Log-Normal Distribution Formula
The log-normal distribution models quantities whose logarithm is normally distributed.
Common in finance, biology, and environmental science.
The Formula
Mean = e^(μ + σ²/2)
Variance = (e^(σ²) − 1) × e^(2μ + σ²)
A random variable X follows a log-normal distribution if ln(X) follows a normal distribution. This arises naturally whenever a quantity is the product of many independent random factors — just as the normal distribution arises from sums. Log-normal distributions are always positive and right-skewed, making them perfect for modeling quantities that cannot be negative.
Variables
| Symbol | Meaning | Note |
|---|---|---|
| x | Random variable (x > 0) | Must be positive |
| μ | Mean of the underlying normal distribution (of ln X) | Can be any real number |
| σ | Standard deviation of the underlying normal distribution | σ > 0 |
| Mean of X | e^(μ + σ²/2) | Always > median = e^μ |
| Median of X | e^μ | More robust than mean for skewed data |
Example 1 — Stock Price Modeling
A stock has log-returns with μ = 0.001 per day and σ = 0.02 per day. What is the expected price after 252 trading days if today's price is $100?
After 252 days: μtotal = 252 × 0.001 = 0.252, σtotal = σ × √252 = 0.02 × 15.87 = 0.317
Expected price = 100 × e^(μ + σ²/2) = 100 × e^(0.252 + 0.050) = 100 × e^0.302
Expected price = 100 × 1.353 = $135.30 after one year
Example 2 — Income Distribution
Incomes in a country are log-normally distributed with μ = 10.5 (ln-dollars) and σ = 0.9. Find the median and mean income.
Median = e^μ = e^10.5 = $36,315
Mean = e^(μ + σ²/2) = e^(10.5 + 0.405) = e^10.905
Mean = $54,176 — the mean exceeds the median because the right tail (very high incomes) pulls the mean up. This is typical of income distributions.
When to Use It
The log-normal distribution is the right choice when:
- Finance: Stock prices, option pricing (Black-Scholes model assumes log-normal returns)
- Biology: Organism sizes, bacteria colony sizes, latency periods of infectious diseases
- Environmental science: Rainfall amounts, concentrations of pollutants
- Engineering: Fatigue life of metal components, particle size distributions
- Social science: Income distribution, city population sizes
- Any quantity that must be positive and results from multiplicative random processes
If your data has a heavy right tail and is always positive, the log-normal distribution is often a better fit than the normal distribution. A quick test: if the data looks roughly normal after taking the logarithm, log-normal is appropriate.