Geometric Mean Calculator
Find the geometric mean of positive numbers, or turn a run of percentage changes — +10%, −5%, +20% — into the true average growth rate per period. The working shows the log method step by step, and the harmonic and arithmetic means sit alongside so you can see which average your data call for.
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How to Use the Geometric Mean Calculator
- 1. Choose what you are averaging. Positive values such as ratios, concentrations or prices, or percentage changes from one period to the next.
- 2. Enter the numbers. For changes, type 10 for +10% and -5 for −5%; a % sign is fine too.
- 3. Read the result. Values give the geometric mean with the harmonic and arithmetic means for comparison; changes give the average growth per period, the misleading simple average, and the total change.
- 4. Check the working. The log method, the product form and a check that the average rate reproduces the total are all shown.
The Geometric Mean Formula
The arithmetic mean adds the values and divides by n. The geometric mean multiplies them and takes the n-th root:
Definition
GM = (x₁ × x₂ × … × xₙ)1/n
Log method
GM = exp((ln x₁ + ln x₂ + … + ln xₙ) ÷ n)
For 2 and 8, GM = √(2 × 8) = √16 = 4, where the arithmetic mean is 5. For the example data — 12, 18, 30, 45 and 200 — the product is 58,320,000 and its fifth root is 35.74, while the arithmetic mean of 61 is dragged upward by the single large value. The calculator uses the log method because it is the same number computed safely: a product of a few hundred ordinary values overflows, but a sum of their logarithms never does.
Average Growth Rates: Why the Geometric Mean Is Right
Growth compounds, so changes multiply. A quantity that grows 10%, shrinks 5% and grows 20% is multiplied by 1.10 × 0.95 × 1.20 = 1.254 — a total rise of 25.4%. The constant rate that gives the same total is the geometric mean of the growth factors minus one: 1.2541/3 − 1 = 7.84% per period.
The simple average of the three changes, 8.33%, overstates it: 1.0833 cubed is 1.2714, so it would turn 100 into 127.14 instead of the 125.40 you actually have. The gap gets dramatic with volatile changes. A rise of 50% followed by a fall of 50% averages 0%, yet 100 becomes 150 and then 75 — the geometric mean reports the truth, −13.4% per period. This is the same calculation as a compound annual growth rate (CAGR).
Which Mean Should You Use?
| Mean | Use it for | Example |
|---|---|---|
| Arithmetic | Quantities that add up | Test scores, heights, daily temperatures |
| Geometric | Quantities that multiply, or data spanning orders of magnitude | Growth rates, investment returns, ratios, index numbers |
| Harmonic | Rates measured over a fixed amount of something | Average speed over equal distances, price–earnings ratios, the F1 score |
The harmonic mean is n ÷ Σ(1/x). Drive 60 km/h out and 40 km/h back over the same distance and your average speed is 2 ÷ (1/60 + 1/40) = 48 km/h, not 50 — you spent longer at the slower speed. For any positive data the three means fall in the order harmonic ≤ geometric ≤ arithmetic.
Frequently Asked Questions
- What is the geometric mean used for?
- It averages quantities that combine by multiplication: growth rates and investment returns over several periods, ratios and fold-changes, and skewed positive data that spans several orders of magnitude, such as bacterial counts or antibody titres. In each case it gives the typical multiplicative value, which the arithmetic mean overstates.
- How do you calculate the geometric mean?
- Multiply the n values together and take the n-th root of the product. Equivalently, average the natural logarithms of the values and exponentiate the result, which is how calculators and software do it. For 4, 9 and 6 the product is 216, and the cube root of 216 is 6.
- Why can't the geometric mean use zero or negative numbers?
- A single zero makes the whole product zero, whatever the other values are, and the logarithm of zero or of a negative number is undefined. With negative values the root may not even be a real number. That is why growth rates are converted to factors such as 0.95 for −5%: the factors stay positive unless a change reaches −100%.
- How do you calculate an average growth rate?
- Convert each percentage change to a growth factor (1 + change ÷ 100), multiply the factors, take the n-th root, and subtract 1. For +10%, −5% and +20% that is 1.254^(1/3) − 1 = 7.84% per period. If you only know the start and end values, the same idea gives CAGR = (end ÷ start)^(1/n) − 1.
- Is the geometric mean always smaller than the arithmetic mean?
- It is never larger. For positive numbers the geometric mean is less than or equal to the arithmetic mean, and they are equal only when every value is the same. The more spread out the values, the bigger the gap.
- When should I use the harmonic mean?
- Use it to average rates when the thing held fixed is in the rate's denominator — average speed over equal distances, average price paid when spending equal amounts each time, or average price–earnings ratios. Machine-learning's F1 score is the harmonic mean of precision and recall, which keeps a high value from hiding a low one.