Log-normal distribution

If X is normally distributed then Y = e^X is log-normal: its logarithm is normal. Equivalently, if you multiply many independent positive random factors together, the product tends to a log-normal by the Central Limit Theorem applied on the log scale.

f (x; μ, σ) = \frac{1}{x σ 2 π} exp (- \frac{( ln x - μ ) ^{2}}{2 σ ^{2}})

Median = e^μ

1.000

Mean = e^(μ+σ²/2)

1.133

μ (log-mean) 0.0 σ (log-std) 0.5 n (samples) 500

If ln X ~ Normal(μ, σ) then X is log-normal. Amber dashes: median. Rose dashes: mean. Large σ pulls the mean far above the median — the signature of right-skewed distributions.

What to notice

Increasing σ stretches the tail dramatically to the right while the mode moves left. The mean pulls far above the median.
Amber dashes mark the median $e^{μ}$ , rose dashes the mean $e^{μ + σ^{2} /2}$ . The gap between them grows with σ — a diagnostic for skew.
The mode (peak of the density) sits at $e^{μ - σ^{2}}$ , lower still.

Why multiplicative processes produce log-normals

Additive processes → normal (Central Limit Theorem). Multiplicative processes → log-normal. Take logarithms: log(X₁ · X₂ · … · Xₙ) = log X₁ + log X₂ + … + log Xₙ, which is a sum. By CLT, this sum approaches normal, so the product approaches log-normal.

This is why log-normal distributions appear wherever quantities grow by percentage shocks: stock prices, income distributions, city populations, biological measurements like body weight, and pandemic spreading factors.

Mean vs median divergence

The mean is pulled right by rare extreme values — a property shared by all right-skewed distributions, but especially severe for the log-normal. Reported “average income” is almost always larger than “typical income” for this reason. The median is a more representative central value for log-normal data.