Geometric vs Arithmetic means

Why is a geometric always smaller or equal to arithmetic means.? And why is it only equal when there is no variability in the observations?

Do you want a mathematical proof?

Yes please!

Here it is for two values, x_1 and x_2. I’ll leave the proof of the general case to you, if you’re keenly interested.

\left(x_1 - x_2\right)^2 \ge 0

Note that it is equal to zero if and only if x_1 = x_2.

x_i^2 - 2x_1x_2+x_2^2 \ge 0

Add 4x_1x_2 to each side.

x_i^2 + 2x_1x_2+x_2^2 \ge 4x_1x_2
\left(x_1 + x_2\right)^2 \ge 4x_1x_2

Take the square root of both sides.

x_1 + x_2 \ge \sqrt{4x_1x_2} = 2\sqrt{x_1x_2}

Note that here we’re assuming that x_1 \ge 0 and x_2 \ge 0. (If only one were negative, then we’d be taking the square root of a negative number. If both were negative, then the left side would be negative while the right side is positive.)

Divide both sides by 2.

\frac{x_1 + x_2}{2} \ge \sqrt{x_1x_2}


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