Log-normal distribution

The story

The Normal comes from adding many small effects; the Log-normal comes from multiplying them. If a quantity grows by a random percentage again and again, its logarithm is a sum of many small pieces and so is roughly Normal, which makes the quantity itself Log-normal. Formally, $X = e^Y$ where $Y$ is Normal with mean $\mu$ and standard deviation $\sigma$ .

Note that $\mu$ and $\sigma$ describe the underlying Normal, not $X$ itself. The median of $X$ is $e^\mu$ , the mean is $e^{\mu + \sigma^2/2}$ and the mode is $e^{\mu - \sigma^2}$ , so in this right-skewed distribution $\text{mode} < \text{median} < \text{mean}$ . Increasing $\sigma$ stretches the right tail dramatically and pulls the mean further away from the median.

Formulas

Support

x > 0

PDF

f(x) = \frac{1}{x\sigma\sqrt{2\pi}}\, e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}}

CDF

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

Expectation

\mathbb{E}[X] = e^{\mu + \sigma^2/2}

Variance

\operatorname{Var}(X) = \left(e^{\sigma^2} - 1\right) e^{2\mu + \sigma^2}

Mode

e^{\mu - \sigma^2}

Parameters

Parameter	What it controls
$\mu$	The mean of $\ln X$ . It sets the scale: the median of $X$ is $e^\mu$ , and shifting $\mu$ by 1 multiplies every value by $e$ .
$\sigma$	The standard deviation of $\ln X$ . It controls the skew: small $\sigma$ gives a near-symmetric hump, large $\sigma$ a sharp peak near zero with a long right tail.

Where it shows up

Household incomes and personal wealth.
Share prices under the standard Black-Scholes model.
The sizes of particles produced by repeated grinding.
Times taken to complete a task, which are positive with occasional very long durations.

Try this

Set $\mu = 0$ and $\sigma = 1$ and shade from 0 to 1. The probability is exactly 0.5, because the median is $e^0 = 1$ . Notice the mean marker sitting out at $e^{0.5} \approx 1.65$ .
Hold $\mu = 0$ and move $\sigma$ from 0.1 up to 1.5. At small $\sigma$ the curve looks almost Normal; at large $\sigma$ the peak crowds towards zero while the tail and the mean run off to the right.
With $\sigma = 1.5$ , switch to Sample means and compare $n = 5$ with $n = 50$ . The averages stay visibly skewed for longer than with most distributions: the Central Limit Theorem always wins, but heavy skew slows it down.

Related distributions

If $X$ is $\text{Log-normal}(\mu, \sigma)$ , then $\ln X$ is Normal with mean $\mu$ and standard deviation $\sigma$ .
Products of independent Log-normals are Log-normal, just as sums of independent Normals are Normal.
Like the Gamma it is positive and right-skewed, but its right tail is heavier.