geometric mean

Geometric mean

$${\bar{x}}_{\text{geo}}=\sqrt[N]{\prod _{i=1}^N{x}_i}$$

Interpretation

The geometric mean is like resizing a rectangle into a square while preserving area:

Changing $a$ affects $b$ multiplicatively.

→ Use it any time values combine multiplicatively (compounding, growth rates, ratios over time).

Growth of +20% +25% -33.3%

E.g. 1000→1200→1500→1000

$$\cancel{\bar{x}=\frac{20+25+33.3}{3}=3.89\%}$$ $${\bar{x}}_{\text{geo}}={\left(\prod _{i=1}^n(1+r_i)\right)}^{1/n}-1=\sqrt[3]{(1.2)(1.25)(0.667)}-1=0\%$$

The +1 converts the growth rates into growth factors (e.g. +20% → 1.2), and the -1 converts the geometric mean of the growth factors back into a growth rate.

What is the formula for the geometric mean of $N$ positive numbers $x_1,\dots ,x_N$?

$${\bar{x}}_{\text{geo}}=\sqrt[N]{\prod _{i=1}^N{x}_i}$$

What is the geometric mean of the growth rates +20%, +25%, and -33.3%?

$${\bar{x}}_{\text{geo}}={\left(\prod _{i=1}^n(1+r_i)\right)}^{1/n}-1=\sqrt[3]{(1.2)(1.25)(0.667)}-1=0\%$$

When should you use the geometric mean instead of the arithmetic mean?

Use the geometric mean any time values combine multiplicatively (compounding, growth rates, ratios over time). The arithmetic mean is for additive combinations.