geometric mean

Geometric mean

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

Interpretation

The geometric mean is the length $c$ of $a$ square, which has the same area as a
rectangle with the width $a$ and length $b$

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

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