Lognormal Distribution
Compute lognormal distribution probabilities, PDF, CDF, mean, median, mode, variance, and skewness with parameters μ and σ. Free online calculator with interactive PDF and CDF curve charts.
About This Calculator
The Lognormal Distribution Calculator computes probabilities and statistics for the lognormal distribution — a continuous probability distribution of a random variable whose natural logarithm follows a normal distribution. Given the scale parameter μ and the shape parameter σ (σ > 0), the calculator evaluates the probability density function (PDF), cumulative distribution function (CDF), mean, median, mode, variance, and skewness at any positive value x. The interactive charts show the full PDF and CDF curves across the distribution's range.
The lognormal PDF formula is f(x; μ, σ) = 1/(xσ√(2π)) · exp(−(ln x − μ)²/(2σ²)) for x > 0. The CDF equals F(x) = Φ((ln x − μ)/σ), where Φ is the standard normal CDF. Key properties: mean = exp(μ + σ²/2), median = exp(μ), mode = exp(μ − σ²), variance = [exp(σ²) − 1]·exp(2μ + σ²), and skewness = [exp(σ²) + 2]·√[exp(σ²) − 1]. The distribution is always positively skewed with support on (0, ∞).
The lognormal distribution is widely used in finance for modeling stock prices and asset returns, in economics for income and wealth distribution, in reliability engineering for failure time analysis, in chemistry for molar mass distributions, in hydrology for rainfall modeling, and in biology for species abundance and growth rates. Its key property — that ln(X) is normally distributed — makes it convenient for data analysis: simply apply a log transformation to use normal-distribution methods, then transform results back.
Regional Notes
The lognormal distribution is a universal mathematical concept used across all regions. In the United States, it appears in finance (option pricing, portfolio theory), actuarial science, and engineering reliability courses. In India, it features in statistics, econometrics, and environmental engineering curricula. In the United Kingdom, it is covered in A-level Further Statistics and university-level courses in econometrics, finance, and engineering. The mathematical formulas and interpretation are identical worldwide.
Frequently Asked Questions
What is the lognormal distribution?
The lognormal (or log-normal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. If Y has a normal distribution, then X = exp(Y) has a lognormal distribution. A lognormal random variable takes only positive values, making it useful for modeling quantities that cannot be negative, such as income, stock prices, asset returns, and failure times.
How do you calculate the PDF of a lognormal distribution?
The probability density function (PDF) of the lognormal distribution is f(x; μ, σ) = 1/(xσ√(2π)) × exp(−(ln x − μ)²/(2σ²)) for x > 0, where μ is the scale parameter and σ > 0 is the shape parameter. The PDF is zero for x ≤ 0 since the lognormal distribution is only defined for positive values.
What are the mean, median, and mode of a lognormal distribution?
For a lognormal distribution with parameters μ and σ, the mean is exp(μ + σ²/2), the median is exp(μ), and the mode is exp(μ − σ²). Note that the median is always less than the mean (since σ² > 0), reflecting the positive skewness of the lognormal distribution. The variance is [exp(σ²) − 1] × exp(2μ + σ²) and the skewness is [exp(σ²) + 2] × √[exp(σ²) − 1].
What is the difference between lognormal and normal distribution?
The key difference is that a normal distribution is symmetric about its mean and can take any real value (negative or positive), while a lognormal distribution is positively skewed and only takes positive values. If X is lognormally distributed with parameters μ and σ, then ln(X) follows a normal distribution with mean μ and standard deviation σ. This relationship means the lognormal distribution has heavier tails on the right side.
What is the CDF of a lognormal distribution?
The cumulative distribution function (CDF) of the lognormal distribution is F(x; μ, σ) = Φ((ln x − μ)/σ) for x > 0, where Φ is the standard normal CDF. This relationship follows directly from the definition: since ln(X) is normally distributed, P(X ≤ x) = P(ln(X) ≤ ln(x)) = Φ((ln x − μ)/σ).
Where is the lognormal distribution used in real-world applications?
The lognormal distribution appears in many fields: in finance for modeling stock prices and asset returns; in economics for income and wealth distribution; in reliability engineering for modeling failure times; in biology for growth rates and species abundance; in hydrology for rainfall and streamflow data; in chemistry for molar mass distributions in polymers; and in social media for modeling comment lengths and user engagement times.
How do you calculate CDF value for a lognormal distribution at a given x?
To calculate the CDF at x for a lognormal distribution with parameters μ and σ, first compute z = (ln(x) − μ)/σ, then evaluate the standard normal CDF Φ(z). The standard normal CDF can be computed using the error function (erf): Φ(z) = 0.5 × [1 + erf(z/√2)]. This calculator uses high-precision approximation methods for accurate results.
What happens to the lognormal distribution shape as sigma changes?
As σ (the shape parameter) increases, the lognormal distribution becomes more positively skewed with a heavier right tail. When σ is small (close to 0), the lognormal distribution approximates a normal distribution. As σ grows, the mean increases (due to the exp(σ²/2) factor), the mode decreases (approaching 0), and the distribution becomes increasingly right-skewed with a long tail of large values.