Lognormal Distribution Calculator

What Is the Lognormal Distribution?

A lognormal distribution describes a continuous random variable $X$ where $\ln(X)$ follows a normal distribution. It applies to positive-valued data such as stock prices, income, and failure times. Compare with the Normal Distribution Calculator for the underlying normal model.

Key Formulas

The probability density function is:

$$f(x) = \frac{1}{x\sigma\sqrt{2\pi}} \exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right)$$

Common measures:

$$\text{Mean} = e^{\mu + \sigma^2/2}, \quad \text{Median} = e^{\mu}, \quad \text{Mode} = e^{\mu - \sigma^2}$$

Example: Stock Price Threshold

With $\mu = 0$ and $\sigma = 1$, the probability that $X \leq 1$ is $P(X \leq 1) = \Phi(0) = 50\%$, where $\Phi$ is the standard normal CDF applied to $(\ln x - \mu)/\sigma$.

Frequently Asked Questions

What do mu and sigma represent in a lognormal distribution?

Mu and sigma are the mean and standard deviation of the natural logarithm of X, not of X itself. They parameterize the underlying normal distribution of ln(X).

Why can lognormal variables only be positive?

Since X = exp(Y) where Y is normally distributed, X is always positive. The exponential function never produces negative or zero values.

How is the lognormal CDF calculated?

The CDF at x is Phi((ln x - mu) / sigma), where Phi is the standard normal cumulative distribution function.

When should I use lognormal instead of normal?

Use lognormal when data is strictly positive and right-skewed, such as asset prices, income distributions, or particle sizes. Use normal for symmetric, unbounded data.

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