Log-Normal Distribution Calculator
The distribution of anything that grows by multiplication — incomes, particle sizes, share prices — where the logarithm is normal and the values are right-skewed.
If ln X is normal with parameters μ and σ, then X is log-normal.
How the log-normal distribution calculator works
If ln X is normal with parameters μ and σ, then X is log-normal. The parameters live on the log scale, so the mean of X is e^(μ+σ²/2) — not e^μ, which is the median.
That gap between mean and median is the whole character of the distribution, and it is why the average income is always well above the typical one.
Formula: median = e^μ; mean = e^(μ + σ²/2)
Worked examples
| Inputs | P(X ≤ x) | Note |
|---|---|---|
| μ = 3, σ = 0.5, x = 30 | 78.883774% | median 20.1, mean 22.8 |
| At the median | 49.999853% | 50% |
| Highly skewed | 60.544433% | the mean is three times the median |
FAQFrequently asked questions
What is a log-normal distribution?
One whose logarithm is normally distributed. It arises whenever a quantity is the product of many independent factors.
Where does it show up?
Incomes, city sizes, particle diameters, share prices, the duration of tasks, biological measurements — anything multiplicative.
Why is the mean above the median?
Because the right tail is long. Exponentiating a symmetric normal stretches the upper half far more than the lower.
Are μ and σ the mean and SD of the data?
No — they belong to the logarithm. The data's mean is e^(μ+σ²/2), which catches people out constantly.
How do I know if my data is log-normal?
Take logs and check for normality — a histogram, a Q-Q plot, or a Shapiro-Wilk test on the logged values.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.