Geometric Mean Calculator The multiplicative average for compounding, ratios, and investment returns
Calculate the geometric mean of 2 to 30 values, or the average compound return of a series of percentage returns. All calculations run locally in your browser.
Enter 2 to 30 numbers to compare the three classic means side by side with the median.
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Browse practical tools for statistics, finance, geometry, and average-selection decisions.
The three Pythagorean means — arithmetic, geometric, and harmonic — each answer a different averaging question.
SectionCAGR (compound annual growth rate) and XIRR are the two standard annualized return metrics in finance.
SectionThe geometric mean theorem says the altitude to a right triangle's hypotenuse is sqrt(pq).
Arithmetic mean (AM) is the sum of all values divided by the count: AM = (x₁ + x₂ + ⋯ + xₙ) / n.
CalculatorWeighted mean is the correct average when each value does not count equally.
CalculatorHarmonic mean (HM) is n divided by the sum of reciprocals: HM = n / (1/x₁ + 1/x₂ + ⋯ + 1/xₙ).
CalculatorVariance measures the average squared distance of each value from the mean.
CalculatorCAGR (compound annual growth rate) is the single annual rate that connects a beginning value to an ending value as if growth were perfectly smooth each year.
CalculatorXIRR (extended internal rate of return) is the annualized rate that makes the net present value of all your dated cash flows equal zero: NPV = Σ [CFᵢ / (1 + r)^(dᵢ/365)] = 0.
CalculatorThe Geometric Mean Altitude Theorem states that the altitude drawn from the right angle of a right triangle to its hypotenuse equals the geometric mean of the two hypotenuse segments it creates.
Arithmetic mean (AM = sum/n) is correct for additive values. Geometric mean (GM = nth root of product) is correct when values multiply or compound.
GuideGeometric mean (GM) is correct when values multiply: growth factors, fold changes, index relatives.
GuideCAGR and geometric mean are the same mathematical operation applied to different input formats.
GuideCAGR = (EV/BV)^(1/n) − 1 is correct when there is exactly one beginning value and one ending value with no interim cash flows.
GuideMean is the arithmetic average (sum/n). Median is the middle value when sorted.
GuideArithmetic mean (AM) is correct for additive values: scores 72+80+88+96 → AM = 84.
Start here
Geometric mean answers a slightly different question depending on what you are averaging. Find your situation below, load one of the example inputs, and the calculator shows the product, the nth root, and the arithmetic mean side by side.
Use this when each value multiplies the one before it: annual revenue growth, population change, or portfolio returns. Enter 2 to 30 positive factors and you get the single factor that, repeated, reproduces the same total product.
With exactly two positive numbers the formula collapses to the square root of their product, so the answer is easy to check by hand. This is the form that appears in right-triangle altitude problems and in most textbook exercises.
Two routes give the same answer: multiply every value and take the nth root, or average the natural logarithms and exponentiate. The log route matters once the product grows past what a computer can store without losing precision.
Percentages become growth factors first: 8% is 1.08, -10% is 0.90, and 0% is 1.00. Subtract 1 from the resulting geometric mean to read the compound return per period. Type the % sign and the conversion happens for you.
Convert fractions to decimals before entering them. A raw zero collapses the whole product to zero, so decide first whether it is a true zero or a 0% change. For frequency-weighted data, weight each ln(value) by its frequency, divide by the total frequency, then exponentiate.
Select an example to load it into the calculator.
For exactly two positive numbers the geometric mean is the square root of their product. These cases show the full working, so you can verify the method before trusting it on a longer list.
| Problem | Input | Calculation | Geometric mean |
|---|---|---|---|
| Geometric mean of 4 and 16 | 4, 16 | sqrt(4 x 16) | 8 |
| Geometric mean of 12 and 16 | 12, 16 | sqrt(12 x 16) | 13.8564 |
| Geometric mean of 16 and 25 | 16, 25 | sqrt(16 x 25) | 20 |
| Geometric mean of a five-value list | 1, 5, 10, 13, 30 | (1 x 5 x 10 x 13 x 30)^(1/5) | 7.21 |
Complete reference guide
This guide explains what the geometric mean is, how to calculate it, when it applies, and where its limits are. The definitions and formulas follow standard references, including Wolfram MathWorld (Geometric Mean), OpenStax Introductory Business Statistics section 2.5 (Geometric Mean), and the NIST Dataplot documentation for the geometric mean.
Geometric mean multiplies every value into a single product, then takes the root whose degree equals the number of values. The result is the one value that, substituted for every observation, would reproduce the same total product.
Example: the geometric mean of 4 and 16 is 8, because 8 x 8 = 64 = 4 x 16. Geometrically, 8 is the side of a square whose area matches a 4-by-16 rectangle — which is why the same formula appears in both statistics and Euclidean geometry.
n is the count of values, and each x must be positive. For large datasets, the log-transform method is numerically safer: average the natural logarithms of all values, then exponentiate. Both paths produce the same geometric mean, but the log method avoids floating-point overflow when the product of many values exceeds what a computer can store.
A rectangle with sides a and b has the same area as a square whose side is their geometric mean.
