Lognormal Distribution Calculator
Lognormal distribution from μ and σ of ln X, or from the arithmetic mean and SD — P(X ≤ x), upper tail, density, mean, median, mode, SD and the 5th–95th percentiles.
How it works
ln X ~ Normal(μ, σ); P(X ≤ x) = Φ[(ln x − μ)/σ]; f(x) = φ[(ln x − μ)/σ] ÷ (xσ); mean = e^(μ + σ²/2); median = e^μ; mode = e^(μ − σ²); SD = mean·√(e^(σ²) − 1); from arithmetic m, s: σ² = ln(1 + s²/m²), μ = ln m − σ²/2
A variable is lognormal when its logarithm is normally distributed — typical of quantities that grow by multiplication: incomes, stock prices, particle sizes, reaction times, pollutant concentrations, city sizes. It is always positive and skewed to the right, so its mean is larger than its median. Many reports give the arithmetic mean and SD instead of μ and σ; the calculator converts them with the method-of-moments formulas from the NIST/SEMATECH e-Handbook.
Worked example
With μ = 1 and σ = 0.5 on the log scale, P(X ≤ 3) = 0.5782 and P(X > 3) = 0.4218; the median is 2.7183, the mean 3.0802 and the mode 2.1170.
Sources: NIST/SEMATECH e-Handbook — Lognormal distribution · SciPy — scipy.stats.lognorm
Frequently asked questions
What are μ and σ?
They are the mean and standard deviation of ln X, not of X itself. The median of X is e^μ and σ controls the skewness.
I only know the mean and SD of my data — what now?
Choose “Arithmetic mean and SD of X”; the calculator converts them: σ² = ln(1 + SD²/mean²) and μ = ln(mean) − σ²/2.
Why is the mean bigger than the median?
The long right tail pulls the mean upward: mean = median × e^(σ²/2). The larger σ, the bigger the gap.
How do I check whether my data are lognormal?
Take the logarithm of every value and test the logs for normality (e.g. with the Shapiro–Wilk test or a normal Q-Q plot).
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This calculator is for educational purposes. Double-check important results before acting on them.