Updated 12 September 2026

Log-normal distribution

If X is normally distributed then Y = e^X is log-normal: its logarithm is normal. A product of independent positive factors can be approximately lognormal when a CLT applies to their log values. For example, iid log factors with finite mean and variance admit a standardized-sum CLT. Independence of arbitrary positive factors alone is insufficient.

What to notice

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

Why multiplicative processes produce log-normals

Taking logarithms changes a product into a sum: log(X₁ · X₂ · … · Xₙ) = log X₁ + … + log Xₙ. If the log factors satisfy an appropriate CLT, their centered, scaled sum approaches normal. This can motivate a lognormal approximation for a finite product; it does not mean the unscaled product converges to one fixed lognormal law.

Multiplicative growth motivates this model. Dependence, time-varying parameters, and extreme tails can invalidate it; inspect the log-scale data before relying on the approximation.

Mean vs median divergence

The mean is pulled right by rare extreme values — a direct consequence of the lognormal mean formula. This is a statement about the lognormal model, not evidence that a particular income dataset follows it. The median locates the halfway point; the mean is needed for expected totals.

The two scales describe the same observations

At μ=0 and σ=1, the log values have mean zero and standard deviation one. On the original scale, the median is 1, the mean is exp(1/2)≈1.648721, and the mode is exp(−1)≈0.367879. The original-scale variance is (e−1)e≈4.670774. The parameters μ and σ belong to ln X, not X.

Switch the scale selector. The first-five-draw table remains the same sample, while the histogram and its density transform together. Simply relabeling a horizontal axis without changing the density would give the wrong areas. The log view uses density per log unit; the original view uses density per original unit.

An exact multiplicative construction

Let independent log factors A and B both be Normal(0,0.5), where 0.5 denotes variance. Then A+B is Normal(0,1), and exp(A)×exp(B)=exp(A+B) is exactly Lognormal(0,1). The product’s median is one and its mean is exp(1/2). This exact result relies on normal log factors; an approximate CLT argument for many other factors needs separate assumptions.

For a fixed worked pair A=0.4 and B=−0.1, the factors are approximately 1.491825 and 0.904837. Their product is exp(0.3)≈1.349859. Adding the original factors instead would define another quantity and generally would not be lognormal.

A central plot can hide important probability

At μ=0, σ=1, P(X>exp(2))=P(Z>2)≈0.022750, and the 95th percentile is exp(1.644854)≈5.180252. Raise μ to 2 and σ to 2: the model mean becomes exp(4)≈54.598, beyond the original plot’s upper limit of 25. The tail calculation and omitted-count notice retain that distinction.

Reference

Random Services: lognormal distribution gives the log transformation, moments, quantiles, and independent-product relationship. The normal lesson explains the sum rule, while Pareto gives a different positive-tail model.