Geometric Mean Calculator
The geometric mean is the nth root of the product of n values and is the appropriate average to use for rates, ratios, and growth rates. Unlike the arithmetic mean, which adds values and divides by the count, the geometric mean multiplies all values and takes the nth root. This calculator computes the geometric mean, arithmetic mean, and harmonic mean for comparison. The geometric mean is essential in finance (compound annual growth rate, investment returns), biology (population growth), and physics (acceleration ratios). All input values must be positive for the geometric mean to be defined.
Geometric mean formula
GM = (x1 * x2 * ... * xn)^(1/n)
or
GM = exp[(ln(x1) + ln(x2) + ... + ln(xn)) / n]
All values must be > 0
Comparing the three means
- Geometric Mean: Used for rates, ratios, and growth. Best for multiplicative data. Always ≤ arithmetic mean.
- Arithmetic Mean: Sum divided by count. Used for additive data like heights, weights, test scores.
- Harmonic Mean: Used for rates where the denominator matters (e.g., speeds, averages of fractions). Always ≤ geometric mean.
- Ordering: Harmonic Mean ≤ Geometric Mean ≤ Arithmetic Mean (for positive numbers).
- Example: If an investment grows by 50% then by 20%, the geometric mean return is 34.16%, not the arithmetic mean of 35%.
Geometric mean: frequently asked questions
What is the geometric mean?
The geometric mean is the nth root of the product of n values: GM = (x1 * x2 * ... * xn)^(1/n). It is used for data that represent rates, ratios, or growth rates, where multiplication is more meaningful than addition.
When should I use the geometric mean instead of the arithmetic mean?
Use the geometric mean for growth rates, investment returns, percentage changes, and any data where the values multiply (e.g., compound interest, population growth, performance multipliers). Use arithmetic mean for simple quantities like heights or weights.
Why must all values be positive?
The geometric mean involves taking roots of products. Negative numbers can produce complex (imaginary) roots, making the calculation undefined or meaningless in most applications. All values must be positive.
How does the geometric mean relate to logarithms?
GM = 10^(mean of log10(values)) or = e^(mean of ln(values)). Taking logarithms converts the multiplication into addition, making the geometric mean equivalent to the arithmetic mean of the log-transformed data.
Is the geometric mean always smaller than the arithmetic mean?
For positive numbers, the geometric mean is always less than or equal to the arithmetic mean (AM-GM inequality). They are equal only when all values are identical. For different values, GM < AM.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: NIST Handbook.
- American Statistical Association: ASA.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.