# Geometric mean

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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.)