x_2 + ... + x_"n")/"n" + ("n"(a - b))/"n"`
= `bar"X" + (a - b)`.    ...[Using (i)]
Hence proved.