Geometric Mean Calculator
The geometric mean is the correct average for anything that multiplies rather than adds: growth rates, investment returns, ratios and index numbers. Using an arithmetic mean on those systematically overstates performance.
Formula
- xᵢ
- = each positive value
- rᵢ
- = each period's growth rate as a decimal
- n
- = the number of values or periods
- Π
- = the product of all terms
Computed internally as exp(mean of logs), which gives the same answer while avoiding numeric overflow on long lists.
How to use this calculator
- Choose whether your data is plain positive numbers or percentage growth rates.
- In rate mode, enter percentages directly — 8 for +8%, −3 for −3%. The calculator converts them to growth factors for you.
- In value mode, all numbers must be positive; a zero or negative value makes the geometric mean undefined.
- Press Calculate. In rate mode the result includes the total compounded change so you can verify the average reproduces reality.
Worked example
Four years of returns: +8%, −3%, +12%, +5%
The arithmetic mean of these rates is 5.5%. But applying 5.5% four times gives a total gain of 23.9%, while the actual sequence produced 23.2%. The arithmetic mean describes a portfolio that did better than yours did.
- Factors: 1.08 × 0.97 × 1.12 × 1.05 = 1.2319776
- Fourth root: 1.2319776^(1/4) = 1.0535392
- Rate: (1.0535392 − 1) × 100 = 5.3539%
Answer: 5.3539% per year — the rate that, compounded four times, reproduces the actual 23.20% total gain.
Why the arithmetic mean overstates growth
Growth compounds. A 50% gain followed by a 50% loss leaves you with 75% of what you started with, not 100% — because the loss applies to a larger base than the gain did. The arithmetic mean of +50% and −50% is 0%, which describes an outcome that did not happen.
The geometric mean of those two rates is −13.4%, and compounding −13.4% twice does reproduce the actual 25% loss. This is not a technicality: the gap between the two means grows with volatility, and for genuinely volatile series it can be enormous. The arithmetic mean of a return series is always greater than or equal to the geometric mean, with equality only when every value is identical.
| Returns | Arithmetic mean | Geometric mean | Actual total |
|---|---|---|---|
| +50%, −50% | 0.00% | −13.40% | −25.00% |
| +8%, −3%, +12%, +5% | 5.50% | 5.35% | +23.20% |
| +10%, +10%, +10% | 10.00% | 10.00% | +33.10% |
| +100%, −50% | 25.00% | 0.00% | 0.00% |
The zero and negative problem
A geometric mean of raw values requires every value to be positive. A single zero makes the product zero and the mean zero regardless of the other values. Negative values make the product's sign depend on how many negatives there are, and an even root of a negative number is not real.
Growth rates get around this because a rate of −3% becomes a factor of 0.97, which is positive. The only impossible case is a rate of −100% or worse, where the factor reaches zero and the value is wiped out — after which no average rate can reconstruct anything. This calculator rejects that case explicitly rather than returning zero.
Beyond finance
The geometric mean is the right average whenever quantities combine multiplicatively. Population growth rates, bacterial doubling, inflation across years, and index numbers all compound the same way returns do.
It is also the correct way to average ratios that have no natural unit — such as normalised benchmark scores where one system is 2× faster on one test and 0.5× on another. The arithmetic mean of 2 and 0.5 is 1.25, implying superiority; the geometric mean is exactly 1, which is the honest answer. This property is why standard benchmark suites specify geometric means.
Relationship to CAGR
The geometric mean of a series of growth rates is the compound annual growth rate for those periods. If you have the period-by-period rates, this calculator gives you CAGR directly. If you only have a starting and ending value plus the number of periods, use the CAGR calculator, which works from the endpoints.
Both approaches give the identical answer when applied to the same data, which is a useful check: the geometric mean of the individual rates should equal the CAGR computed from the first and last values.
Important considerations
- All values must be positive in value mode; use rate mode for changes that can be negative.
- The geometric mean is always less than or equal to the arithmetic mean, with equality only for identical values.
- A rate of −100% or worse makes an average growth rate meaningless.
- The geometric mean is the correct average for anything that compounds.
- It is not appropriate for quantities that add, such as counts or totals.
Common mistakes to avoid
- Averaging returns arithmetically. This systematically overstates performance, and the overstatement grows with volatility.
- Including a zero. One zero forces the geometric mean of raw values to zero.
- Entering rates as factors in rate mode. Enter 8 for +8%, not 1.08.
- Using it for additive quantities. Heights, weights and counts should use the arithmetic mean.
- Confusing it with the harmonic mean. Rates over a fixed distance or a fixed amount need the harmonic mean instead.
Frequently asked questions
What is the geometric mean?
The nth root of the product of n values. Unlike the arithmetic mean, which is built on addition, the geometric mean is built on multiplication — which makes it the correct average for anything that compounds.
When should I use it instead of a normal average?
Whenever your values multiply rather than add: investment returns, growth rates, inflation, ratios and index numbers. Use the arithmetic mean for additive quantities such as heights, weights or totals.
Why is the geometric mean always lower?
It is a mathematical result (the AM–GM inequality) that the arithmetic mean is never less than the geometric mean for positive numbers, with equality only when all values are identical. The more spread out the values, the wider the gap.
Can I use negative numbers?
Not as raw values — an even root of a negative product is not a real number. Switch to rate mode, where a −3% rate becomes a positive factor of 0.97 and the calculation works normally.
Is the geometric mean the same as CAGR?
Yes, for a series of period growth rates they are the same figure. CAGR is normally computed from a start value, an end value and a number of periods; the geometric mean gets there from the individual period rates instead.
Why do benchmark suites use the geometric mean?
Because it treats speed-up ratios symmetrically. Being 2× faster on one test and 2× slower on another gives a geometric mean of exactly 1 — no net advantage — whereas the arithmetic mean would report 1.25 and falsely imply superiority.
Related calculators
Tools people usually open alongside this one.
Average Calculator
Mean, median, mode and range from one list of numbers, with the sorted data shown.
Open calculator
Harmonic Mean Calculator
The correct average when rates apply over a fixed distance, quantity or budget.
Open calculator
Weighted Average Calculator
Average values that carry different importance — credits, portfolio sizes, category weights.
Open calculator
Percentage Change Calculator
One tool for rises and falls — a signed percentage change measured from the original value.
Open calculator