Geometric Mean Calculator

What the geometric mean calculator does

The geometric mean is the average for numbers that multiply together rather than add up. This calculator finds it from your data, along with the count of numbers used.

It is the right average for growth rates, ratios, and anything that compounds, where the ordinary average quietly gives the wrong answer. Below is how it works and exactly when it is the one to use.

How to use it

1. Enter your numbers in the box, separated by commas, spaces, or new lines. They should all be positive.
2. Press Calculate to see the geometric mean and how many numbers it used, or Reset to clear it.

How the geometric mean is worked out

Where the ordinary average adds your numbers and divides, the geometric mean multiplies them and takes a root:

Geometric mean = the n-th root of (all the values multiplied together)

So for three numbers you multiply them and take the cube root, for four numbers the fourth root, and so on. The effect is to find the value that, if every number were the same, would give the same overall product. That is what makes it the honest average for things that build on each other.

When to reach for the geometric mean

The clearest case is growth over time. If an investment returns different amounts each year, or a population grows at different rates, the arithmetic mean of those rates overstates the real result, because growth compounds. The geometric mean gives the single steady rate that lands you in exactly the same place, which is the same idea as a compound annual growth rate.

It is also the correct average for ratios and for numbers on different scales, like combining several index scores into one. Whenever your numbers are multipliers rather than plain amounts, the geometric mean is the one that keeps the math honest.

A worked example: growth that compounds

Take the three numbers 1, 3, 9. Multiplied together they make 27, and the cube root of 27 is 3. Notice the pattern: 1 to 3 to 9 each step multiplies by three, and 3 is the value sitting right in that multiplicative middle.

Compare that to the arithmetic mean of the same three numbers, which is (1 + 3 + 9) ÷ 3 = 4.33. For data that grows by multiplying, the geometric mean of 3 is the truer centre, and the gap between the two is exactly why the choice of average matters here.

Positive numbers only, and the rounding

The geometric mean only makes sense for positive numbers. Multiplying in a zero would drag the whole product to zero, and a negative would break the root entirely, so this is one average where a stray zero or minus sign gives a result you cannot use. If your data includes returns that dip below zero, convert them to growth factors first, so a loss of 5 percent becomes 0.95 rather than a negative.

The result is shown to six decimal places with trailing zeros trimmed, which is plenty of precision for rates and ratios.

Questions people ask

What is the geometric mean?

The n-th root of all your values multiplied together. It is the average for numbers that compound or multiply, rather than the arithmetic mean, which is for numbers that add.

When should I use it instead of the normal average?

For growth rates, investment returns, ratios, and values on different scales. In those cases the arithmetic mean overstates the result, and the geometric mean gives the correct steady figure.

Why does it need positive numbers?

Because the calculation multiplies everything and takes a root. A zero forces the whole result to zero, and a negative makes the root undefined, so the geometric mean is only meaningful for positive values.

Is the geometric mean always smaller than the arithmetic mean?

For positive numbers that are not all identical, yes, it is always a little smaller. The two are equal only when every value in the set is the same.

References

A quick note on where the methods here come from. The geometric mean and the kinds of data it suits are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook that covers the geometric mean alongside the other averages.

1. NIST/SEMATECH e-Handbook of Statistical Methods (measures of location). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (measures of the center of the data). https://openstax.org/details/books/introductory-statistics-2e