Lognormal Distribution Calculator

μ and σ are not your data’s mean and standard deviation

This is the error that makes lognormal calculators dangerous, and it is invisible in the output.

μ is the mean of ln X. σ is the standard deviation of ln X. Neither is a property of your data on its own scale. They use the familiar symbols because the underlying normal distribution owns them, and that familiarity is exactly what causes the substitution.

Take the tool’s default: μ = 3, σ = 0.8. The data those describe has a median of 20.1 and a mean of 27.7. Someone with data averaging 27.7 who types 27.7 into the μ box is describing a distribution whose median is e²⁷⋅⁷, which is about a trillion. Nothing in the answer looks wrong — probabilities are still between 0 and 1.

So the tool accepts either form and converts. Given the data’s mean and standard deviation it applies the method of moments: σ² = ln(1 + CV²) and μ = ln(mean) − σ²/2, where CV is the coefficient of variation. The suite asserts the round trip returns the original parameters to ten decimal places.

A quick sanity check whichever way you enter it: eᵘ should be near the middle of your data, because eᵘ is the median. If it is nowhere near, the parameters are on the wrong scale.

One more consequence worth knowing: because σ is a log-scale quantity, it is unitless and behaves like a percentage. σ = 0.1 means roughly ±10% variation; σ = 0.8 means the spread is enormous. Comparing σ across datasets in different units is legitimate in a way comparing standard deviations is not.

Three centres, always in the same order

For any lognormal, mode < median < mean. Strictly, with no exceptions, and the gaps are formulaic rather than incidental.

Mode = e⁽ᵘ⁻σ²⁾ — the peak, the single most likely value. Median = eᵘ — half above, half below. Mean = e⁽ᵘ⁺σ²ˈ²⁾ — the balance point, dragged up by the tail.

The ratio of mean to median is exactly eσ²ˈ², depending only on σ. At the tool’s default that is 1.377, so the mean sits 38% above the median. At σ = 1.5 the mean is more than three times the median.

The number that makes this concrete: only about 34% of a lognormal population lies above its own arithmetic mean at σ = 0.8 — the tool prints the exact figure for your parameters. Two thirds of the population is “below average”, which is not a paradox, just what a right tail does.

This is why national statistics agencies publish median income and median house prices. The mean is a real quantity and it answers a different question — total wealth divided by headcount — and for “what does a typical person have?” the median is the honest answer.

The Mean, Median and Mode Calculator reads skew directly off the gap between the first two. A lognormal is the textbook case of that gap being large.

Why multiplication produces this shape

The normal distribution arises when many small independent effects add. The lognormal arises when they multiply, and the reason is a one-line consequence of that.

Multiply many independent factors and take logarithms: the product becomes a sum of logs. The central limit theorem applies to that sum, so ln X tends to normal — which is precisely the definition of X being lognormal.

So the question “why is this data skewed?” often answers itself: because the process is proportional. Incomes grow by percentage raises. Investments compound. Cities grow in proportion to their size. Particle sizes result from repeated fragmentation, each step removing a fraction.

Two consequences follow that are worth having.

Taking logs often makes the analysis ordinary. If X is lognormal then ln X is exactly normal, so a t-test, a confidence interval or a regression on the logged data is on solid ground where the same method on raw values is not. This is why economists and biologists log so much of their data — not a trick, a change to the scale on which the process actually operates.

But interpret carefully after back-transforming. Exponentiating the mean of the logs gives the geometric mean, which equals the median — not the arithmetic mean. That is usually what you want for skewed data, and it is worth stating rather than leaving implicit. The Geometric Mean Calculator covers that quantity in its own right.

How to tell it from a normal

Three checks, and the first rules out a normal immediately in most real cases.

Can the quantity be negative? A lognormal is strictly positive by construction. A normal is not, so fitting one to incomes or durations puts probability mass below zero — which is impossible, and the error grows as the spread does. If a fitted normal predicts negative values for a quantity that cannot be negative, that alone settles it.

Compare the mean with the median. A normal has them equal. A large gap in the positive direction is the lognormal signature, and the ratio estimates σ directly: mean/median = eσ²ˈ².

Plot the logarithm. The decisive test. If ln X looks normal on a histogram or a Q-Q plot, X is lognormal. This is cheap and far more informative than a formal test, for the reasons the normal page sets out about normality testing.

