Calculate Geometric Mean

Online calculator to calculate the geometric mean of a data series

Geometric Mean Calculator

Geometric Mean

The geometric mean is the n-th root of the product of n positive numbers. It is particularly suitable for growth rates and ratios.

Enter Data
Positive data values (separated by spaces or semicolons)
Result
Geometric Mean:
Properties of Geometric Mean

Important: x̄geom ≤ x̄arith (always less than or equal to arithmetic mean). Only defined for positive numbers.

Multiplicative For Growth Rates Positive Values Only

Geometric Mean Concept

The geometric mean is the n-th root of the product.
Ideal for relative changes and ratios.

Product → n-th Root 2 3 4 5 6 × × × × Product 720 ⁵√ 3.7279

■ Input Values ■ Product ■ Geometric Mean

What is the Geometric Mean?

The geometric mean is a specialized measure of position with important applications:

  • Definition: n-th root of the product of n positive numbers
  • Calculation: ⁿ√(x₁ · x₂ · ... · xₙ)
  • Requirement: All values must be positive (> 0)
  • Property: Always ≤ arithmetic mean
  • Application: Ideal for growth rates, returns, ratios
  • Advantage: Considers multiplicative relationships

Calculating the Geometric Mean

Calculation is done in three steps:

1. Multiply

Form the product of all values:
P = x₁ · x₂ · ... · xₙ

2. Count

Determine the count:
n = Number of values

3. Extract Root

Take the n-th root:
x̄geom = ⁿ√P

Applications of Geometric Mean

The geometric mean is particularly important for:

Finance
  • Calculate average growth rates
  • Average returns over multiple periods
  • Compound interest calculations
  • Investment performance measurement
Science
  • Biology: Population growth
  • Physics: Velocities and ratios
  • Geometry: Mean proportional
  • Statistics: Log-normal distribution

Formulas for Geometric Mean

Geometric Mean
\[\overline{x}_{geom} = \sqrt[n]{\prod_{i=1}^{n} x_i}\]

n-th root of the product of all values

Explicit Form
\[\overline{x}_{geom} = \sqrt[n]{x_1 \cdot x_2 \cdot ... \cdot x_n}\]

Explicit representation with multiplication

Logarithmic Form
\[\overline{x}_{geom} = \exp\left(\frac{1}{n}\sum_{i=1}^{n}\ln(x_i)\right)\]

Practical for numerical calculations of large values

For Two Values
\[\overline{x}_{geom} = \sqrt{x_1 \cdot x_2}\]

Special case: Mean proportional of two numbers

Symbol Explanations
\(\overline{x}_{geom}\) Geometric mean
\(\prod\) Product sign
\(x_i\) Individual data value
\(n\) Number of values
\(\sqrt[n]{}\) n-th root
\(\ln\) Natural logarithm

Example Calculations for Geometric Mean

Example 1: Basic Calculation
Data: 2, 3, 4, 5, 6

Calculate: Geometric mean of 5 values

1. Form Product
\[P = 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6\] \[= \color{blue}{720}\]

Multiply all values

2. Determine Count
\[n = 5\]

Count the data values

3. Extract Root
\[\overline{x}_{geom} = \sqrt[5]{720}\] \[\approx \color{blue}{3.7279}\]

5-th root of the product

Example 2: Average Growth Rate
Growth factors: 1.1, 1.2, 0.9, 1.15

Calculate: Average growth factor over 4 years

Calculation
\[\overline{x}_{geom} = \sqrt[4]{1.1 \cdot 1.2 \cdot 0.9 \cdot 1.15}\] \[= \sqrt[4]{1.3662} \approx \color{blue}{1.0811}\]

Geometric mean of the factors

Interpretation

Average growth: 8.11% per year
(Factor 1.0811 corresponds to +8.11%)

Why not arithmetic?
The arithmetic mean would give 1.0875, but that doesn't match the actual final value!

