Log-Normal Distribution

What Is a Log-Normal Distribution?

A log-normal distribution is a probability distribution for a variable that can only take positive values. A variable is log-normal when the natural logarithm of that variable follows a normal distribution. If you take the logarithm of each value, the results form the familiar bell curve, but the original values do not.

Because exponentiation reverses the logarithm, the original values are never negative. The shape is also skewness|right-skewed: most outcomes cluster toward the lower end, while a smaller number of very large values stretch the tail to the right. The distribution is defined by two parameters, the mean and the standard deviation of the underlying normal distribution of logarithms.

How It Works

If X follows a log-normal distribution, then:

ln(X) ~ Normal(μ, σ²)

Several useful properties follow from this relationship:

• The median of X equals e^μ.
• The mean of X equals e^(μ + σ²/2), so the mean sits above the median.
• Values can grow without a fixed upper limit but cannot fall below zero.

Why It Matters to Investors

Many financial models assume that stock prices follow a log-normal distribution. The reasoning is that investors tend to think in percentage changes rather than dollar changes, and a price that cannot drop below zero fits this pattern. The Black-Scholes model, a widely used framework for pricing options, relies on this assumption.

The assumption has limits. Real market returns often show larger moves and fatter tails than a log-normal model predicts, and volatility itself changes over time. For this reason, analysts generally treat log-normal models as useful approximations rather than exact descriptions of market behavior. Understanding the shape also helps explain why the average outcome can differ noticeably from the typical one.