The log-normal distribution
A variable is log-normal when its logarithm follows a normal distribution. It is positive and skewed to the right, with a long tail of large values. Many quantities that result from multiplying many small factors are log-normal: incomes, city sizes, particle sizes, stock prices, reaction times, pollutant concentrations, and the time to repair equipment.
Parameters
The log-normal distribution is usually defined by μ and σ, the mean and standard deviation of ln X, not of X itself. The calculator also accepts the mean and SD of X directly and converts them:
σ² = ln(1 + s² ÷ m²) and μ = ln m − σ² ÷ 2
Key formulas
P(X ≤ x) = Φ((ln x − μ) ÷ σ), where Φ is the standard normal CDF
Median = e^μ, mean = e^(μ + σ²/2), mode = e^(μ − σ²)
SD = √[(e^σ² − 1) e^(2μ + σ²)]
Because of the skew, mode < median < mean.
Example
μ = 0 and σ = 0.5, for x = 1.5:
- z = ln 1.5 ÷ 0.5 ≈ 0.81, so P(X ≤ 1.5) ≈ 0.79;
- median 1, mean ≈ 1.133, mode ≈ 0.779;
- SD ≈ 0.604.
Working on the log scale
To analyze log-normal data, take the logarithm of each value, then use normal methods:
- the geometric mean e^μ estimates the median;
- the geometric standard deviation e^σ is a multiplicative spread: about 68% of values fall between median ÷ e^σ and median × e^σ.
When to suspect log-normality
- The data are positive and the histogram has a long right tail.
- The SD is large compared with the mean.
- Effects are multiplicative: a 10% change matters equally at every level.
Uses
- Finance: under the Black–Scholes model, future stock prices are log-normal.
- Environment: concentrations of pollutants or radon are often summarized with geometric means.
- Biology and medicine: incubation periods, drug concentrations and many lab values.
- Engineering: repair times and fatigue life, as an alternative to the Weibull distribution.
The normal distribution calculator handles the symmetric case; the Weibull calculator is another common choice for failure times.