0.398942
0.5
1.648721
4.670774
2.161197
1
0.367879
1.310832
5.180252
0.628832
0.398942
0.5
1.648721
4.670774
2.161197
1
0.367879
1.310832
5.180252
0.628832
The Lognormal Distribution Calculator computes the probability density function and key statistics for the lognormal distribution, a continuous distribution widely used to model quantities that are the product of many independent positive random factors. A random variable X follows a lognormal distribution if its natural logarithm ln(X) follows a normal distribution with mean μ and standard deviation σ. This multiplicative structure makes the lognormal distribution the natural counterpart of the normal distribution for positive, right-skewed data.
The lognormal distribution appears throughout science, finance, and engineering. In finance, stock prices and asset returns are classically modeled as lognormal — the Black-Scholes option pricing model assumes geometric Brownian motion, which produces lognormal prices. In environmental science, pollutant concentrations, particle sizes, and rainfall intensities often follow lognormal patterns. Income distributions within populations are approximately lognormal, as are the sizes of cities, organisms, and mineral deposits. In reliability engineering, the lognormal models failure times for processes dominated by fatigue crack growth or chemical degradation.
A distinctive feature of the lognormal distribution is the relationship between its mean, median, and mode. The mean is always greater than the median, which is always greater than the mode, reflecting the distribution's right skew. The median has the elegant formula e^μ, making it independent of the variance parameter. The difference between mean and median grows with σ, so highly dispersed lognormal data can have means that are substantially larger than typical (median) values — a pattern frequently observed in income and wealth data.
The lognormal distribution has important connections to other distributions. As σ approaches 0, it converges to a point mass at e^μ. The product of independent lognormal variables is also lognormal, analogous to how the sum of normals is normal. The distribution has finite moments of all orders, with the n-th moment being e^(nμ + n²σ²/2). These properties make it particularly tractable for analytical work in finance and engineering.
This calculator takes the parameters μ and σ of the underlying normal distribution (not the mean and standard deviation of X itself) and evaluates the PDF at a specified point. It also computes the distribution's mean, variance, median, and mode, providing a complete statistical summary.
The lognormal PDF is:
$$f(x; \mu, \sigma) = \frac{1}{x\sigma\sqrt{2\pi}} \exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right), \quad x > 0$$
The key statistics are derived from the moment-generating function of the normal distribution:
$$\text{Mean} = e^{\mu + \sigma^2/2}, \quad \text{Variance} = (e^{\sigma^2} - 1) \cdot e^{2\mu + \sigma^2}$$
$$\text{Median} = e^{\mu}, \quad \text{Mode} = e^{\mu - \sigma^2}$$
Note the ordering: Mode < Median < Mean for all σ > 0, reflecting the right skew inherent in lognormal distributions.
The PDF indicates how likely a particular value is. For the lognormal, the PDF peaks at the mode and has a heavy right tail. The mean is pulled rightward by extreme values and can be much larger than the median for large σ. The median (e^μ) is often the most representative central value for skewed data. The mode is the most frequently occurring value. The variance grows exponentially with σ², so even modest increases in σ dramatically increase the spread. If your data span several orders of magnitude, this is a strong indicator of lognormality.
Inputs
Results
Lognormal(0, 0.5) centered at median=1.0 with mean=1.13 pulled right by skew. At x=1.2, the PDF is 0.655, still in the high-density region.
Inputs
Results
With μ=3, σ=1.2, median particle size is ~20 units but mean is ~43 due to heavy right tail. The mode at ~4.8 shows most particles are much smaller than the mean.
Use the lognormal when your data are positive, right-skewed, and multiplicatively generated. Strong indicators include: data spanning several orders of magnitude, a histogram that is skewed right but becomes symmetric after log-transformation, and a process where values grow by random percentage changes (multiplicative growth). Common applications include stock prices, income, particle sizes, latency times, and biological measurements like cell counts or organ weights.
No. The parameters μ and σ are the mean and standard deviation of ln(X), not of X itself. The actual mean of X is e^(μ + σ²/2) and the standard deviation is √[(e^(σ²) − 1) · e^(2μ + σ²)]. To convert from observed data: if your data have sample mean m and sample variance v, then σ² = ln(1 + v/m²) and μ = ln(m) − σ²/2. This distinction is a common source of confusion.
Stock prices change by percentage returns (multiplicative factors) over each time period. If daily returns are independent and identically distributed, the central limit theorem applied to their logarithms implies that log-prices are approximately normal, making prices lognormal. This is the foundation of the Black-Scholes model. The lognormal is particularly suitable because it is bounded below by zero (prices cannot be negative) and is right-skewed (large gains are possible but losses are bounded).
The normal distribution is symmetric, defined on all real numbers, and arises from additive processes. The lognormal is right-skewed, defined only for positive values, and arises from multiplicative processes. If X is lognormal, then ln(X) is normal. The normal has equal mean, median, and mode; the lognormal has Mode < Median < Mean. The lognormal also has heavier right tails — extreme high values are more probable than under a normal distribution.
The coefficient of variation (CV = σ_X / μ_X) for the lognormal has the elegant formula CV = √(e^(σ²) − 1), which depends only on σ and not on μ. This means the relative spread is constant regardless of the distribution's location on the positive axis. For small σ, CV ≈ σ. For σ = 1, CV ≈ 1.31 (131% relative variation). This property makes the CV a natural dispersion measure for lognormal data.
Take the natural logarithm of your data and test whether the logged values are normally distributed. Methods include: Q-Q plot of log-data against normal quantiles (should be a straight line), Shapiro-Wilk test on the log-transformed data, Kolmogorov-Smirnov test comparing the empirical CDF to a fitted lognormal CDF, or visual inspection of a histogram of log-data for symmetry and bell shape. A probability plot on log-probability paper is the classic graphical approach.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Normal Distribution Calculator
Probability Distributions
Standard Normal Distribution Calculator
Probability Distributions
Poisson Distribution Calculator
Probability Distributions
Exponential Distribution Calculator
Probability Distributions
Uniform Distribution Calculator
Probability Distributions
Geometric Distribution Calculator
Probability Distributions
