Lognormal Distribution Calculator

No. Mu and sigma are the mean and standard deviation of the natural logarithm of the random variable, ln(X), not of X itself. μ acts as the location parameter on the log scale, and e^μ is the scale parameter of X—its median—while σ sets the shape. The mean of X is e^μ+σ²/2, and its standard deviation is e^μ+σ²/2·√(e^σ² − 1). Use the Mean & SD of X option if those are the numbers you have.

Use the cumulative distribution function formula F(x) = Φ((ln x − μ) ÷ σ). Compute z = (ln x − μ) ÷ σ and look up Φ(z) in a standard normal distribution table. That is P(X ≤ x). For the right tail use 1 − Φ(z), and for an interval subtract the two CDF values: Φ(z_b) − Φ(z_a). The calculator follows the same steps and lists them next to the output, so you can check your own work.

The long right tail contains a few very large values that pull the average—the expected value—upward, while the median only depends on the middle observation. The ratio of mean to median is e^σ²/2, so the gap grows quickly as σ increases. Variance, skewness, and kurtosis rise even faster with σ, which is why a lognormal model with a large σ produces occasional extreme values.

No. The density is zero for x ≤ 0. If your dataset contains zeros, consider a shifted (three-parameter) lognormal, a mixture model with a separate probability of zero, or a different distribution such as the gamma. Negative values usually mean the quantity is not a good fit for a lognormal model at all.

The geometric mean is e^μ, identical to the median. The geometric standard deviation is e^σ, a unitless multiplier: about 68% of values lie between median ÷ e^σ and median × e^σ—roughly the 16th to 84th percentile—and about 95% within a factor of e^2σ.

The calculator draws standard normal values z with the Box–Muller method and returns e^μ+σz. Each click of New sample produces a fresh draw, so small samples will vary noticeably around the theoretical mean and median. That is ordinary sampling variation in statistics: the sample mean and median only settle close to the population values as n grows.

Not directly. For a continuous distribution the probability of any exact value is zero; f(x) is a density per unit of x and can be greater than 1. Probabilities come from areas under the curve, which is what the Probability mode calculates. To go the other way—from a probability to a value—use Quantile mode, which inverts the CDF.

The normal CDF uses a complementary error function approximation accurate to about 1 part in 10 million, and the inverse uses Acklam's rational approximation with a relative error near 10⁻⁹. That is far more precise than the uncertainty in input parameters estimated from real data. When μ and σ come from a small sample, a confidence interval for each parameter at your chosen confidence level says more about the risk of a wrong answer than extra decimal places do.