Q: What is the geometric mean?

The geometric mean of n positive numbers is the nth root of their product: GM = (x1 x x2 x ... x xn)^(1/n). It is particularly suited for data that grows multiplicatively (rates of return, growth rates, ratios). For two numbers, it is the square root of the product: GM(4, 9) = sqrt(36) = 6.

Q: When should you use the geometric mean instead of arithmetic?

The geometric mean is preferable for: financial returns over multiple periods, percentage growth rates, ratios and indices, data on different scales, and any quantity that compounds multiplicatively. The arithmetic mean overestimates the average return of volatile investments.

Q: Why is the geometric mean always less than or equal to the arithmetic mean?

This is the AM-GM inequality, a fundamental mathematical theorem. For positive numbers, the arithmetic mean is always >= the geometric mean, with equality only when all numbers are equal. The difference is larger when values are more dispersed.

Geometric Mean Calculator

Calculate the geometric, arithmetic, and harmonic mean of a series of numbers. Comparison of the three means with AM-GM inequality explanation.

How to Calculate the Geometric Mean

The geometric mean is a measure of central tendency particularly suited for data that grows multiplicatively, such as rates of return, growth rates, and ratios. This calculator computes the geometric mean, comparing it with the arithmetic and harmonic means.

Formula

For n positive numbers x1, x2, ..., xn: Geometric Mean = (x1 x x2 x ... x xn)^(1/n)

To avoid overflow problems with large numbers, the calculator uses the equivalent logarithmic formula: GM = exp(sum(ln(xi)) / n)

The Three Means Compared

For any series of positive numbers (not all equal), the AM-GM-HM inequality always holds: Harmonic Mean <= Geometric Mean <= Arithmetic Mean

Equality holds only when all values are identical. The greater the dispersion, the larger the difference between the three means.

The Geometric Mean in Finance

In finance, the geometric mean is essential for calculating the compound annual growth rate (CAGR). If an investment returns +50% the first year and -33% the second, the arithmetic average is +8.5%, but the capital actually returns to the starting point. The geometric mean correctly captures this effect.

Practical Applications

Beyond finance, the geometric mean is used in biology (population growth rates), acoustics (decibel averages), geometry (side of a square with the same area as a rectangle), photography (exposure time averages), and economics (Fisher price index).

Frequently Asked Questions

What is the geometric mean?▼

The geometric mean of n positive numbers is the nth root of their product: GM = (x1 x x2 x ... x xn)^(1/n). It is particularly suited for data that grows multiplicatively (rates of return, growth rates, ratios). For two numbers, it is the square root of the product: GM(4, 9) = sqrt(36) = 6.

When should you use the geometric mean instead of arithmetic?▼

The geometric mean is preferable for: financial returns over multiple periods, percentage growth rates, ratios and indices, data on different scales, and any quantity that compounds multiplicatively. The arithmetic mean overestimates the average return of volatile investments.

Why is the geometric mean always less than or equal to the arithmetic mean?▼

This is the AM-GM inequality, a fundamental mathematical theorem. For positive numbers, the arithmetic mean is always >= the geometric mean, with equality only when all numbers are equal. The difference is larger when values are more dispersed.