Multiply the values together and take the nth root. For growth rates, turn each into a multiplier first: +50% is 1.5, −50% is 0.5. Their product is 0.75, and the square root of that is 0.866 — a compound rate of −13.4% a year. Averaging the rates arithmetically gives 0%, which is a number nothing actually earned.
The +50% then −50% case
Put 100 in. It grows 50% to 150. It falls 50% to 75. You are down a quarter.
The arithmetic mean of +50 and −50 is 0%, which would imply you finished where you started. It is a correct average of the two NUMBERS and a wrong answer about the money, because the second year multiplies the first rather than adding to it.
The geometric mean of 1.5 and 0.5 is √0.75 = 0.866, or −13.4% a year. Apply that twice: 100 × 0.866 × 0.866 = 75. It reproduces the actual result, which is the entire test of whether an average of rates is the right one.
Why the arithmetic mean flatters
The arithmetic mean of a set of growth multipliers is never smaller than the geometric mean, and is equal only when every value is identical. So averaging returns arithmetically ALWAYS overstates them, never understates.
The gap grows with volatility. Two steady 10% years have no gap at all; a +50 and a −50 have thirteen points of it. That is why a volatile fund can advertise a good arithmetic average while its investors are down.
It is also why regulators require compound figures in fund literature. The arithmetic mean is not a lie about the arithmetic; it is an answer to a question nobody asked.
CAGR is this, renamed
Compound annual growth rate is the geometric mean of the annual multipliers, minus one. Same calculation, different name, and finance uses the name.
It answers: what single steady rate would have produced this end result from this start? By construction it reproduces the total, which is what makes it comparable between investments.
It also hides the path. Two investments with the same CAGR can have had entirely different years, and one of them may have been unholdable. CAGR is the right summary and it is only a summary.
When to use it on plain values
Outside growth rates, the geometric mean is the right average whenever the quantities are RATIOS or multiply together.
Aspect ratios, price indices, dilution factors, and anything where doubling and halving ought to cancel. The geometric mean of 2 and 8 is 4, which sits proportionally between them — twice the first and half the second. The arithmetic mean, 5, does not.
A useful test: if multiplying every value by 10 should multiply the answer by 10, and halving some values should cancel doubling others, you want the geometric mean.
Why it needs positive numbers
A zero makes the product zero, and the nth root of zero is zero — whatever the other values were. That is technically correct and not an average of anything.
Negatives are worse: an even root of a negative product has no real value at all, so the answer would exist or not depending on how many values you happened to have.
In the rate reading it makes sense. A −100% period leaves nothing, and no subsequent growth rate can recover from nothing, so there is no meaningful average afterwards. This page says that rather than returning a number.
The inequality, and what the gap measures
For any positive data: harmonic ≤ geometric ≤ arithmetic ≤ root mean square. Equality happens only when every value is identical.
So the gap between two of these means is itself a measure of spread. Two averages that agree tell you the data barely varies; a wide gap tells you it varies a lot, before you compute any variance.
The arithmetic-geometric inequality is one of the oldest results in mathematics and has a proof short enough to fit in a paragraph. This site checks it computationally on every generated series rather than quoting it.
Sources and methodology
The geometric mean is standard; these set out where it is required rather than merely preferred.
Method. Values are converted to multipliers where rates were given, then the mean is computed as exp(mean of the logs) so a long series cannot overflow before the root is taken. The answer is then raised to the power n and compared with the actual product; the page reports that comparison rather than assuming it. The arithmetic mean is computed alongside and its implied total is shown for contrast. That engine is verified on every change against 61 hand-written assertions, including that the geometric mean raised to n reproduces the product across two thousand generated series, and that it never exceeds the arithmetic mean. The count and the per-case breakdown are published on the formula verification page.
Read the guide
For rates over a fixed distance or quantity — speeds, prices per unit — the Harmonic Mean Calculator is the right one. For the forward direction, turning a rate into a final amount, the Compound Interest Calculator does that.