How to Calculate the Geometric Mean

Takes the nth root of the product of n values. It is the right average for growth rates and multipliers: the average of ×1.2 and ×1.8 is not ×1.5 but about ×1.47. All values must be positive.

Explanation

The geometric mean multiplies all the values together and takes the $n$ -th root of the product.

G = \sqrt[n]{x_1 \times x_2 \times \cdots \times x_n}

Use it whenever the quantities compound by multiplication, such as growth rates or investment returns. Adding and dividing gives the wrong answer for those.

Example

Suppose something grows by a factor of 1.2 in the first year and 1.8 in the second. What is the average yearly factor?

G = \sqrt{1.2 \times 1.8} = \sqrt{2.16} = 1.4697

Growing by 1.4697 twice gives $1.4697^2 = 2.16$ , exactly the true two-year growth of $1.2 \times 1.8 = 2.16$ .

The arithmetic mean $(1.2 + 1.8) \div 2 = 1.5$ would imply $1.5^2 = 2.25$ , overstating the growth.

Watch out

Every value must be positive. A zero or a negative value leaves the root undefined
The arithmetic mean is always greater than or equal to the geometric mean, with equality only when all the values are the same
If growth is quoted as "up 20%, up 80%", convert to the multipliers 1.2 and 1.8 before taking the geometric mean