Geometric Mean CalculatorFree online math calculator — instant results, no sign-up
Calculate the geometric mean of positive values.
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About the Geometric Mean Calculator
The Geometric Mean Calculator finds the multiplicative average of a list of positive values by taking the nth root of their product. For the default values 2, 4, 6, 8, 10, it returns a geometric mean of 5.2103421694. This statistic is useful when ratios or proportional differences matter more than additive differences.
- Values
- Enter the positive numbers to average, using consistent units for measurements or no units for ratios; the default list is 2, 4, 6, 8, 10.
Formula and method
G = (x₁ × x₂ × … × xₙ)^(1/n) = exp((1/n) × Σᵢ₌₁ⁿ ln(xᵢ)); for the default values, G = (2 × 4 × 6 × 8 × 10)^(1/5) = 3840^(1/5) ≈ 5.2103421694.
- GThe geometric mean of the entered values.
- xᵢThe positive value at position i in the list.
- nThe total number of entered values.
- iThe index identifying each value, from 1 through n.
- ΣSummation over all entered values.
- lnThe natural logarithm.
- expThe exponential function, which reverses the natural logarithm.
Count the values, multiply them together, and take the root corresponding to that count. For 2, 4, 6, 8, 10, there are five values and their product is 3840; the fifth root is approximately 5.2103421694. An equivalent computational approach averages the natural logarithms of the values and then exponentiates that average, avoiding the potentially very large or small intermediate product.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Values2, 4, 6, 8, 10
Results
- Geometric mean5.2103421694
How to use the Geometric Mean Calculator
- 1Enter your list in Values, or keep the default list 2, 4, 6, 8, 10.
- 2Check that every entry is strictly positive and that measurements use consistent units.
- 3Run the calculation to obtain the geometric mean.
- 4Read the Geometric mean result; with the default inputs, it is 5.2103421694.
Reading your result
The Geometric mean output is the single positive value that, multiplied by itself once for each input, reproduces the product of the inputs before rounding. For the default list, 5.2103421694 is approximately the number whose fifth power equals 3840. If all inputs are measurements in the same unit, the result has that unit; if they are dimensionless ratios, the result is dimensionless.
Limitations
This tool assumes a nonempty list of strictly positive values and gives every entry equal weight. Zero and negative values are outside its supported domain. Use it for data with a meaningful ratio scale, not quantities whose zero point is arbitrary, such as Celsius temperatures. Raw percentage changes must be converted to positive growth factors before calculating a compounded average. Displayed results are rounded, and finite numerical precision can affect extreme inputs.
Common questions
Why is the geometric mean of 2, 4, 6, 8, 10 lower than their arithmetic mean?+
Their arithmetic mean is 6, while their geometric mean is approximately 5.2103421694. For positive values, the geometric mean never exceeds the arithmetic mean; they are equal only when all entries are identical.
Can I enter zero or negative values?+
No. This calculator supports strictly positive inputs. Although a product-based extension can assign a geometric mean of zero to some nonnegative lists, that is outside this tool's stated domain, and logarithms of zero or negative values are not defined as finite real numbers.
How do I use the geometric mean for percentage growth rates?+
Convert each percentage rate r to a growth factor using 1 + r/100, then enter those factors. Convert the resulting geometric mean G back to a percentage with (G − 1) × 100%. For an average per-period growth rate, each factor should cover an equal-length period and must be positive.
Does this calculator support a weighted geometric mean?+
No separate weights are provided: each entered value contributes equally. A weighted geometric mean requires averaging the logarithms with specified weights rather than assigning every entry the same weight.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.