Geometric Mean Calculator
Calculate the geometric mean and harmonic mean of a set of numbers.
Note: All values must be positive for geometric and harmonic means.
What Is the Geometric Mean Calculator?
The Geometric Mean Calculator computes the geometric mean of a set of positive numbers — the nth root of their product. The geometric mean is the appropriate average for multiplicative data such as growth rates, ratios, financial returns, and scale-invariant quantities.
Formula
How to Use
Enter a set of positive numbers separated by commas. The calculator multiplies them all together, takes the nth root, and returns the geometric mean. It also shows the arithmetic mean for comparison, and the ratio between them (always GM ≤ AM).
Example Calculation
Three growth rates: 1.10, 1.25, 0.90 (10% growth, 25% growth, 10% decline). GM = (1.10 × 1.25 × 0.90)^(1/3) = 1.2375^(1/3) = 1.0734 → average growth rate of 7.34% per period.
Understanding Geometric Mean
The geometric mean is the correct average for data that is multiplicative in nature — growth rates, ratios, and any quantities where the appropriate 'middle value' lies in the multiplicative sense rather than the additive sense. While the arithmetic mean is the appropriate average for additive quantities (like heights or temperatures), the geometric mean is used for rates and proportions.
In finance, the geometric mean of returns gives the compound annual growth rate (CAGR), which is the single constant growth rate that would produce the same final wealth as the actual series of varying returns. Using the arithmetic mean of returns overestimates actual investment performance due to the effects of volatility drag.
The geometric mean appears throughout science and engineering: it is used to compute the equivalent radius of an ellipse (GM of semi-axes), the preferred center frequency of a bandpass filter (GM of cutoff frequencies), the central measure in log-normal distributions (which model many natural phenomena), and the basis for Pythagorean means in music theory.
Frequently Asked Questions
When should I use geometric mean instead of arithmetic mean?
Use geometric mean for data that is multiplicative or ratio-based: investment returns, population growth rates, infection rates, pH values, and any data spanning multiple orders of magnitude.
Why can't I use geometric mean with zero or negative numbers?
The product of values includes zero if any value is zero, making the GM zero regardless of other values. Negative products may have no real nth root. Use a suitable transformation (e.g., add a constant) for such data.
What is the relationship between geometric and arithmetic mean?
By the AM-GM inequality, the arithmetic mean is always ≥ the geometric mean (with equality only when all values are equal). The geometric mean is always the more conservative estimate for averaging growth-type data.
How is geometric mean used in finance?
The geometric mean of (1+rₜ) across periods gives the compound annual growth rate (CAGR), the true average investment return over time, accounting for compounding effects.
Is this calculator free?
Yes, completely free with no account required.
Related Tools
Average Calculator — Mean, Median & Mode
Calculate the mean, median, and mode of a set of numbers.…
Weighted Average Calculator
Calculate the weighted average of values with different weights.…
Variance Calculator — Population & Sample
Calculate population variance and sample variance from a dataset with step-by-st…