Lognormal Distribution

A right-skewed distribution for positive values, ideal for modeling multiplicative processes and naturally occurring phenomena with inherent variability.

Distribution Overview

What is the Lognormal Distribution:

The lognormal distribution is a continuous probability distribution where the logarithm of the variable follows a normal distribution. If X is lognormally distributed, then ln(X) is normally distributed. This creates a right-skewed distribution that is bounded by zero and extends infinitely to the right, making it ideal for modeling positive quantities that result from multiplicative processes.

Key Properties:

• • Right-skewed with long upper tail
• • Support: positive real numbers (0, +∞)
• • Parameters: μ and σ of underlying normal
• • Mode < Median < Mean
• • Natural log transformation yields normality

Applications:

• • Financial asset prices and returns
• • Income and wealth distributions
• • Failure times in reliability analysis
• • Particle size distributions
• • Environmental concentration data

Mathematical Properties:

Lognormal probability density function

= mean of the underlying normal distribution of ln(x)

= standard deviation of the underlying normal distribution

Defined for x > 0.

Statistical Characteristics

Key Statistics:

• • Mean: e^(μ + σ²/2)
• • Median: e^μ
• • Mode: e^(μ - σ²)
• • Variance: (e^σ² - 1) × e^(2μ + σ²)
• • Skewness: Always positive
• • Kurtosis: > 3 (leptokurtic)

Transformation Property:

Key Relationship:

If X ~ Lognormal(μ, σ²), then ln(X) ~ Normal(μ, σ²)

• • Enables normality-based analysis
• • Simplifies parameter estimation
• • Allows confidence interval construction

Reading the Lognormal Density Curve

Example Output & Result Interpretation

Worked Example:

A supplier's particle-size measurements (in microns) are ln-transformed and tested for normality. The ln-transformed sample gives a mean of μ = 1.20 and standard deviation of σ = 0.35.

• Median (back-transformed): e^μ = e^1.20 ≈ 3.32 microns

• Mean: e^(μ + σ²/2) = e^(1.20 + 0.061) ≈ 3.53 microns

• Mode: e^(μ - σ²) = e^(1.20 - 0.1225) ≈ 2.93 microns

Result Interpretation:

✅ Lognormal Fit Supported When:

• • ln(X) passes normality tests (Shapiro-Wilk, Anderson-Darling)
• • Probability plot of ln(X) is approximately linear
• • Data is strictly positive and right-skewed

❌ Reconsider Fit When:

• • ln(X) shows residual skewness or heavy tails
• • Data contains zero or negative values
• • Q-Q plot of ln(X) shows systematic curvature

Analysis & Testing

Example Data Layout for Lognormal Distribution Analysis

Lognormal data analysis showing original values, natural log transformation, standardized scores, and back-transformation validation. The ln-transformed data should follow normal distribution patterns. Sample Size Recommendation: Minimum 30-50 observations for reliable lognormal distribution fitting and parameter estimation.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.