Log-Normal Distribution Calculator
Compute the PDF, CDF, mean, and variance of a log-normal distribution.
Enter the log-scale mean and standard deviation to evaluate at any positive x.
The log-normal distribution describes a variable X whose natural logarithm is normally distributed. If ln(X) follows Normal(μ, σ), then X follows LogNormal(μ, σ). Every log-normal value is strictly positive, which makes it useful whenever you are modeling something that cannot go below zero.
The PDF is f(x) = exp(−(ln(x) − μ)² / (2σ²)) / (x · σ · √(2π)) for x > 0.
The CDF is F(x) = Φ((ln(x) − μ) / σ), where Φ is the standard normal CDF.
Where log-normal comes from. Just as the Central Limit Theorem says the sum of many independent random factors tends toward a normal distribution, the product of many independent positive random factors tends toward a log-normal distribution. Take the log of a product and you get a sum of logs, which goes normal. Exponentiate back, and you get log-normal. That is why so many natural quantities follow this shape: anything that compounds multiplicatively (stock returns, biological growth, particle aggregation, network latencies) ends up log-normal.
Mean is not μ. Despite being called the log-scale mean, μ is not the mean of X. The actual mean is exp(μ + σ²/2), which is always larger than exp(μ). The median of X is exp(μ). The mode is exp(μ − σ²). The variance is (exp(σ²) − 1) · exp(2μ + σ²). The gap between mean and median is exactly why log-normal looks asymmetric: it has a long right tail. With larger σ, the distribution spreads out more and the mean pulls further above the median.
Where it shows up:
| Field | Use |
|---|---|
| Finance | Stock prices; Black-Scholes option pricing assumes log-normal returns |
| Biology | Organism sizes, bacterial colony counts, disease latency periods |
| Engineering | Fatigue life of metal parts, particle size in grinding/aerosols |
| Earth science | Rainfall amounts, earthquake intensities, pollutant concentrations |
| Social science | Income, city population sizes, file sizes on disk |
| Computing | Web response times, request latency tails |
A useful rule of thumb: if your data is always positive, always has a long right tail, and looks roughly normal after you take the log, log-normal is probably the right model.
Parameter estimation. μ and σ are the mean and standard deviation of the log-transformed data, not of the raw data. When fitting a log-normal, take ln of your sample, then compute the ordinary sample mean and standard deviation of those values. Plugging in the raw mean and standard deviation directly is the most common mistake.
Relationship to other distributions. The log-normal is closely related to the normal: any probability or quantile calculation for a normal variable translates by taking exp() of the result. It is also related to the Pareto distribution in the tail. Both are used to model wealth distributions, but log-normal models the body of the distribution while Pareto better fits the extreme right tail.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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