Log-Normal Distribution Calculator
The log-normal distribution models positive quantities whose logarithm is normally distributed, which is common for incomes, biological measurements, particle sizes and many multiplicative processes. This calculator takes the two parameters mu and sigma of the underlying normal distribution and a value x, then returns the probability density at x, the cumulative probability up to x, and the distribution's mean, median and variance. The cumulative function is computed with the error function and the moments use their exact closed forms, so the results are precise rather than estimated.
Log-normal distribution formula
PDF f(x) = 1 / (x * sigma * sqrt(2*pi)) * exp(-(ln x - mu)^2 / (2*sigma^2))
CDF F(x) = 0.5 * (1 + erf((ln x - mu) / (sigma * sqrt(2))))
Median = exp(mu)
Mean = exp(mu + sigma^2 / 2)
Variance = (exp(sigma^2) - 1) * exp(2*mu + sigma^2)
The value x must be positive and sigma must be greater than zero.
Log-normal context
- mu and sigma are the mean and standard deviation of the natural log of the variable, not of the variable itself.
- The mean always exceeds the median, reflecting the right skew of the distribution.
- Common applications include incomes, file sizes, particle sizes and survival times.
- If ln x is normal, then x is log-normal; the two are linked by the exponential transformation.
- For sigma near zero the distribution is nearly symmetric around exp(mu).
Log-normal distribution: frequently asked questions
What is the log-normal distribution?
A positive random variable is log-normally distributed if its natural logarithm is normally distributed. It is described by two parameters: mu, the mean of the log values, and sigma, the standard deviation of the log values. It models quantities that are products of many positive factors, such as incomes and particle sizes.
How are the PDF and CDF computed here?
The CDF at x is 0.5 times one plus the error function of (ln x minus mu) divided by sigma times the square root of two. The PDF is one over (x times sigma times the square root of two pi) times the exponential of minus (ln x minus mu) squared over two sigma squared. Both require x to be positive.
What are the mean, median and variance?
The median is e to the mu. The mean is e to the power (mu plus half sigma squared), which is always greater than the median. The variance is (e to the sigma squared minus one) times e to the power (two mu plus sigma squared). The calculator reports all of these.
Are mu and sigma the same as the data's mean and standard deviation?
No. Mu and sigma are the mean and standard deviation of the logarithm of the variable, not of the variable itself. If you have the mean and variance of the raw data, you must convert them to mu and sigma before using them here.
Why is the log-normal distribution skewed to the right?
Because the logarithm is symmetric but the exponential transformation stretches large values and compresses small ones, the resulting distribution has a long right tail. This is why the mean exceeds the median, a hallmark of right-skewed data like income.
Official sources
- U.S. NIST/SEMATECH e-Handbook of Statistical Methods: Engineering Statistics Handbook, log-normal distribution.
- NIST Digital Library of Mathematical Functions: Error Functions.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.