A caution in the other direction: at small σ a lognormal looks almost normal. Below about σ = 0.3 the skew is mild and either model fits acceptably in the middle. The difference shows up in the tails, which is where the consequential questions usually live — so “it looks normal enough” is a statement about the middle and not about the risk.

Heavy, but not as heavy as people assume

The lognormal is right-skewed with a long tail, and it is worth being precise about how long, because it sits between two families that get conflated with it.

Heavier than a normal, lighter than a power law. All its moments exist — every mean, variance and higher moment is finite, which is not true of a Pareto with a small exponent. So it is a heavy-tailed distribution in the everyday sense without being one in the technical sense that breaks the central limit theorem.

That distinction matters in practice. Income and wealth data are often better described as lognormal in the body and Pareto in the extreme tail — the bulk follows a multiplicative growth process, and the top fraction of a percent follows something more extreme. Fitting a single lognormal to the whole range will understate the very top.

The same applies to financial risk. Black-Scholes assumes prices are lognormal, which is why it produces closed-form option prices. Real returns have heavier tails than that, and the volatility smile observed in option markets is precisely the market pricing in a departure the model does not contain.

The practical position: the lognormal is an excellent default for positive, right-skewed, multiplicative data, and it should be treated with suspicion specifically at the extreme upper tail, which is often where the money or the risk is.

Where it genuinely fits

The test is whether the generating process is proportional rather than additive.

Incomes and wealth. Growth by percentage raises and compounding returns. The classic case, and the origin of the mean/median divergence in every income statistic.

Durations. Time to complete a task, time to repair, length of a hospital stay. All positive, all right-skewed, and all subject to multiplicative delays.

Sizes. Particle diameters after repeated fragmentation, file sizes, city populations. Each step scales what came before.

Concentrations in biology and environmental science. Antibody titres, pollutant concentrations, bacterial counts — conventionally analysed on the log scale for exactly this reason.

Asset prices. A geometric random walk produces lognormal prices, which is the foundation of Black-Scholes and much of quantitative finance — with the tail caveat above.

Where it does not fit: anything that can be zero or negative, anything bounded above, and anything additive. Temperatures in Celsius, profit and loss, and test scores out of 100 are all wrong for different reasons.

Reporting it without misleading

Four habits, each of which prevents a specific misreading.

Lead with the median. It is the typical value, it equals the geometric mean, and it is what most readers assume “average” means. Give the arithmetic mean alongside if the total matters — for a payroll budget it does.

Give a percentile range rather than a symmetric interval. “Median 20, 10th to 90th percentile 7 to 56” describes the shape. “27.7 ± 26” implies symmetry the data does not have and a lower bound near zero that is misleading.

Quote σ as the spread if the audience is technical. Being unitless, it is comparable across datasets in a way a raw standard deviation is not.

Say which scale a summary is on. “Mean of the logs” and “log of the mean” are different numbers, and the first exponentiates to the median while the second does not. This is the reporting version of the parameter trap at the top of this page, and it is just as easy to get wrong.

Sources and methodology

References for the distribution and the parameter conversion.

• NIST/SEMATECH e-Handbook — the lognormal distribution and its parameterisation
• Limpert, Stahel & Abbt, “Log-normal Distributions across the Sciences”, BioScience 51(5), 2001 — on how widely the shape occurs and how often it is mishandled
• R documentation for dlnorm — whose meanlog and sdlog are named to prevent exactly the substitution this page warns about
• Normal Distribution Calculator — what ln X follows, and the additive counterpart
• Geometric Mean Calculator — which equals this distribution’s median
• How we calculate — methodology

Method. Probabilities are computed as Φ((ln x − μ)/σ) using the higher-precision normal CDF, so the tails stay accurate where a polynomial approximation would not. The conversion from a data mean and standard deviation uses the method of moments, σ² = ln(1 + CV²) and μ = ln(mean) − σ²/2, and the suite asserts the round trip returns the original parameters to ten decimal places — which is the check that makes the two input modes genuinely interchangeable rather than approximately so. It also asserts the strict ordering mode < median < mean across a range of σ, and that the mean-to-median ratio is exactly eσ²ˈ². That engine is verified on every change against 219 assertions. The count and the per-case breakdown are published on the formula verification page.

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