Geometric Standard Deviation: Complete Guide

When to Use Geometric Standard Deviation

Geometric standard deviation (GSD) is the appropriate measure of spread for data that is multiplicative rather than additive—like growth rates, ratios, concentrations, or any log-normally distributed measurements.

Consider stock returns: a gain of 10% followed by a loss of 10% doesn't return you to break-even (you'd have 99% of original). These multiplicative relationships require geometric statistics instead of arithmetic ones.

Understanding Log-Normal Data

Data is log-normally distributed when its natural logarithm follows a normal distribution. Common examples include:

• Stock prices and investment returns over time
• Income and wealth distributions
• Particle sizes in aerosols and pharmaceuticals
• Bacterial colony counts and viral loads
• Environmental pollutant concentrations
• Antibody titers and drug concentrations

The key characteristic: processes that involve repeated multiplication generate log-normal distributions, just as repeated addition generates normal distributions.

Formula and Calculation

Or more simply: take the natural log of all values, calculate the regular standard deviation, then exponentiate.

import numpy as np
from scipy import stats

def geometric_sd(data):
    """Calculate geometric standard deviation"""
    log_data = np.log(data)
    sd_log = np.std(log_data, ddof=1)
    return np.exp(sd_log)

def geometric_mean(data):
    """Calculate geometric mean"""
    return stats.gmean(data)

# Example: Antibody titers (highly variable, log-normal)
titers = [64, 128, 256, 128, 512, 64, 256]
gm = geometric_mean(titers)
gsd = geometric_sd(titers)
print(f"Geometric Mean: {gm:.1f}")
print(f"Geometric SD: {gsd:.2f}")

Interpreting GSD Values

Unlike arithmetic SD which is in the same units as your data, GSD is a multiplicative factor—a ratio. A GSD of 2.0 means the data typically varies by a factor of 2.

• GSD = 1.0:No variation (impossible in practice)
• GSD ≈ 1.2:Low variability (±20% typical)
• GSD ≈ 2.0:Moderate variability (data doubles/halves)
• GSD ≈ 3.0:High variability (spans an order of magnitude)

Real-World Applications

GSD vs Regular SD

Using arithmetic SD on log-normal data gives misleading results: