Descriptive statistics
Geometric Mean Calculator
The right average for things that multiply: growth rates, returns, ratios and index numbers.
Have a spreadsheet? Drop a CSV here
Or . CSV, TSV or plain text. You pick which column is which after it loads — the names in your file do not have to match ours. The file is read in your browser and never uploaded.
How to read this result
Use the geometric mean whenever the quantities multiply. Growth rates, investment returns, population changes, index numbers, ratios of any kind. The arithmetic mean answers "what would each value have to be for the total to be the same"; the geometric mean answers "what would each value have to be for the product to be the same", and for growth it is the product that matters.
It is always less than or equal to the arithmetic mean, with equality only when every value is identical. That is the AM-GM inequality, and it is why average returns quoted arithmetically always flatter: +50% then −50% averages to 0% arithmetically and leaves you 25% down, which the geometric mean correctly reports as −13.4% per period.
Every value must be positive. The geometric mean multiplies them all together, so a zero makes it zero and a negative makes it undefined. For growth rates that means working with factors (1.08 for +8%) rather than percentages, and a rate of −100% or worse cannot be averaged this way.
Computed in log space. The product of a few hundred numbers overflows or underflows a double long before the mean does anything interesting, so this calculator averages the logarithms and exponentiates once at the end.
The formula
The two forms are identical mathematically and not numerically — the second is what a computer should use.
Compound growth
The geometric mean of the per-period growth factors, minus one.
Worked example
Five years of returns
An investment grows by 8%, then 15%, then falls 8%, then grows 21%, then 5%. What was the average annual return?
As growth factors: 1.08, 1.15, 0.92, 1.21, 1.05
- Multiply the factors together.1.08 × 1.15 × 0.92 × 1.21 × 1.05 = 1.4519
- Take the fifth root.1.4519^(1/5) = 1.0774
- Subtract one.geometric mean return = 7.74% per year
- Compare with the arithmetic mean of the factors.(1.08 + 1.15 + 0.92 + 1.21 + 1.05)/5 = 1.082 → 8.2%
The arithmetic average overstates the return by half a percentage point a year. Compounded over a decade, that gap is a real amount of money — which is why fund returns are required to be quoted geometrically.
Checked against R's exp(mean(log(x))).
The calculator above is loaded with these numbers by the Load the worked example button.
Assumptions, and when to use something else
All values must be strictly positive. The quantities must be ones that compound multiplicatively: growth factors, ratios, index numbers. Averaging temperatures or test scores geometrically is arithmetically possible and meaningless.
For growth over unequal time periods, the simple geometric mean is not enough — weight the logarithms by the length of each period, or work from the start and end values with the total elapsed time, which is what the second mode does.
- The values add rather than multiplyArithmetic meanTotals, counts and measurements average arithmetically.
- Some values should count moreWeighted averageFor unequal weights, periods or portfolio sizes.
- You want the change between two figuresPercentage changeThe simple change, and how it differs from a compound rate.
Questions people ask
When should I use the geometric mean instead of the arithmetic mean?
Whenever the values multiply rather than add: growth rates, returns, ratios, index numbers, and anything expressed as “times larger”. The test is simple — if chaining the values together is the meaningful operation, average them geometrically.
Why can't the geometric mean handle negative numbers?
Because it takes the nth root of a product. With one negative value the product is negative and its even root does not exist in the real numbers. For growth, express the rates as factors — −8% becomes 0.92 — and the problem disappears unless a value loses everything.
What is the difference between the geometric mean and CAGR?
None, mathematically. CAGR is the geometric mean of the growth factors minus one, presented as a rate. The second mode above computes it from the start and end values, which is usually more convenient.
What is the harmonic mean for?
Rates over a fixed distance — average speed over a route, average price per unit bought with a fixed budget. It is reported beside the other two above, and it is always the smallest of the three.