Lognormal Distribution Calculator
A random variable X follows a lognormal distribution if its natural logarithm ln(X) is normally distributed. Enter the mean μ and standard deviation σ of ln(X) to get the full summary of X — mean, median, mode, variance, skewness, and the PDF and CDF at any point x.
Expected value = exp(μ + σ²/2)
- 1
σ²
0.5² = 0.25 - 2
σ² / 2
0.25 ÷ 2 = 0.125 - 3
μ + σ² / 2
0 + 0.125 = 0.125 - 4
Mean E[X] = exp(μ + σ²/2)
exp(0.125) = 1.1331The mean of a lognormal variable always exceeds its median exp(μ) due to right-skew.
How does this calculator work?
Set μ (mean of ln X) and σ (std dev of ln X, must be > 0). The lognormal mean is exp(μ + σ²/2), median is exp(μ), mode is exp(μ − σ²). Enter x > 0 to get the PDF and CDF (= Φ((ln x − μ)/σ)) at that point.
Formula
How this is calculated
If X is lognormally distributed with parameters μ and σ, then Y = ln(X) follows a normal distribution with mean μ and standard deviation σ. This is why μ is called the "log-mean" and σ the "log-standard deviation" — they describe the underlying normal variable, not X itself. Because the log-normal can only take positive values and is right-skewed, it models many real-world phenomena well: income distributions, stock prices, biological measurements, and equipment lifetimes.
The probability density function f(x) = [1 / (x·σ·√(2π))]·exp[−(ln x − μ)²/(2σ²)] is zero at x = 0 and rises to a peak at the mode exp(μ − σ²), then decays slowly. The mean lies to the right of the median (which equals exp(μ)), and the median lies to the right of the mode. Larger σ produces a more skewed, heavy-tailed distribution. The CDF at x is simply Φ((ln x − μ) / σ), where Φ is the standard normal CDF, computed here via the Abramowitz–Stegun erf approximation.
Note that μ and σ are the parameters of the natural log of X, not of X itself. The true mean of X is exp(μ + σ²/2), which is always larger than the median exp(μ). The calculator assumes a continuous, infinite-support distribution; there is no finite upper bound on X.
Frequently asked questions
μ and σ describe the normal distribution of ln(X). The actual mean of X is exp(μ + σ²/2), which is larger than exp(μ), and the actual standard deviation of X is √[(exp(σ²) − 1)·exp(2μ + σ²)]. For small σ the two sets of parameters are close; for large σ they diverge significantly.
Because X = e^Y is an exponential transformation of Y. Large values of Y produce disproportionately large values of X, so the right tail of the distribution is much heavier than the left tail, which is bounded below by zero. The degree of right-skew is captured by the skewness formula (exp(σ²) + 2)·√(exp(σ²) − 1).
Yes. μ is the mean of ln(X), which can be any real number. A negative μ simply means the typical log-value of X is below zero, so the median of X is exp(μ) < 1. X itself remains strictly positive.
Also known as
TG we-Calculate Editorial Team. (2026). Lognormal Distribution Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/lognormal-distribution-calculator
TG we-Calculate Editorial Team. "Lognormal Distribution Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/lognormal-distribution-calculator.
TG we-Calculate Editorial Team, "Lognormal Distribution Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/lognormal-distribution-calculator
@misc{wecalculate_lognormal_distribution_calculator, title = {Lognormal Distribution Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/lognormal-distribution-calculator}}, year = {2026}, note = {TG we-Calculate} }
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