Log-Normal Distribution Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The probability density of the log-normal distribution is calculated with f(x) = 1/(x×σ×√(2π)) × exp(−(ln(x)−μ)²/(2σ²)), defined only for x > 0. For the standard parameters μ=0 and σ=1, the density at x=1 is exactly 1/√(2π) ≈ 0.3989, the same value as the standard normal density at 0.
Explanation
The log-normal distribution describes a random variable X whose natural logarithm, ln(X), follows a classic normal distribution with parameters μ and σ. It appears naturally wherever a phenomenon results from the MULTIPLICATION of many independent factors rather than their addition — for example the compound growth of an income or a stock price, the size of particles from successive grinding, or certain survival times in biology — whereas the classic normal distribution models phenomena where many influences ADD UP (such as independent measurement errors that cancel out on average), the same symmetric distribution used by our normal-distribution percentile tools. It also differs from the exponential distribution, which likewise models a strictly positive quantity (a waiting time) but with a constant decay rather than an asymmetric peak, and from the heavier-tailed Pareto distribution. A direct consequence of this multiplicative origin is that the log-normal distribution is asymmetric (unlike the symmetric bell curve of the normal distribution): it has a long tail toward high values, never negative or zero values, which makes it a much better model than the normal distribution for quantities that are strictly positive by nature, such as an income or a price. The parameters μ and σ are not directly the mean and standard deviation of X itself, a frequent confusion: they are the mean and standard deviation of the LOGARITHM of X, which is why the actual expected value of X, exp(μ + σ²/2), is always greater than exp(μ) (which would be the median of the distribution) — a gap all the more marked the larger σ is, because of the curve's asymmetry. This distinction between μ (the logarithm's parameter) and the actual expected value of X is the most common pitfall when using this distribution, unlike the normal distribution where the two coincide.
Example: standard log-normal distribution (μ=0, σ=1)
Inputs
Value x = 1. Parameters μ = 0, σ = 1.
Calculation
f(1) = 1/(1×1×√(2π)) × exp(−(ln(1)−0)²/(2×1²)) = 1/√(2π) × exp(0) = 1/√(2π) ≈ 0.3989.
Result
The density at x=1 is about 0.3989. For these same parameters, the expected value of X is √e ≈ 1.6487 and its variance is (e−1)×e ≈ 4.6708 — both greater than μ=0 would suggest, because of the distribution's asymmetry.
Frequently asked questions
Why is the expected value of X not simply exp(μ)?
Because the log-normal distribution is asymmetric: its long tail toward high values pulls the mean upward, beyond exp(μ), which actually corresponds to the MEDIAN of the distribution (the value splitting it into two equal halves), not its mean. The extra term exp(σ²/2) in the expected-value formula captures precisely this asymmetry effect: the larger σ is, the wider the gap between median and mean.
How do you tell a normal distribution from a log-normal one on real data?
A simple visual clue: if the data histogram is symmetric and bell-shaped, the normal distribution is probably appropriate; if it is concentrated toward low values with a long tail toward high values (never negative values), the log-normal distribution is often a much better candidate. A more rigorous method is to apply the logarithm to the data and check whether the result follows a symmetric bell-shaped distribution — if so, the original data follows a log-normal distribution by the very definition of that distribution.
Does this formula work for x=0 or negative x?
No, the log-normal distribution is defined only for strictly positive values of x, since it relies on ln(x), which does not exist for x ≤ 0. This is in fact one of the reasons this distribution is preferred over the normal distribution for modeling quantities that can physically never be negative, such as a price, a mass, or a duration.