What is a lognormal distribution?
A lognormal distribution describes a random variable X whose natural logarithm ln(X) follows a normal distribution. It is defined for positive values only (x > 0). It is commonly used to model phenomena like stock prices, income distributions, and survival times where values are always positive and right-skewed. See also our calculate Beta Distribution.
What do μ (mu) and σ (sigma) represent in a lognormal distribution?
In a lognormal distribution X ~ LogN(μ, σ), μ is the mean and σ is the standard deviation of the underlying normal variable ln(X) — not of X itself. The actual mean of X is e^(μ + σ²/2), and the median is e^μ. Always ensure σ > 0.
How do I compute a left-tail probability P(X < x)?
Select 'P(X < x)' from the calculation mode dropdown, enter your μ and σ parameters, and enter the x value. The calculator returns the cumulative probability that the random variable is less than x. This equals Φ((ln(x) − μ) / σ), where Φ is the standard normal CDF.
How do I compute a right-tail probability P(X > x)?
Select 'P(X > x)' from the dropdown and provide x, μ, and σ. The right-tail probability is simply 1 − P(X < x). It represents the probability that the random variable exceeds x. You might also find our use the Gumbel Distribution Calculator useful.
How do I find a percentile (inverse CDF) for a lognormal distribution?
Select 'Find x from probability p', enter a probability between 0 and 1 in the p field (e.g. 0.8 for the 80th percentile), and provide μ and σ. The calculator returns the x value such that P(X < x) = p. Internally this is computed as x = e^(μ + σ · Φ⁻¹(p)).
What is the probability density function (PDF) of a lognormal distribution?
The PDF is f(x) = [1 / (x · σ · √(2π))] · exp(−(ln(x) − μ)² / (2σ²)) for x > 0. It gives the relative likelihood of the variable taking a specific value. The calculator displays f(x) at your entered x value alongside the probability result.
How are the mean, median, and mode of a lognormal distribution calculated?
For X ~ LogN(μ, σ): the mean is e^(μ + σ²/2), the median is e^μ, and the mode is e^(μ − σ²). The variance is (e^(σ²) − 1) · e^(2μ + σ²). Note that the mean is always greater than the median, which is always greater than the mode for a lognormal distribution.
Can x be zero or negative in a lognormal distribution?
No. A lognormal distribution is only defined for strictly positive values (x > 0). The probability P(X ≤ 0) = 0. If you enter x = 0 or a negative value, the result is undefined. This is one of the key properties that makes the lognormal distribution suitable for modeling prices, sizes, and other inherently positive quantities.