Example 3: Comparison Arithmetic vs. Geometric Mean
Data: 1, 2, 8
Arithmetic Mean
\[\overline{x}_{arith} = \frac{1+2+8}{3}\] \[= \frac{11}{3} \approx \color{blue}{3.667}\]

Sum divided by count

Geometric Mean
\[\overline{x}_{geom} = \sqrt[3]{1 \cdot 2 \cdot 8}\] \[= \sqrt[3]{16} \approx \color{blue}{2.520}\]

3-rd root of the product

Important Insight

x̄geom ≤ x̄arith: The geometric mean (2.520) is always less than or equal to the arithmetic mean (3.667).
Equality holds only when all values are identical.
Difference increases with greater dispersion of values.

Mathematical Foundations of Geometric Mean

The geometric mean is an important measure of position with special properties, used especially in multiplicative relationships.

Properties of Geometric Mean

The geometric mean has characteristic mathematical properties:

  • Inequality: x̄geom ≤ x̄arith (Inequality of arithmetic and geometric means)
  • Positive values only: Defined only for xᵢ > 0, otherwise product = 0 or undefined
  • Multiplicative Invariance: Scaling all values by factor c: x̄geom(c·X) = c · x̄geom(X)
  • Logarithmic Property: ln(x̄geom) = (1/n)Σln(xᵢ) - arithmetic mean of logarithms
  • Robustness: Less sensitive to large values than arithmetic mean

When Geometric vs. Arithmetic Mean?

Use Geometric Mean
  • Growth rates: Average growth over multiple periods
  • Returns: Average investment return
  • Ratios: Average of quotients or factors
  • Multiplicative Processes: When values are multiplied together
  • Logarithmic Scales: With exponential relationships
Use Arithmetic Mean
  • Additive Processes: Sums of independent quantities
  • Linear Relationships: When values are added
  • Symmetric Distributions: With normally distributed data
  • Average Values: Typical everyday calculations
  • Statistical Tests: Basis for many parametric tests

Practical Example: Investment Return

Problem:

An investment develops over 3 years as follows:
Year 1: +20% (factor 1.20)
Year 2: -10% (factor 0.90)
Year 3: +15% (factor 1.15)

Wrong Method (Arithmetic):

x̄arith = (1.20 + 0.90 + 1.15) / 3 = 1.0833 → +8.33% per year
But: 100€ · 1.20 · 0.90 · 1.15 = 124.20€ ≠ 100€ · 1.0833³ = 127.21€

Correct Method (Geometric):

x̄geom = ³√(1.20 · 0.90 · 1.15) = ³√1.242 = 1.0749 → +7.49% per year
Correct: 100€ · 1.20 · 0.90 · 1.15 = 124.20€ = 100€ · 1.0749³ = 124.20€ ✓

Relationship to Other Means

Harmonic Mean

x̄harm ≤ x̄geom ≤ x̄arith
For velocities and rates

Geometric Mean

Middle position
For growth rates

Arithmetic Mean

Largest value
For additive quantities

Special Applications

Geometry
  • Mean Proportional: For a and b, x̄geom = √(a·b) is the mean proportional
  • Similar Triangles: Altitude theorem and leg theorem
  • Rectangle-Square: Area-equivalent square to rectangle
Statistics
  • Log-Normal Distribution: When ln(X) is normally distributed
  • Index Numbers: Average price and quantity indices
  • Concentration Measurement: Geometric mean in inequality measures
Summary

The geometric mean is the correct measure of position for multiplicative relationships such as growth rates, returns, and ratios. It is always less than or equal to the arithmetic mean and only defined for positive values. In finance, it is essential for correctly calculating average returns over multiple periods. The logarithmic representation makes it numerically stable even with very large or very small values.

More statistics functions

Arithmetic Mean • Contraharmonic Mean • Covariance • Empirical distribution CDF • Deviation • Five-Number Summary • Geometric Mean • Harmonic Mean • Inverse Empirical distribution CDF • Kurtosis • Log Geometric Mean • Lower Quartile • Median • Pooled Standard Deviation • Pooled Variance • Skewness (Statistische Schiefe) • Upper Quartile • Variance