Question

If `barx` represents the mean of n observations x₁, x₂, ..., x_n, then value of `sum_(i = 1)^n (x_i - barx)` is ______.

Options

• –1

• 0

• 1

• n – 1

Answer 1

If `barx` represents the mean of n observations x₁, x₂, ..., x_n, then value of `sum_(i = 1)^n (x_i - barx)` is 0.

Explanation:

The formula of the mean `(barx)` is:

`barx = (sum_(i = 1)^n x_i)/n`

`sum_(i = 1)^n x_i = nbarx`  ...(I)

Here, n is total number of observation.

The value of `sum_(i = 1)^n (x_i - barx)` is calculated as follows:

`sum_(i = 1)^n(x_i - barx) = sum_(i = 1)^n x_i - sum_(i = 1)^n barx`

Now, from equation (I), we get

`sum_(i = 1)^n (x_i - barx) = nbarx - sum_(i = 1)^n barx`

= `nbarx - barx sum_(i = 1)^n 1`

= `nbarx - nbarx`

= 0