If a quantity’s logarithm is normally distributed, the quantity itself is lognormal — right-skewed, always positive, with a long tail of occasional large values. Incomes, particle sizes, claim amounts and time-to-repair data all wear this shape. This Lognormal Distribution Calculator takes the log-scale parameters μ and σ plus a point x, and returns the density, the cumulative probability, and the distribution’s mean, median and mode — which for a lognormal are three genuinely different numbers.
Worked Example: The Standard Case μ = 0, σ = 1
With μ = 0 and σ = 1 at x = 1, the calculator reports PDF 0.3989, CDF 0.5000, mean 1.6487, median 1 and mode 0.3679. Every figure hand-checks. The CDF at x = 1 is Φ((ln 1 − 0)/1) = Φ(0) = 0.5 — the median is e^μ = 1 by definition, so exactly half the mass lies below it. The PDF at the median is 1/(1 × 1 × √2π) = 0.3989, the standard normal density at zero wearing a 1/x jacket. The mean e^(μ + σ²/2) = e^0.5 ≈ 1.6487 sits far above the median: the right tail, stretching toward infinity, drags the average up while the median stays at the typical value. The mode e^(μ − σ²) = e^−1 ≈ 0.3679 completes the classic ordering mode < median < mean that diagnoses right skew at a glance.
The Formulas, and the One Idea Behind Them
Everything flows from X = e^Y with Y ~ Normal(μ, σ²). The density: f(x) = [1 / (x σ √2π)] · exp(−(ln x − μ)² / (2σ²)) — the normal density of ln x, divided by x because stretching the axis by exponentiation thins the density by exactly that factor. The CDF needs no new machinery: F(x) = Φ((ln x − μ)/σ), the standard normal CDF at the z-score of ln x, which is how this calculator evaluates it (via the error function). The summaries: median = e^μ, mean = e^(μ + σ²/2), mode = e^(μ − σ²), variance = (e^(σ²) − 1) · e^(2μ + σ²). Note what σ does: at σ = 0.2 the three summaries nearly coincide and the shape is almost symmetric; at σ = 1 they spread by a factor of e in each direction; at σ = 2 the mean is e² ≈ 7.4 times the median. Multiplicative noise compounds — that is the whole personality of the distribution.
Where the Lognormal Earns Its Place
The central limit theorem has a multiplicative twin: when a quantity results from many small percentage changes multiplied together — investment returns compounding, particles repeatedly fracturing, doses diluting through stages — its logarithm adds those changes, turns normal, and the quantity turns lognormal. Hence its dominions: equity prices in the Black–Scholes model (returns are normal in logs, prices lognormal), income and wealth distributions, insurance claim sizes, environmental concentrations and latency/repair times in engineering. Fitting real data runs through the log door: take natural logs of your positive data, compute their ordinary mean and SD, and those are μ̂ and σ̂ — then this calculator turns the fitted model back into probabilities like P(X ≤ x) for planning thresholds.
Common Mistakes
Confusing the parameters: μ and σ describe ln(X), not X — reporting “the mean is μ” understates the true mean e^(μ+σ²/2), the most persistent lognormal error in practice. Using base-10 logs where the formulas assume natural logs (convert: μ_ln = μ₁₀ × ln 10). Asking for probabilities at x ≤ 0, where the distribution has no support at all. And summarizing skewed data with the mean alone — for a lognormal, the median is the typical value and the mean is a tail-weighted planning figure; quoting one where the decision needs the other misleads in a predictable direction.
Recovering the Parameters From Percentiles
The formulas also run backwards, which is how practitioners fit the distribution from published summaries. Suppose a quantity is reported with median 1 and mean 1.6487: the ratio mean/median = e^(σ²/2) = 1.6487 gives σ²/2 = ln(1.6487) = 0.5, so σ = 1 — and μ = ln(median) = 0, recovering this page’s example exactly. Percentiles convert through the normal quantile function: the p-th percentile is e^(μ + z_p σ), so for the standard case the 95th percentile is e^1.6449 ≈ 5.18 — a value five times the median, in a distribution where half of everything sits below 1. Holding the median and the 95th percentile side by side is the fastest honest summary of a lognormal: typical case, and bad case, in two numbers.
Related Tools
Continue with the Beta Distribution Calculator, the Weibull Distribution Calculator, the MSE Calculator, or browse the full Statistics Calculators hub.
Frequently Asked Questions
What is a lognormal distribution?
A distribution whose logarithm is normal. It is defined for positive values only, is right-skewed, and arises whenever many small multiplicative effects compound — prices, incomes, particle sizes, repair times.
What are the mean and median of a lognormal?
Median = e^μ; mean = e^(μ + σ²/2). The mean always exceeds the median (for σ > 0), and the gap grows with σ — at μ = 0, σ = 1, median 1 versus mean 1.6487.
How do I fit a lognormal to my data?
Take natural logs of the data, then use their ordinary sample mean as μ and their sample standard deviation as σ. The lognormal parameters are exactly the normal parameters of the logged data.
How is P(X ≤ x) computed?
As Φ((ln x − μ)/σ) — log the point, standardize with μ and σ, and read the standard normal CDF. This calculator performs exactly those steps.
Why is the mode below the median?
The density’s peak sits at e^(μ − σ²), pulled left of the median e^μ because the 1/x factor in the density tilts mass toward smaller values even as the tail stretches right.
Can a lognormal take negative values?
Never. Its support is x > 0 — exponentiation cannot produce zero or negatives. Data containing zeros or negatives cannot be lognormal as-is.