# Geometric mean

Geometric mean returns or growth are calculated by taking account of compounding.

(Contrasted with the arithmetic mean, which ignores compounding).

### Example 1: Positive returns or growth

The geometric mean return calculated from sample returns of 4%, 5% and 6% is given by:

(1.04 x 1.05 x 1.06)^{(1/3)} - 1

= **4.9968%**.

### Relationship between geometric mean and arithmetic mean

When returns or growth are positive, geometric means are smaller figures than arithmetic means.

In Example 1 above, the arithmetic mean is:

(4% + 5% + 6%) / 3 = **5.0000%**

*The geometric mean of +4.9968% is a smaller positive number than the arithmetic mean of +5.0000%.*

On the other hand, when returns or growth are *negative*, the geometric mean is a larger negative number - further away from zero - than the arithmetic mean.

### Example 2: Negative returns or decline

The geometric mean return calculated from three *negative* sample returns of -(4)%, -(5)% and -(6)% is given by:

( (1 - 0.04) x (1 - 0.05) x (1 - 0.06) )^{(1/3)} - 1

(0.96 x 0.95 x 0.94)^{(1/3)} - 1

= **-(5.0035)%**.

The negative geometric mean of -(5.0035)% is a larger negative number - further away from zero - than the arithmetic mean of -(5.0000)%.

(The arithmetic mean of the negative returns of -(4)%, -(5)% and -(6)% is the three items added together and divided by 3.)