The Log-Normal Distribution: When Logarithms Are Normal

Many real quantities cannot go below zero and have a long tail stretching to the right: incomes, house prices, waiting times, the sizes of particles, the concentrations of chemicals in blood. A bell curve fits them badly, since it would predict negative values and a symmetric spread. Yet they often become beautifully bell-shaped after taking logarithms. A variable whose logarithm follows a normal distribution is said to be log-normally distributed, and recognizing it turns hard-looking skewed data into ordinary normal distribution problems.

The Definition

If Y follows a normal distribution with mean μ and standard deviation σ, then X = e^Y follows a log-normal distribution. Equivalently, if X is log-normal, then ln(X) is normal. The parameters μ and σ describe the distribution of the logarithm, not of X itself, which is a frequent source of confusion. Because an exponential is always positive, X can never be negative or zero, and because exponentiation stretches large values, the distribution is right-skewed. A larger σ produces a heavier right tail.

Mean, Median, and Mode

The three measures of centre no longer coincide, and they fall in a fixed order:

The median is simply e^μ, because the median of the logs, μ, maps straight through the increasing exponential function. The mean sits above the median, pulled up by the long right tail, exactly the pattern described in skewness and kurtosis. The variance is (e^σ² − 1) × e^{2μ + σ²}, and the coefficient of variation depends on σ alone: √(e^σ² − 1).

Worked Example

Problem: Household incomes in a region are modelled as log-normal, with a median of $50,000 and σ = 0.6 on the log scale. Find the mean and mode, the probability that a household earns more than $100,000, the 90th percentile, and the probability of earning between $30,000 and $80,000.

Step 1 — The parameters. The median gives μ = ln(50,000) ≈ 10.820, with σ = 0.6.

Step 2 — Mean and mode.

• Mean = 50,000 × e^0.6²/2 = 50,000 × e^0.18 ≈ $59,861
• Mode = 50,000 × e^−0.6² = 50,000 × e^−0.36 ≈ $34,884

The typical (most common) household earns about $34,900, the middle household earns $50,000, and the average is nearly $60,000, pulled up by high earners. The standard deviation is about $39,400, giving a coefficient of variation of √(e^0.36 − 1) ≈ 0.66.

Step 3 — Probability of earning more than $100,000. Work on the log scale with a z-score, as in the z-score method:

P(Z > 1.155) ≈ 0.124. About 12.4% of households earn more than $100,000.

Step 4 — The 90th percentile. The 90th percentile of a standard normal is 1.2816, so on the log scale the value is μ + 1.2816σ. Exponentiating: 50,000 × e^{1.2816 × 0.6} = 50,000 × e^0.769 ≈ $107,873. By symmetry on the log scale, the 10th percentile is 50,000 × e^−0.769 ≈ $23,175. Notice the asymmetry: the 90th percentile is $57,873 above the median, while the 10th is only $26,825 below it. On the log scale the distances are equal, but in dollars the upside is far bigger.

Step 5 — Probability between $30,000 and $80,000. The z-scores are ln(30,000 ÷ 50,000) ÷ 0.6 ≈ −0.851 and ln(80,000 ÷ 50,000) ÷ 0.6 ≈ 0.783. The probability is Φ(0.783) − Φ(−0.851) ≈ 0.783 − 0.197 = 0.586, so roughly 59% of households fall in that band.

Why Multiplication Produces It

The normal distribution arises when many small effects are added, as the central limit theorem explains. The log-normal arises when many small effects are multiplied. Taking logarithms turns a product into a sum, so the log of a product of many independent positive factors behaves like a sum of many independent terms and tends toward normality. That is why the log-normal is a natural model for quantities built up through repeated proportional changes: a fortune growing by random percentages each year, a particle repeatedly breaking into pieces, or a bacterial population multiplying by random factors. It also explains why relative comparisons ("20% higher") suit such data better than absolute differences.

Fitting and Checking

Fitting the distribution to data is simple: take logarithms of every observation and treat them as a normal sample. The mean of the logs estimates μ, and their standard deviation estimates σ. Exponentiating the mean of the logs gives the geometric mean, which estimates the median. For the four values 20, 40, 80, and 160, the logs average to a mean whose exponential is 56.57, the geometric mean covered in weighted and geometric mean, and not the arithmetic mean of 75.

To check the fit, plot the logged data on a Q-Q plot or run a normality test such as the Shapiro-Wilk test on the logs. If the logs look normal, the log-normal model is reasonable. Similarly, applying a log transformation before running a t-test on right-skewed data is a common way to satisfy normality assumptions, with results then read in terms of ratios rather than differences.

Real-World Uses

Economics: incomes and wealth in the bulk of a population, with the far upper tail often heavier than a log-normal predicts.

Biology and medicine: concentrations of substances in blood, incubation periods of diseases, and organism sizes often follow log-normal patterns, and analysts report geometric means and ratios.

Engineering and reliability: repair times and material strengths are commonly modelled with a log-normal, alongside the Weibull distribution.

Common Mistakes

Treating μ and σ as the mean and standard deviation of X. They are the mean and standard deviation of ln(X). The mean of X is e^{μ + σ²/2}, not μ.

Averaging the raw values and calling it typical. With skewed data, the arithmetic mean overstates the typical case. The median or geometric mean usually communicates better.

Taking the logarithm of zero or negative values. The transformation requires strictly positive data, so zeros must be handled deliberately, not silently dropped.

Assuming any right-skewed data is log-normal. Several skewed shapes look similar. Always check the logged data before relying on the model.

How is the log-normal different from a normal distribution?

A normal variable is symmetric and can be negative. A log-normal variable is strictly positive and right-skewed, and it is the exponential of a normal variable. Its logarithm is exactly normal.

Why is the mean larger than the median?

Exponentiation stretches the upper tail, so a few very large values pull the average above the middle value. The gap widens as σ grows.

Can I use a t-test on log-normal data?

Directly on the raw data it is risky when skewness is strong. A common approach is to take logs first and run the t-test on the logged values, then back-transform the difference to a ratio of geometric means.

What happens to the distribution as σ shrinks?

As σ approaches zero the skew disappears, the mean, median, and mode converge, and the distribution looks nearly normal around its centre.