Lognormal Distribution Calculator
Enter log-scale parameters μ and σ to evaluate lognormal probabilities and descriptive statistics.
PDF f(x): 0.152614
CDF F(x): 0.917171
Survival S(x): 0.082829
Mean: 1.1331
Median: 1.0000
Mode: 0.7788
Variance: 0.3647
How to Use This Calculator
- Input μ and σ from the underlying normal distribution of ln X.
- Specify the x value where you need the PDF or CDF.
- Review summary statistics (mean, median, mode, variance).
- Use results in reliability, finance, or environmental modeling contexts where lognormal data arise.
Formula
f(x) = [1 / (xσ√(2π))] · exp(-(ln x − μ)² / (2σ²)), x > 0
F(x) = Φ((ln x − μ) / σ)
Mean = exp(μ + σ²/2) • Median = exp(μ) • Mode = exp(μ − σ²)
Φ denotes the standard normal cumulative distribution function.
Frequently Asked Questions
What does μ represent?
μ is the mean of ln X, not of X itself. Lognormal parameters describe the associated normal distribution of log-transformed data.
Can σ be zero?
σ = 0 collapses the distribution to a point mass at exp(μ). The PDF becomes degenerate.
Where does the lognormal distribution arise?
It models multiplicative processes such as stock prices, income distributions, and time-to-failure data.