Log-Normal Distribution

What it is

A log-normal distribution describes a positive-valued variable whose natural logarithm is normally distributed. If Y is normally distributed with mean μ and variance σ² (Y ~ N(μ, σ²)), then X = e^Y follows a log-normal distribution. Log-normal variables cannot be negative and typically exhibit positive skew (a long right tail).

Why it matters

Log-normal models are useful when multiplicative effects or compounding drive outcomes. They are widely used in finance (e.g., stock prices), biology, engineering, and environmental studies where values grow proportionally and must remain positive.

Relationship to the normal distribution

• Taking the natural log of a log-normal variable yields a normal distribution.
• Conversely, exponentiating a normal variable produces a log-normal variable.
• Normal distributions are symmetric and allow negative values; log-normal distributions are asymmetric and strictly positive.

Key properties and formulas

For X ~ LogNormal(μ, σ²) where μ and σ refer to the mean and standard deviation of ln(X):

• Support: X > 0
• Median: median(X) = e^μ
• Mean: E[X] = e^{μ + σ²/2}
• Mode: mode(X) = e^{μ – σ²}
• Variance: Var(X) = (e^{σ²} − 1) e^{2μ + σ²}

Probability density function:
f(x) = (1 / (x σ √(2π))) exp(−(ln x − μ)² / (2σ²)), for x > 0

Cumulative distribution function:
F(x) = Φ((ln x − μ) / σ), where Φ is the standard normal CDF.

Applications in finance

• Stock prices are often modeled as log-normal because returns (log returns) are approximately normal, while prices must remain positive.
• Expected compound returns and price distributions over time are naturally represented with log-normal assumptions.
• The log-normal assumption implies skewness and a long right tail, meaning large positive outcomes are possible but rare.

Using Excel

Excel provides the LOGNORM.DIST function to work with log-normal distributions:
LOGNORM.DIST(x, mean, standard_dev, cumulative)

• x: value at which to evaluate the function (must be > 0)
• mean: mean of ln(x) (μ)
• standard_dev: standard deviation of ln(x) (σ), must be > 0
• cumulative: TRUE to return the CDF, FALSE to return the PDF

Example:
– PDF at x: =LOGNORM.DIST(x, μ, σ, FALSE)
– CDF at x: =LOGNORM.DIST(x, μ, σ, TRUE)

Practical notes

• Check whether modeling the raw variable or its log is more appropriate. If multiplicative shocks or compound growth are present, log-normal is often better than normal.
• Parameter estimates for μ and σ are typically obtained by taking the natural log of observed positive samples and computing the sample mean and standard deviation.
• Because the mean and median differ when σ ≠ 0, reporting both can clarify central tendency for skewed data.

Key takeaways

• A log-normal distribution models positive-valued data whose log is normally distributed.
• It is skewed right, suitable for multiplicative processes and compound growth.
• Common in finance for modeling prices; use log returns for normality-based analyses.
• Excel’s LOGNORM.DIST handles PDF and CDF calculations given μ and σ of ln(X).