Lognormal Distribution

NIST/SEMATECH Section 1.3.6.6.9 Lognormal Distribution

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\mu

: 0.00-22

\sigma

: 1.000.13

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Formulas

Probability Density Function

f(x) = \frac{1}{x\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right), \quad x > 0

Cumulative Distribution Function

F(x) = \frac{1}{2}\left[1 + \operatorname{erf}\!\left(\frac{\ln x - \mu}{\sigma\sqrt{2}}\right)\right]

Properties

Mean

e^{\mu + \sigma^2/2}

Variance

(e^{\sigma^2} - 1)\,e^{2\mu + \sigma^2}

Overview

The lognormal distribution describes a random variable $X$ whose logarithm $\ln X$ is normally distributed with parameters $\mu$ and $\sigma$ . It is commonly used to model positive-valued data with right skew, such as income, stock prices, and particle sizes.

Related Distributions

Normal Distribution Weibull Distribution Gamma Distribution