If X̅ is the mean of x1, x2, x3, ...x_n, then for a ≠ 0, the mean of (ax1, ax2, ax3, ... axn, \(\rm\frac{x_{1}}{a}, \frac{x_{2}}{a}, \frac{x_{3}}{a}, \ldots \frac{x_{n}}{a}\) is:

Given:

The mean of x₁, x₂, x₃, ... x_n is X̅.

We need to find the mean of the transformed values:

(ax₁, ax₂, ax₃, ..., ax_n) and (x₁/a, x₂/a, x₃/a, ..., x_n/a).

Formula used:

The mean of a set of values {x₁, x₂, ..., x_n} is:

X̅ = (x₁ + x₂ + ... + x_n) / n

For a transformation involving multiplication by a constant **a** and division by **a**:

Calculation:

Mean of {ax₁, ax₂, ..., ax_n}:

⇒ (ax₁ + ax₂ + ... + ax_n) / n

⇒ a (x₁ + x₂ + ... + x_n) / n

⇒ aX̅

Mean of {x₁/a, x₂/a, ..., x_n/a}:

⇒ (x₁/a + x₂/a + ... + x_n/a) / n

⇒ (1/a) (x₁ + x₂ + ... + x_n) / n

⇒ (1/a) X̅

∴ The mean of the given transformed values is (a + 1/a)X̅.