Log Normal Distribution Calculator

Calculate probabilities, percentiles, and generate visualizations for Log Normal distributions. Essential for statistics, finance, and data science.

Distribution Parameters
-2 0 3
μ
Mean of the underlying normal distribution
0.1 1 2
σ
Standard deviation (must be > 0)
Standard (μ=0, σ=1)
Stock Returns
Income Distribution
Narrow Distribution
Probability
Percentile
Range
Random Sample
Calculate PDF and CDF at this value
0.3989
0.5000
Find x such that P(X ≤ x) = p
7.389
The 95th percentile
0.3834
Probability that X is between a and b
Number of random values (1-10,000)
For reproducible results
Calculating...
Distribution Statistics

Understanding Log Normal Distribution

The log-normal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. It's widely used in various fields including finance, economics, and engineering.

Mathematical Definition:

If Y = ln(X) follows a normal distribution with mean μ and variance σ², then X follows a log-normal distribution with parameters μ and σ.

X ~ LogNormal(μ, σ²) ⇔ ln(X) ~ N(μ, σ²)

Key Properties

Property Formula Description
Support x ∈ (0, ∞) Only positive values
Mean exp(μ + σ²/2) Expected value of X
Variance [exp(σ²) - 1] · exp(2μ + σ²) Spread of the distribution
Mode exp(μ - σ²) Most frequent value
Median exp(μ) 50th percentile
Skewness (exp(σ²) + 2)√(exp(σ²) - 1) Always positive (right-skewed)

Applications

1

Finance: Stock prices, asset returns, and option pricing models often follow log-normal distributions.

2

Economics: Income distribution, firm sizes, and city populations are often modeled with log-normal distributions.

3

Engineering: Failure analysis, particle sizes, and material strength often follow log-normal distributions.

4

Biology: Species abundance, incubation periods of infectious diseases, and physiological measurements.

Relationship with Normal Distribution

The log-normal distribution is intimately connected to the normal distribution through the logarithmic transformation:

  • If X ~ LogNormal(μ, σ²), then ln(X) ~ N(μ, σ²)
  • If Y ~ N(μ, σ²), then exp(Y) ~ LogNormal(μ, σ²)
  • Multiplication of independent log-normal variables results in another log-normal variable

Calculator Features:

  • Calculate PDF, CDF, and inverse CDF (percentiles)
  • Generate random samples from log-normal distribution
  • Visualize the distribution with interactive charts
  • Compute key statistics: mean, variance, median, mode
  • Export results and generated data for further analysis

Frequently Asked Questions

The normal distribution is symmetric and can take any real value (positive or negative), while the log-normal distribution is skewed to the right and only takes positive values. If the logarithm of a variable is normally distributed, then the variable itself follows a log-normal distribution.

Use a log-normal distribution when your data:
  • Is strictly positive (cannot be zero or negative)
  • Is right-skewed (has a long tail to the right)
  • Results from multiplicative processes (rather than additive)
  • Has a natural lower bound at zero but no upper bound
Common examples include stock prices, income distributions, and particle sizes.

Given a sample x₁, x₂, ..., xₙ from a log-normal distribution:
  1. Take the natural logarithm of each value: yᵢ = ln(xᵢ)
  2. Calculate the mean of the yᵢ values: μ̂ = (1/n) ∑ yᵢ
  3. Calculate the standard deviation of the yᵢ values: σ̂ = √[(1/(n-1)) ∑ (yᵢ - μ̂)²]
These μ̂ and σ̂ are the maximum likelihood estimates of the parameters.

Yes, the log-normal distribution is commonly used in risk assessment and financial risk management. It's particularly useful for modeling:
  • Value at Risk (VaR) calculations
  • Insurance claim amounts
  • Project cost overruns
  • Environmental pollutant concentrations
Its right-skewed nature makes it suitable for modeling rare but extreme events.

  • μ (location parameter): Affects the scale of the distribution. Increasing μ shifts the entire distribution to the right (higher values). The median is exp(μ).
  • σ (shape parameter): Affects the shape and spread. Increasing σ makes the distribution more skewed and spread out. Variance increases exponentially with σ².
  • When σ is small, the log-normal distribution approximates a normal distribution after a log transformation.