The Geometric Mean

Log-scale averaging for multiplicative data: ratios, growth rates, titres

Introduction

The geometric mean is the arithmetic mean on the log scale, back-transformed to the original scale. It is the appropriate summary for variables that combine multiplicatively rather than additively: antibody titres, concentrations, growth rates, investment returns. For log-normal data, the geometric mean is the median and is thus a natural measure of typicality.

Prerequisites

The reader should know the arithmetic mean and be comfortable with natural logarithms.

Theory

For positive values \(x_1, \ldots, x_n\):

\[\mathrm{GM}(x) = \left(\prod x_i\right)^{1/n} = \exp\!\left(\frac{1}{n} \sum \log x_i\right).\]

The second form is numerically stable and matches how the geometric mean is computed in practice. Key properties:

• \(\mathrm{GM}(x) \leq \bar{x}\), with equality only when all \(x_i\) are equal (AM-GM inequality, a consequence of Jensen).
• For log-normal data, the geometric mean equals the population median.
• The ratio of the geometric mean to the arithmetic mean is a simple measure of skewness.

Assumptions

• All values must be strictly positive; a single zero or negative value renders the geometric mean undefined.
• For inference (confidence intervals), log-normality of the data is typically assumed.

R Implementation

library(psych)

set.seed(2026)
titres <- 2^sample(4:12, 50, replace = TRUE)  # antibody titres in 2^n form

mean(titres)
median(titres)
psych::geometric.mean(titres)
exp(mean(log(titres)))

# 95% CI for the geometric mean
n <- length(titres)
log_x <- log(titres)
log_mean <- mean(log_x)
log_se   <- sd(log_x) / sqrt(n)
exp(log_mean + c(-1, 1) * qt(0.975, n - 1) * log_se)

Output & Results

[1] 1157         # arithmetic mean
[1] 512          # median
[1] 588          # geometric mean
[1] 588          # (equivalent computation)

[1] 441 782      # 95% CI for GM

The geometric mean (588) sits close to the median (512) and well below the arithmetic mean (1157). This is typical of log-normal or multiplicative data: the arithmetic mean is pulled up by the upper tail.

Interpretation

Report the geometric mean for titres, viral loads, drug concentrations, and any other multiplicatively combining variable. In a paper: “the geometric mean antibody titre was 588 (95% CI 441-782)”.

Practical Tips

• Non-positive values require imputation or exclusion; adding a small constant (e.g., 0.5) before log is common but distorts small values.
• The geometric mean of ratios is the exponent of the mean of log-ratios; reporting log-scale effects is cleaner.
• For growth rates, the geometric mean equals the arithmetic mean of \((1 + r_i)\) values minus 1.
• In meta-analysis of ratios, log transformation + normal approximation is the standard pipeline.
• When data are centred around 1 (e.g., hazard ratios), the difference between AM and GM is small.