Question 1

What is the log-normal distribution?

Accepted Answer

A variable X is log-normally distributed if Y = ln(X) is normally distributed. This happens when X is the product of many independent positive random variables (multiplicative processes), just as the sum of many variables is normal. The distribution is always right-skewed (positive skew), defined only for x > 0, and has a heavier right tail than the normal distribution. Skewness and kurtosis increase with σ.

Question 2

What are the mean, median, and mode of a log-normal distribution?

Accepted Answer

For X ~ LogNormal(μ, σ²): Mean = e^(μ + σ²/2), Median = e^μ, Mode = e^(μ − σ²). Note: Mean > Median > Mode — all three differ (unlike the symmetric normal). The mean is always larger than the median for σ > 0, reflecting the right skew. If σ is small (σ < 0.3), all three are approximately equal and X looks nearly symmetric around e^μ.

Question 3

How do I estimate μ and σ from my data?

Accepted Answer

If you have raw data values x₁, x₂, …, xₙ: (1) Take the log of each value: y_i = ln(x_i). (2) Compute the sample mean of {y_i}: μ̂ = (1/n) Σ y_i. (3) Compute the sample std dev of {y_i}: σ̂ = sqrt[Σ(y_i − μ̂)²/(n−1)]. Verify the fit by plotting a histogram of {y_i} — it should look approximately normal. Use the normality test calculator on the log-transformed data.

Question 4

What are real-world examples of log-normal distributions?

Accepted Answer

Log-normal distributions appear in: (1) Finance — stock prices, asset returns over long periods, income distributions, wealth. (2) Biology — organism sizes, species abundance, protein expression levels. (3) Engineering — fatigue life of materials, repair times, response latency in computing. (4) Epidemiology — dose-response relationships, incubation periods. (5) Environmental science — particle sizes, pollutant concentrations. Any multiplicative growth process tends to produce log-normal distributions.

Question 5

Log-normal vs normal distribution — when to use which?

Accepted Answer

Use normal when data is symmetric and can be negative (temperatures, measurement errors, test scores). Use log-normal when data is strictly positive and right-skewed (prices, income, lifetimes). A practical test: log-transform your data and check normality. If ln(X) passes a normality test, X is log-normally distributed. If X is log-normal and σ < 0.3, the normal distribution is often a good approximation.

Question 6

What is the variance of a log-normal distribution?

Accepted Answer

Variance of X = (e^(σ²) − 1) × e^(2μ + σ²). This simplifies to: Var(X) = Mean² × (e^(σ²) − 1). The coefficient of variation (CV = SD/Mean) = sqrt(e^(σ²) − 1), which depends only on σ, not μ. For small σ: CV ≈ σ. For σ = 0.5: CV ≈ 0.53 (53%). For σ = 1: CV ≈ 1.31 (131%).

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