Take the values 1, 5, 10, 13, and 30. Their product is 1 × 5 × 10 × 13 × 30 = 19,500. There are five values, so take the fifth root: 19,500^(1/5) ≈ 7.2112. The arithmetic mean of the same values is 59 ÷ 5 = 11.8. The geometric mean is lower because it reduces the pull of the large value, 30. For a simpler check, 4 and 16 have a product of 64, and the square root of 64 is 8.
| Step | Calculation | Result |
|---|---|---|
| Product | 1 x 5 x 10 x 13 x 30 | 19,500 |
| Root | 19,500^(1/5) | 7.21 |
| Arithmetic comparison | (1 + 5 + 10 + 13 + 30) / 5 | 11.80 |
Enter 2 to 30 numbers. In values mode, use positive numbers. In returns mode, each return must be -100% or greater. A return of exactly -100% is a complete loss and gives an average return of -100%.
An 8% return becomes 1.08, a -10% return becomes 0.90, and 0% becomes 1.00.
Consider five period returns: 6%, 7%, 8%, −35%, and 16%. Their arithmetic mean is 0.4%, which suggests a small gain. Multiply the growth factors instead: 1.06 × 1.07 × 1.08 × 0.65 × 1.16 ≈ 0.923602. The fifth root of that product is about 0.98423, a geometric average return of about −1.58% per period. Starting with 1,000, the sequence ends at about 923.60, a loss. Compounding 1,000 at −1.58% for five periods reaches the same ending value, but compounding at the arithmetic mean of 0.4% does not. This example describes past returns only. It does not predict future performance.
Enter period returns to compare the actual sequence with paths at a constant geometric or arithmetic mean rate.
The geometric mean of growth factors gives the constant per-period return that turns the starting value into the ending value over the same number of periods. That makes it useful for summarizing a historical sequence of returns. Requirements differ between frameworks and organizations, so check the specific rules that apply to your situation rather than assuming one method is mandated everywhere.
Quantities that change by multiplication, such as a population or price level over several periods, can be summarized with the average growth factor. For example, growth rates of 8%, −10%, and 11% give a combined factor of 1.08 × 0.90 × 1.11 = 1.07892. The cube root is about 1.0256, an average of about 2.56% per period.
When results are expressed as ratios to a reference, such as relative speed or relative cost, the geometric mean gives a consistent summary. The ranking of compared items does not depend on which item serves as the reference. Ratios are dimensionless, so this works even when the raw measurements use different units, provided each one is normalized first.
For two positive lengths a and b, the geometric mean √(ab) is the side of a square with the same area as an a × b rectangle. In a right triangle, the altitude to the hypotenuse is the geometric mean of the two segments it divides the hypotenuse into. Both results are lengths, because the inputs are lengths in the same unit.
The geometric mean is the nth root of the product of n values. It is the typical value for quantities that combine by multiplication, such as growth factors and ratios. Enter 2 to 30 numbers. The steps below show how the calculator gets the result.
Enter between 2 and 30 numbers.
Combine the values by multiplication, then take the nth root.
Read the result in the right units and period.
Drag the volatility slider to see the AM-GM inequality in action. As spread increases, arithmetic mean holds steady while geometric mean drops — demonstrating why volatile returns always compound to less than the arithmetic average suggests.
Widen symmetric factors around 1 and watch the arithmetic mean hold steady while the geometric mean falls.
Choose by how the quantities combine. Use the arithmetic mean when values add up, such as averaging independent measurements. Use the geometric mean when values multiply, such as growth factors and ratios.
Enter 2 to 30 numbers to compare the three classic means side by side with the median.
The geometric mean of n positive values is the nth root of their product: GM = (x₁ × x₂ × … × xₙ)^(1/n). An equivalent form is GM = exp(average of ln xᵢ). For example, the geometric mean of 4 and 16 is √64 = 8.
Use returns mode. Each number is read as a percentage, with or without a % sign. Convert each return r to the factor 1 + r/100, take the geometric mean of the factors, and subtract 1. For 8%, −10%, and 11%, the factors multiply to 1.07892, and the average return is about 2.56% per period. Do not enter raw negative percentages in values mode.
Values mode accepts only strictly positive numbers. A zero would make the geometric mean 0, and negative values can produce results that are not real numbers. In returns mode, returns from −100% upward are allowed. Exactly −100% gives an average return of −100%, and anything lower is rejected.
Choose by how the quantities combine. Use the arithmetic mean when values add up, such as averaging independent measurements. Use the geometric mean when values multiply, such as growth factors and ratios. For positive numbers, the geometric mean is never larger than the arithmetic mean, and the two are equal only when all values are equal. The gap between them reflects spread in the data. It does not tell you what kind of data you have.
For a meaningful result, yes, or else they should all be dimensionless. If every input is in the same unit, the geometric mean is in that unit too. If units differ, first convert each measurement to a ratio against a reference. Then take the geometric mean of those dimensionless ratios.
The calculator computes results locally in your browser with JavaScript, and the numbers you enter are not transmitted. This statement covers the calculator inputs only. It does not describe every part of the website.
Yes. Both names describe the same product-root average: multiply the values together, then take the root that matches how many values you have.
Enter the returns as percentages such as 8%, -10%, 11%, or as factors such as 1.08, 0.90, 1.11. The geometric mean reported is the compound growth factor per period; subtract 1 to read it as a percentage return.
Formulas follow standard definitions from Wolfram MathWorld, OpenStax Introductory Business Statistics, and the NIST Dataplot reference. Returns mode converts each return to a growth factor, takes the nth root of the product, and subtracts 1. All calculations run in your browser.
Read our calculation methods and references