Geometric Mean Calculator

Calculate the geometric mean of multiple values. Compare with arithmetic and harmonic means. Step-by-step solutions included.

Enter Your Values

Enter positive numbers separated by commas, spaces, or newlines.

Result

Geometric Mean
5.656854
Arithmetic Mean
7.5
Harmonic Mean
4.266667
Number of Values (n) 4
Product of All Values 1,024
Sum of All Values 30
Min Value 2
Max Value 16

Step-by-Step Solution

GM = (2 x 8 x 4 x 16)^(1/4) = 1024^0.25 = 5.656854

What Is the Geometric Mean?

The geometric mean is a type of average that indicates the central tendency of a set of numbers by using the product of their values rather than their sum. For n positive numbers, the geometric mean is the nth root of their product. It is particularly useful when comparing values that have different units or scales, or when dealing with rates of change and ratios.

Formulas and Comparisons

Geometric Mean

The nth root of the product of n values.

GM = (a1 * a2 * ... * an)^(1/n)

Arithmetic Mean

The sum of values divided by the count.

AM = (a1 + a2 + ... + an) / n

Harmonic Mean

The reciprocal of the arithmetic mean of reciprocals.

HM = n / (1/a1 + 1/a2 + ... + 1/an)

Mean Inequality

For positive numbers, the means are always ordered.

HM ≤ GM ≤ AM

When to Use the Geometric Mean

The geometric mean is the preferred average in many real-world situations. It is particularly appropriate when:

  • Growth rates: Calculating average investment returns, population growth rates, or compound interest.
  • Ratios and proportions: When values represent ratios or percentages.
  • Log-normal distributions: Data that is positively skewed, such as income distributions.
  • Normalized data: Combining different metrics into a single score (e.g., HDI).

Geometric Mean vs Arithmetic Mean

The arithmetic mean can be misleading when dealing with growth rates or multiplicative processes. For example, if an investment gains 100% one year and loses 50% the next, the arithmetic mean return is 25%, but the actual net gain is 0%. The geometric mean correctly gives 0% average return in this case.

Properties of the Geometric Mean

  • The geometric mean is always less than or equal to the arithmetic mean (AM-GM inequality).
  • It is only defined for positive numbers (or an even number of negative values).
  • If any value is zero, the geometric mean is zero.
  • The logarithm of the geometric mean equals the arithmetic mean of the logarithms.
  • It is more robust to outliers than the arithmetic mean.

Example Applications

Finance: A stock returns +10%, -5%, +20%, +15% over four years. The average annual return is the geometric mean of 1.10, 0.95, 1.20, and 1.15, which gives approximately 9.7% per year, accurately reflecting compounding.

Science: The geometric mean is used to average pH values, decibel levels, and other logarithmic scales where the arithmetic mean would be inappropriate.