Geometric Mean Calculator The multiplicative average for compounding, ratios, and investment returns
Geometric mean is the nth root of the product of n positive values — the correct average when data compounds rather than adds. A portfolio returning +50% then losing -50% has an arithmetic mean of 0% but a geometric mean of -13.4% per year, matching the actual $750 ending value on $1,000. Enter 2-30 growth factors or percentage returns and get geometric mean, arithmetic mean, product, count, and step-by-step work — computed locally in your browser.
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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 appears as a physical length in right-triangle geometry, not only as a statistic.
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
Geometric mean is the nth root of the product of n positive values. It is the mathematically correct average whenever data compounds, multiplies, or represents ratios — which is why the CFA Institute curriculum, GIPS performance standards, and actuarial practice all require geometric mean for multi-period investment return reporting.
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.
For the dataset 1, 5, 10, 13, 30: the product is 19,500 and there are 5 values. Geometric mean = 19,500^(1/5) = 7.21 — notably below the arithmetic mean of 11.80 because the smallest value pulls the product-root average down more sharply than it affects the additive average.
| 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 |
Percentage losses convert cleanly into positive growth factors (-10% becomes 0.90), but raw negative numbers and raw zeros require context. A raw zero collapses the entire product to zero. A 0% return, however, converts to factor 1.00 — a flat period, not a wipeout.
An 8% return becomes 1.08, a -10% return becomes 0.90, and 0% becomes 1.00.
When gains and losses compound sequentially, arithmetic average overstates actual performance. A portfolio earning 6%, 7%, 8%, -35%, and 16% has an arithmetic mean of +0.40% per year — yet the actual ending value is $923.60 on a $1,000 start. Geometric mean reports -1.58% per year, exactly reproducing that ending balance.
| Period | Start | Return | End |
|---|---|---|---|
| Year 1 | $1,000.00 | 6% | $1,060.00 |
| Year 2 | $1,060.00 | 7% | $1,134.20 |
| Year 3 | $1,134.20 | 8% | $1,224.94 |
| Year 4 | $1,224.94 | -35% | $796.21 |
| Year 5 | $796.21 | 16% | $923.60 |
CAGR is geometric mean applied between two endpoint values. Multi-period return attribution, GIPS-compliant performance reporting, and fund-versus-benchmark analysis all rely on geometric mean to avoid the volatility drag that arithmetic mean ignores.
Census-to-census population growth, GDP deflator chains, and epidemiological incidence ratios all multiply across periods. Arithmetic averaging overstates long-run growth because it misses the compounding interaction between consecutive periods.
When performance is expressed as a ratio to a benchmark — for example, a fund returning 1.12x the index — geometric mean preserves the multiplicative structure. Arithmetic mean of ratios biases the summary toward extreme outperformance periods.
The altitude of a right triangle equals the geometric mean of the two hypotenuse segments (h = sqrt(p x q)). The same relationship underlies ISO 216 paper sizing, the 16:9 aspect ratio standard, and horizon-distance formulas in surveying.
The calculator converts percentage returns into growth factors (8% becomes 1.08, -10% becomes 0.90), multiplies all valid positive values into a single product, then takes the nth root. The result is the one factor that, repeated n times, reproduces the same cumulative product — which is exactly why geometric mean outperforms arithmetic mean for any compounding dataset.
Type growth factors like 1.08, 0.97, 1.15 — or percentage returns like 8%, -10%, 11%. The calculator auto-detects the % sign and converts each return into its factor form.
All valid positive factors are multiplied into a single product, then the calculator extracts the nth root — where n equals the count of values entered.
The result panel displays geometric mean alongside arithmetic mean, product, and count — so you can immediately see the AM-GM gap and judge whether your data is multiplicative.
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.
Use geometric mean when values multiply across periods: investment returns, population growth rates, price-to-earnings ratios, and fold changes. Use arithmetic mean when values sum to a meaningful total: test scores, temperatures, and counts. Quick diagnostic: if AM is noticeably higher than GM for your dataset, the data is likely multiplicative and arithmetic mean would overstate the true center.
Selected scenario
Each step preserves the multiplicative relationship, so the final geometric mean acts as a representative single-period growth factor — not an additive summary that would distort compounded outcomes.
| Use case | Input style | Why it fits |
|---|---|---|
| Growth factors | 1.08, 1.12, 0.97, 1.15 | Population, revenue, and index changes multiply from period to period. |
| Investment returns | 1.20, 0.92, 1.06, 1.11 | Returns compound over time, so product-based averaging is more representative. |
| Ratios and indexes | 0.86, 1.04, 1.22, 1.10 | Ratios are scale-relative, making geometric mean less sensitive to one large value. |
The geometric mean is the nth root of the product of n positive values. It is useful when values multiply or compound, such as growth rates, ratios, and investment returns.
Yes. Enter returns like 8%, -10%, 11%. The calculator converts them into factors: 1.08, 0.90, 1.11 before calculating the geometric mean.
The gap shows how uneven the values are. For positive values, arithmetic mean is always greater than or equal to geometric mean.
No. The calculation runs in your browser. The values you type into the calculator are not sent to a server.
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.