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Geometric Mean Calculator

Free geometric mean calculator: the multiply-compounded average of positive numbers, shown beside the arithmetic mean so you can see the compounding gap. Runs in your browser.

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Deep dive guides

A LITTLE CONTEXT · 4 MIN READGeometric Mean vs Arithmetic Mean
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  • Geometric Mean vs Arithmetic Mean — One average adds, one multiplies: the geometric mean formula, volatility drag on returns, the AM–GM inequality, and when each is correct.

How geometric mean calculator works

The geometric mean is the average for numbers that multiply. Where the arithmetic mean adds n numbers and divides by n, the geometric mean multiplies them and takes the nth root — computed here as exp(mean of the natural logs) so large lists stay numerically stable. For 4 and 9 the product is 36 and √36 = 6: the geometric mean lands exactly halfway on the multiplication scale, not the addition scale.

That difference matters most for growth rates. An investment that gains 10% one year and loses 10% the next has an arithmetic mean return of 0% — but the actual outcome is 1.1 × 0.9 = 0.99, a 1% loss. The geometric mean of the growth factors, √(1.1 × 0.9) ≈ 0.995, reports the honest −0.5% average per year. Anytime quantities compound — investment returns, population growth, interest, file compression ratios — the geometric mean is the one that matches reality.

The tool also shows the arithmetic mean and the ratio between the two. That ratio is never below 1 (the AM–GM inequality) and grows with the spread of the data: identical numbers give a ratio of exactly 1, while wildly varying numbers pull the two means far apart. A large gap is a warning sign that volatility is eating your average — the wider the swings around the same arithmetic mean, the lower the geometric outcome.

Frequently asked questions

What is the geometric mean formula?
The geometric mean of n positive numbers is the nth root of their product: (x₁ × x₂ × … × xₙ)^(1/n). Equivalently, exp of the average of the natural logs — the form calculators actually use, because it stays accurate when the product would overflow.
Why can't the geometric mean handle zero or negative numbers?
The formula takes logarithms, and logs are only defined for strictly positive values. A single zero would also drag the whole product to zero regardless of the other numbers, and negatives break the root's meaning. For data containing zeros or negatives, use the arithmetic mean or a different measure.
When should I use the geometric mean instead of the arithmetic mean?
Whenever the quantities multiply rather than add: percentage growth over successive periods, ratios, index returns, discount factors, or speeds over equal distances (the harmonic-mean case is the inverse pattern). For already-added quantities like heights or salaries, the arithmetic mean is correct.
Why is the geometric mean always ≤ the arithmetic mean?
That is the AM–GM inequality: for positive numbers the arithmetic mean is at least the geometric mean, with equality only when all the numbers are equal. The gap widens with spread — which is why volatile investments trail their 'average' return, an effect called volatility drag.
What does a 10% gain followed by a 10% loss actually average?
Not zero. The multiplicative path is 1.10 × 0.90 = 0.99, so you end with 99% of what you started with — a 1% total loss. The geometric mean of the factors, √0.99 ≈ 0.995, reports the true average change per period: −0.5%, not 0%.

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