Right-skewed distribution where log(X) is normal. Income, asset prices, file sizes.
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Log-normal Distribution
A log-normal distribution describes a variable X whose natural logarithm is normally distributed. The result on the original (untransformed) scale is a right-skewed curve: it rises steeply from zero, peaks early, then decays into a long right tail. The mode (the peak) sits to the left; the median is to the right of the mode; the mean is further right still. This separation of mode, median, and mean is a defining visual signature of positive skew.\n\nThe distribution arises naturally when a quantity is the product of many independent positive factors, by the multiplicative version of the central limit theorem. Household income, asset prices, city populations, biological response times, and software request latencies all approximate log-normal distributions.
The three labeled vertical lines each answer a different question. The mode marks the most common value (the peak of the curve). The median marks the 50th percentile: half of all values fall below it. The mean is the arithmetic average, pulled rightward by the long tail. When mean exceeds median by a wide margin, as it does for income data, the average is an unreliable summary: most individuals earn less than the average.\n\nA common confusion is between log-normal and normal distributions. The normal curve is symmetric; its mean, median, and mode coincide. The log-normal curve is asymmetric; the three markers split apart. If a histogram of your data looks like a roughly symmetric bell after a log transformation but skewed before it, the log-normal is likely the better model.
The probability density function of a log-normal random variable X with parameters mu and sigma is:\n\nf(x) = (1 / (x * sigma * sqrt(2*pi))) * exp(-(ln(x) - mu)^2 / (2 * sigma^2)) for x > 0\n\nHere mu and sigma are the mean and standard deviation of ln(X), not of X itself. The mean of X on the original scale is exp(mu + sigma^2 / 2); the median is exp(mu); the mode is exp(mu - sigma^2). The gap between them grows with sigma, meaning more skew. Aitchison and Brown (1957) provide the definitive statistical treatment of the log-normal family.
The Lognormal Distribution , Aitchison, J. and Brown, J. A. C. (1957) - Cambridge University Press. The classical reference for statistical theory of the log-normal distribution.
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Source & accuracy
This log-normal distribution is an editorial illustration built to represent the concept accurately. Where it shows figures, they are typical or representative values chosen to make the relationship clear, not a single underlying dataset. The diagram and its explainer are reviewed and maintained centrally, and updated over time as understanding improves.
Exponential growth with log-normal outcomes
A log-normal distribution describes a variable whose logarithm follows a normal distribution. If you take the log of a log-normally distributed variable, you get a regular bell curve. This relationship emerges naturally when a quantity grows multiplicatively (percentage changes accumulate) rather than additively.
Common examples include personal income (compound salary increases), asset prices (returns compound over time), file sizes on a computer (copies and variants multiply), and particle sizes from fragmentation. Whenever you expect geometric growth or proportional change to dominate, the log-normal distribution often fits well.
Right-skewed shape and mode-mean-median ordering
The log-normal distribution is always right-skewed, with a long upper tail. This skewness means the mode (peak) occurs to the left of the median, which occurs to the left of the mean, creating the ordering mode < median < mean. Extreme outliers in the upper tail pull the mean rightward, sometimes dramatically.
This ordering contrasts with the normal distribution, where mode, median, and mean coincide. The log-normal distribution also has the property that it cannot go negative (since exp(x) > 0 always), making it suitable for purely positive quantities like prices and weights.
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Reference
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A free, embeddable, animated log-normal distribution for any website.
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