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Fred C. Leone coauthored the 1961 paper The Folded Normal Distribution by F. C. Leone and others with L. S. Nelson and R. B. Nottingham. The original paper lists Leone at the Case Institute of Technology. Its motivating problem was industrial measurement in which a difference or deviation is retained but its algebraic sign is lost or disregarded.^[ref]

The resulting folded normal distribution combines two possible signed observations at each positive magnitude. If $X$ has normal distribution density $f_X$ and $Y=|X|$, thenThe negative side is reflected onto the positive side. It is not simply deleted. At zero signed mean this gives the half-normal distribution; a nonzero offset changes the shape while the observation remains nonnegative.

For a concrete constructed example, suppose a measuring device displays distance from a target rather than signed displacement. A true mean displacement $\mu =2$ and its mirror $\mu =-2$, at the same standard deviation, give identical recorded-distance distributions. The joint paper treats the distribution and estimation problem induced by that recording convention. Its tables and example connect the mathematical transformation to analysis of measurements.^[ref]

The work illustrates a broader principle of statistical modelling: specify what the instrument records before selecting the data distribution. A measurement transformation can alter the observed shape without any change in the underlying physical variability.