Translate lognormal parameters before reporting an arithmetic mean

Define the task

Record which logarithm was used and whether a number describes the log scale or the original scale. A lognormal distribution with natural-log mean m and log SD s has median exp(m). This distribution describes strictly positive outcomes. Structural zeros and a measurement detection limit need explicit handling before adopting it.

Carry out the check

The arithmetic mean is exp(m+s²/2), and the coefficient of variation is the square root of exp(s²) minus one. These population formulas differ from simply exponentiating a sample average and labeling the result an arithmetic mean. Keep uncertainty in estimated parameters separate from the spread of individual observations.

Worked hypothetical example

Consider a hypothetical positive response whose log mean is zero and log SD is log(2). Its median is one, its geometric SD factor is two, and its arithmetic mean is approximately 1.2715. Increasing log SD while retaining the same log mean raises the arithmetic mean while keeping the median fixed. This is a useful consistency check when two tables appear to disagree about the center. Verify log bases before interpreting such a discrepancy.

Method checked 11 November 2026 against R documentation: The Log-Normal Distribution.

Open the lognormal population moment map.

Download the lognormal reporting reconciliation.