Question

Let `barx` be the mean of x₁, x₂, ..., x_n and `bary` the mean of y₁, y₂, ..., y_n. If `barz` is the mean of x₁, x₂, ..., x_n, y₁, y₂, ..., y_n, then `barz` is equal to ______.

Options

• `barx + bary`

• `(barx + bary)/2`

• `(barx + bary)/n`

• `(barx + bary)/(2n)`

Answer 1

Let `barx` be the mean of x₁, x₂, ..., x_n and `bary` the mean of y₁, y₂, ..., y_n. If `barz` is the mean of x₁, x₂, ..., x_n, y₁, y₂, ..., y_n, then `barz` is equal to `underlinebb((barx + bary)/2)`.

Explanation:

Given, `sum_(i = 1)^n x_i = nbarx` and `sum_(i = 1)^n y_i = nbary`  ...(i) `[∵ barx = (sum_(i = 1)^n x_i)/n]`

Now, `barz = ((x_1 + x_2 + ... + x_n) + (y_1 + y_2 + ... + y_n))/(n + n)`

= `(sum_(i = 1)^n x_i + sum_(i = 1)^n y_i)/(2n)`

= `(nbarx + nbary)/(2n)`  ...[From equation (i)]

= `(barx + bary)/2`