# Geometric Mean Calculator

Written by:
PK

Here is a geometric mean calculator or geometric average calculator, which will calculate the geometric mean of a list of numbers you input.

Enter any numbers – including decimals, and fractions – and the tool will return the average.

## What is the Geometric Average or Geometric Mean?

The geometric mean or average of a set of numbers is a number which represents or describes the central tendency by taking the nth root of the product of n numbers. The geometric mean only works for positive numbers.

In general, an average (or mean) of a list of numbers is one number that represents or summarizes the whole set. The most common average is the arithmetic mean – a number we get by adding numbers together and dividing by the length of our number list.

The geometric mean is useful when comparing values of different weights, or – as in investment returns – averaging across rates or returns. For more on working with percentage returns, see the worked geometric mean example below.

## Formula for Geometric Mean

The formula for the geometric mean is:

geometric\ mean = \sqrt[n]{x_1*x_2*x_3...x_{n-1}*x_n}

In the formula:

• n: The number of numbers you are averaging together (that is, the length of the list)
• x_1, x_2, etc.: The terms/numbers

### Example Geometric Mean Calculation

The geometric mean is very useful in investing. While your first instinct may be to reach for the arithmetic mean, there are some major downfalls.

Imagine a stock which returns the following over 3 years: -50%, +30%, +20%. What's your average annual return?

\sqrt{0.5*1.3*1.2}\\\sqrt{0.78}\\\approx (.92051 - 1)*100\\average\ annual\ return \approx -7.95\%

(Note we converted .92051 to a negative percentage here by subtracting 1 or 100%, then multiplying by 100)

If you entered those numbers into the arithmetic average, you'd get the wrong answer:

wrong\ annual\ return=\frac{0.5+1.3+1.2}{3}\\\frac{3}{3} = (1-1)*100\\wrong\ annual\ return=0\%

Using the wrong average showed no change – an average 0% return a year. Of course, for every $100 you invested you now have$78... proving you should pick the correct mean.

## Using the Geometric Average Calculator

In the text box at the top, enter a list of numbers to average. As noted, you can toggle between Numbers and Percentages. If you use Percentages, merely put the percentages in (such as -30, 50, etc.) and the tool will convert them to numbers for you.

You could alternatively rewrite your percentages as numbers (divide by 100 and add '1').

When happy, simply hit the Compute Geometric Mean button below and DQYDJ will find the geometric average.

Next, try the harmonic mean calculator. Next, visit our other calculators and tools.

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