Lognormal Distribution Calculator
A quantity is lognormal when its logarithm is normally distributed, which happens whenever a value is the product of many small independent factors. Enter mu and sigma (the mean and standard deviation on the log scale) and an evaluation point x. The calculator returns the probability density at x, the cumulative probability that a draw is at most x, and the mean and variance of the variable itself. It models income, stock prices, particle sizes, and biological measurements. The cumulative function is computed with a high-accuracy error-function approximation.
Lognormal distribution formulas
f(x) = e^(-(ln x - mu)^2 / (2 sigma^2)) / (x * sigma * sqrt(2 pi)) for x > 0
F(x) = 0.5 * [1 + erf((ln x - mu) / (sigma * sqrt(2)))] for x > 0
Mean = e^(mu + sigma^2 / 2)
Variance = (e^(sigma^2) - 1) * e^(2 mu + sigma^2)
The median of the variable is e^mu. Because mu and sigma describe the log of the variable, the mean of the variable itself is always larger than its median.
Worked example and notes
- With mu = 0, sigma = 0.5, the median is e^0 = 1.
- The mean of the variable is e^(0.125), about 1.1331.
- At x = 1 the cumulative probability is 0.50, since x equals the median.
- The density at x = 1 is about 0.7979.
- Larger sigma produces a longer right tail and a larger gap between mean and median.
Lognormal distribution: frequently asked questions
What is the lognormal distribution?
A positive random variable is lognormally distributed if its natural logarithm follows a normal distribution. The parameters mu and sigma are the mean and standard deviation of the log of the variable, not of the variable itself. It models quantities that result from the product of many small positive factors.
What are the density and cumulative function?
For x greater than 0 the density is exp(minus (ln x minus mu) squared over 2 sigma squared) divided by (x sigma times the square root of 2 pi). The cumulative function is the standard normal CDF evaluated at (ln x minus mu) over sigma, computed here with the error function.
How do I get the mean of the variable itself?
The mean of the lognormal variable is exp(mu plus sigma squared over 2). The median is exp(mu) and the mode is exp(mu minus sigma squared). The mean exceeds the median because the distribution is right-skewed.
What is the variance of a lognormal variable?
The variance is (exp(sigma squared) minus 1) times exp(2 mu plus sigma squared). Because the distribution has a long right tail, the variance grows rapidly as sigma increases.
What are valid inputs?
The log-scale standard deviation sigma must be strictly positive; mu can be any real number. The evaluation point x can be any real number, but the density and cumulative value are 0 for x at or below 0, since the lognormal is supported on the positive reals.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Lognormal distribution.
- NIST Digital Library of Mathematical Functions: Error functions.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.