What it means
A normal distribution is symmetric around its mean, and exponentiating a normally distributed variable changes the shape and scale: negative and positive values on the log scale become positive values on the original scale, and large positive log values create especially large original values. This makes the lognormal model relevant to quantities affected by multiplicative changes, because a sequence of proportional factors becomes a sum when expressed as logarithms.
That can motivate considering a lognormal model, but does not prove every process with percentage changes meets its assumptions. The NIST reference defines the distribution through the logarithm and describes different parameterisations.
In the common zero-location form, mu is the mean of the natural logarithm and sigma is its standard deviation, and they are not the ordinary mean and standard deviation of the original data. The median of that form is exp(mu) and its arithmetic mean is exp(mu + sigma squared divided by two), which is greater than the median when sigma is positive.
The difference matters for budgets and service measures, because a few very large positive values can raise the arithmetic average above the central experience. A manager using the mean as a typical case should understand how the right tail affects that number, and reporting a median alongside the mean can make the distribution easier to interpret without pretending the two measures answer the same question.
The standard two-parameter model requires strictly positive observations, and zero and negative values cannot be fed into its natural-log transformation. NIST also discusses a shifted form with a location parameter, but adding a shift is a modelling choice that must be justified and documented.
Lognormal does not mean the original values form a normal bell curve, since a histogram on the original scale and one on the log scale can look quite different. The relevant check concerns normality on the transformed scale, together with how well the model represents important outcomes.
NIST identifies reliability and failure-time modelling among common applications, so a potential use is to represent positive durations with a long right tail. The distribution should still be compared with alternatives and tested for the specific process rather than selected from its name alone.
Financial applications need particular caution, because modelling a positive price with a lognormal distribution is different from claiming that every observed return follows that distribution. Returns can be negative, and real market behaviour can depart from simplified models.
The fitted model is also limited by the data period and process, since changes in operations, measurement or customer mix can make a historical fit less useful, and a mathematically neat distribution should not hide a change in what generated the observations.
In practice
Real-world examples.
Example
A fictional maintenance team analyses positive repair durations. It considers a lognormal model because a few repairs take much longer than most, then checks the transformed data rather than assuming right skew is sufficient proof.
Example
A manager compares an average invoice size with its median. A long right tail explains why a small number of very large invoices can lift the mean above the size of a typical invoice.
Example
An analyst includes zero-valued observations in a log transformation and encounters a problem. They review the data and model choice rather than silently dropping the zeros or adding an unexplained constant.
Formula
Calculation
For X = exp(Y), with Y normally distributed with mean mu and standard deviation sigma, median(X) = exp(mu) and mean(X) = exp(mu + sigma squared / 2).
Using invented parameters mu = 0 and sigma = 1, the median is exp(0) = 1, while the mean is exp(0.5), approximately 1.6487. The units are hypothetical. These calculations illustrate the difference between mean and median, not a fitted estimate for a real business.Case study
Seen in the real world.
In this fictional case, Birch Repairs uses average repair time to set every customer's expectation. Most jobs finish quickly, but occasional complex work produces much longer durations and lifts the average. The analyst considers a lognormal model and inspects the positive durations and their logarithms. The team compares the fit with alternatives and distinguishes a central estimate from a high-duration planning scenario.
It documents exclusions rather than hiding unusual jobs. The service report now explains both typical experience and the right-tail risk. The case shows how a distribution can clarify planning only when its assumptions and the underlying process are checked.
Watch out
Common mistakes.
- Confusing log-scale parameters with the original data's mean and standard deviation.
- Assuming every right-skewed dataset must be lognormal.
- Dropping zeros or negative values without explaining how that changes the data and model.
Questions
People also ask.
Is a lognormal variable normally distributed?
Its natural logarithm is normal; the original variable generally is not.
Can the standard two-parameter form include negative values?
No. Its original values are strictly positive.
Are mean and median the same?
No. With positive log-scale variance, the arithmetic mean exceeds the median.
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