Lognormal Distribution Density Calculator

Background.

Plenty of quantities cannot go below zero, bunch up at modest values and trail off in a long right-hand tail: the sizes of particles in a powder, the hours needed to repair a machine, a share price a few years out. When a quantity is built by multiplying many small, independent factors rather than adding them, its logarithm drifts towards the normal bell curve, and the quantity itself follows a lognormal distribution.

That relationship is the whole definition. X is lognormal when ln X is normal with mean μ and standard deviation σ, and those log-scale figures are the two parameters entered here — not the mean and spread of X itself. The distinction catches people out. With μ = 0.5 and σ = 0.8, the median of X is e^0.5, about 1.65, but the tail drags the mean up to about 2.27, while the single most likely value, the mode, is only about 0.87.

The result is the probability density at one point x: the height of the curve there. It is not a probability. A density is measured per unit of x, can exceed 1 when the distribution is narrow, and becomes a probability only when multiplied by a width — over a short interval, density × width approximates the chance of landing inside it.

The page does not fit μ and σ to data, give cumulative or interval probabilities, or test whether a lognormal model suits your measurements. It evaluates the curve for parameters you already have, and leaves the judgement about whether they describe your data to you.

What is lognormal distribution density calculator?

The lognormal probability density function describes how probability is spread across the positive values of a random variable X whose natural logarithm is normally distributed. It is defined only for x > 0, is skewed to the right, and is fixed by two parameters: μ, the mean of ln X, and σ, the standard deviation of ln X. Its value f(x) is a probability per unit of x, and the total area under the curve is 1.

How to use this calculator.

1. 01Enter μ and σ on the natural-log scale. If they were worked out from base-10 logarithms, multiply both by 2.3026 first.
2. 02If you know only the ordinary mean m and standard deviation s of X, convert them: σ² = ln(1 + s²/m²) and μ = ln m − σ²/2.
3. 03Enter the point x where you want the density. It must be positive, because the lognormal curve does not exist at zero or below.
4. 04Read the result as a height, not a chance. For the probability of a narrow band around x, multiply the density by the band's width.

The formula.

f(x) = exp(−(ln x − μ)² ÷ (2σ²)) ÷ (x σ √(2π)). The exponential is the familiar normal-curve shape applied to ln x instead of x: it equals 1 when ln x = μ and falls away as the gap grows, faster when σ is small. The denominator holds the normal curve's constant σ√(2π) and one extra factor, x. That 1/x is the change-of-variable term — mapping the even log scale back onto ordinary numbers squeezes the curve at large x and lifts it at small x. It is why the peak, the mode at e^(μ − σ²), sits below the median e^μ, which in turn sits below the mean e^(μ + σ²/2). A larger σ fattens the right tail and pushes those three further apart; changing μ slides the whole curve along the log scale without altering its shape there.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST/SEMATECH e-Handbook, Lognormal Distribution.
2. [2]Aitchison and Brown, The Lognormal Distribution (PRINT).
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.