The geometric mean is the nth root of the product of n values. It’s particularly useful for data representing rates, growth rates, and proportional changes.

When to Use Geometric Mean

  • Growth rates: Average annual growth rates
  • Ratios: Investment returns over multiple periods
  • Proportional data: Data where values are multiplied, not added
  • Percentage changes: Average rate of increase/decrease

Advantage over arithmetic mean: Less affected by extreme values in certain contexts.

Geometric Mean for Ungrouped Data

Formula:

GM = ⁿ√(x₁ × x₂ × ... × xₙ)
GM = (x₁ × x₂ × ... × xₙ)^(1/n)

Or using logarithms (practical):

GM = 10^(Σ log(xᵢ) / n)  or  e^(Σ ln(xᵢ) / n)

Example: Investment returns over 3 years: 10%, 15%, 8%

GM = ³√(1.10 × 1.15 × 1.08) GM = ³√(1.3662) GM ≈ 1.1096 or 10.96% average annual return

Geometric Mean for Grouped Data

Formula:

GM = ⁿ√(x₁^f₁ × x₂^f₂ × ... × xₖ^fₖ)

where:
xᵢ = class midpoint
fᵢ = frequency
n = Σfᵢ (total frequency)

Using logarithms (practical):

GM = e^(Σ(fᵢ × ln(xᵢ)) / Σfᵢ)

Key Characteristics

  • Always less than or equal to arithmetic mean (AM)
  • Useful for normalized/indexed data
  • Cannot be used with zero or negative values (for logarithm method)
  • Represents the “middle” rate of change

For positive values:

Harmonic Mean ≤ Geometric Mean ≤ Arithmetic Mean

HM ≤ GM ≤ AM