Geometric Mean Calculator
Calculate geometric mean with multiple methods and step-by-step solutions
Calculate Geometric Mean
You can enter up to 30 numbers. Geometric mean requires positive numbers only.
Example Calculation
Basic Example: GM of 4 and 9
Numbers: 4, 9
Product: 4 × 9 = 36
Geometric Mean: √36 = 6
Verification: √(4 × 9) = √36 = 6
Investment Growth Example
Annual Growth Rates: 10%, 20%, -5% (converted to 1.10, 1.20, 0.95)
Product: 1.10 × 1.20 × 0.95 = 1.254
Geometric Mean: ∛1.254 ≈ 1.0784
Average Growth Rate: 7.84% per year
Calculation Methods
Direct Method
GM = (x₁ × x₂ × ... × xₙ)^(1/n)
Most intuitive approach
Logarithmic Method
GM = antilog(Σlog(xᵢ)/n)
Prevents overflow for large numbers
Formula Method
Uses mathematical notation
Shows symbolic representation
Common Applications
Finance
Average growth rates, compound returns
Geometry
Rectangle side from area, proportions
Science
Signal processing, bacterial growth
Statistics
Central tendency for ratio data
Key Properties
Only defined for positive numbers
Always between min and max values
HM ≤ GM ≤ AM (AM-GM inequality)
Multiplicative property: GM(ax₁, ax₂, ...) = a × GM(x₁, x₂, ...)
Understanding Geometric Mean
What is Geometric Mean?
The geometric mean is a type of average that's calculated by multiplying all values together and then taking the nth root (where n is the count of values). It's particularly useful when dealing with values that represent rates, percentages, or ratios.
When to Use Geometric Mean?
- •When data values represent rates of growth or change
- •For averaging ratios or percentages
- •When values span several orders of magnitude
- •In finance for calculating average returns
Mathematical Formula
GM = √[n]{x₁ × x₂ × ... × xₙ}
= (x₁ × x₂ × ... × xₙ)^(1/n)
Relationship with Other Means
Logarithmic relationship:
log(GM) = (log(x₁) + log(x₂) + ... + log(xₙ))/n
This means the log of the geometric mean equals the arithmetic mean of the logs.
Geometric Mean vs Other Means
| Mean Type | Formula | Best Used For |
|---|---|---|
| Arithmetic | (x₁ + x₂ + ... + xₙ)/n | General averaging, linear data |
| Geometric | (x₁ × x₂ × ... × xₙ)^(1/n) | Growth rates, ratios, proportions |
| Harmonic | n/(1/x₁ + 1/x₂ + ... + 1/xₙ) | Rates, speeds, unit rates |