Log Normal Distribution — Definition, Formula & Examples

A log normal distribution describes a positive random variable whose natural logarithm follows a normal distribution. It produces a right-skewed curve and commonly models quantities like income, stock prices, and biological measurements that cannot be negative.

A continuous random variable $X$ has a log normal distribution with parameters $\mu$ and $\sigma^2$ if $Y = \ln(X)$ is normally distributed with mean $\mu$ and variance $\sigma^2$ . The support of $X$ is $(0, \infty)$ .

Key Formula

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

Where:

$x$ = Value of the random variable (must be positive)
$\mu$ = Mean of the natural logarithm of X
$\sigma$ = Standard deviation of the natural logarithm of X

How It Works

To work with a log normal distribution, you transform the variable by taking its natural logarithm, which converts it into a normal distribution. You can then apply all the familiar normal distribution techniques — z-scores, probability tables, confidence intervals — to

\ln(X)

. When you need results back in the original scale, you exponentiate. The parameters

\mu

and

\sigma

refer to the mean and standard deviation of

\ln(X)

, not of

X

itself.

Worked Example

Problem: The natural log of a company's daily revenue follows a normal distribution with μ = 6 and σ = 0.5. Find the probability that revenue on a given day is less than $200.

Step 1: Transform to log scale: Take the natural log of the threshold value.

\ln(200) \approx 5.298

Step 2: Compute the z-score: Use the normal distribution parameters for ln(X).

z = \frac{5.298 - 6}{0.5} = \frac{-0.702}{0.5} = -1.404

Step 3: Look up the probability: Using a standard normal table or calculator, find P(Z < −1.404).

P(X < 200) = P(Z < -1.404) \approx 0.0802

Answer: There is approximately an 8.0% chance that daily revenue falls below $200.

Why It Matters

Log normal distributions appear throughout finance, biology, and engineering. Stock prices are often modeled as log normal in the Black-Scholes option pricing model. Environmental scientists use them to model pollutant concentrations, where values are strictly positive and heavily right-skewed.

Common Mistakes

Mistake: Interpreting μ and σ as the mean and standard deviation of X itself.

Correction: These parameters describe ln(X), not X. The actual mean of X is exp(μ + σ²/2), and its variance is [exp(σ²) − 1]·exp(2μ + σ²).