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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 XX has a log normal distribution with parameters μ\mu and σ2\sigma^2 if Y=ln⁡(X)Y = \ln(X) is normally distributed with mean μ\mu and variance σ2\sigma^2. The support of XX is (0,∞)(0, \infty).

Key Formula

f(x)=1x σ2π exp⁡ ⁣(−(ln⁡x−μ)22σ2),x>0f(x) = \frac{1}{x\,\sigma\sqrt{2\pi}}\,\exp\!\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right), \quad x > 0
Where:
  • xx = Value of the random variable (must be positive)
  • μ\mu = Mean of the natural logarithm of X
  • σ\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)\ln(X). When you need results back in the original scale, you exponentiate. The parameters μ\mu and σ\sigma refer to the mean and standard deviation of ln⁡(X)\ln(X), not of XX 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)≈5.298\ln(200) \approx 5.298
Step 2: Compute the z-score: Use the normal distribution parameters for ln(X).
z=5.298−60.5=−0.7020.5=−1.404z = \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)≈0.0802P(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).

Related Terms