Compute PDF, CDF, percentiles, and moments for the lognormal distribution with interactive density curve, percentile table, and properties reference.
The lognormal distribution calculator computes probabilities, percentiles, and statistical properties for the lognormal distribution — a continuous distribution whose logarithm is normally distributed. If X is lognormal with parameters μ and σ, then ln(X) ~ Normal(μ, σ²).
The lognormal distribution naturally models quantities that arise from multiplicative processes: stock prices, income distributions, particle sizes, failure times, biological measurements, and file sizes. Its characteristic right skew means the mean exceeds the median, which exceeds the mode.
Enter the location parameter μ (log-scale mean) and scale parameter σ (log-scale standard deviation) to compute PDF, CDF, interval probabilities, a visual density curve, a detailed percentile table, and a properties reference card. The same parameter pair also makes it easy to compare the median, mean, and mode, which is useful when the data is strongly right-skewed.
The lognormal distribution appears in finance, biology, environmental science, reliability engineering, and anywhere positive, right-skewed data is produced by multiplicative effects. This calculator gives you PDF/CDF evaluation, percentiles, a density curve, and a properties table without making you do the logarithms by hand.
It is a practical fit when the mean, median, and mode are all meaningfully different and the raw values cannot go below zero.
PDF: f(x) = (1/(xσ√(2π))) exp(−(ln(x)−μ)²/(2σ²)). Mean = e^(μ+σ²/2). Median = e^μ. Mode = e^(μ−σ²). Variance = (e^(σ²)−1)e^(2μ+σ²).
Result: P(X ≤ 1) = 0.5000 (50%), PDF f(1) = 0.3989
With μ = 0 and σ = 1, the median of the lognormal is e^0 = 1, so exactly 50% of the distribution lies below x = 1. The PDF at x = 1 peaks at 0.3989.
Just as the normal distribution emerges from sums of independent variables, the lognormal emerges from products. If X = Y₁ × Y₂ × ... × Yₙ where Yᵢ are positive independent variables, then ln(X) = ln(Y₁) + ... + ln(Yₙ), which by the CLT tends to normal. Hence X tends to lognormal.
The Black-Scholes option pricing model assumes stock prices follow geometric Brownian motion, making future prices lognormally distributed. The parameters μ (drift) and σ (volatility) determine expected returns and risk. Understanding lognormal percentiles is crucial for Value at Risk (VaR) calculations.
Pollutant concentrations, bacterial colony counts, and particle size distributions often follow lognormal models. Environmental regulations frequently use geometric mean and geometric standard deviation rather than arithmetic ones, reflecting the underlying lognormal nature of the data.
Last updated:
μ and σ are parameters of the underlying normal distribution of ln(X). The actual mean of X is e^(μ+σ²/2), which is larger than e^μ (the median). Similarly, the actual std dev depends on both μ and σ in a complex way.
Take the natural log of all data points, then compute the sample mean and sample standard deviation of the logged values. Those are your estimates of μ and σ.
Income results from multiplicative factors — education, experience, industry, etc. The central limit theorem for products leads to lognormal distributions, explaining the characteristic right skew of income data.
No, the lognormal is strictly positive (X > 0) since it's the exponential of a normal variable. This makes it appropriate for inherently positive quantities like prices, times, and sizes.
The normal is symmetric and can take any real value. The lognormal is right-skewed and strictly positive. If ln(X) is normal, then X is lognormal. For small σ, the lognormal approaches normality near its peak.
The geometric mean equals the median: e^μ. This makes the geometric mean a natural "typical value" for lognormal data, more representative than the arithmetic mean.