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Failure of sign recovery from magnitude-only calibration data occurs when opposite signed calibration models induce exactly the same distribution of every recorded magnitude. Under this symmetry, more repetitions can estimate the magnitude distribution accurately without deciding the missing sign. This is the limitation relevant to sign-sensitive measurements that resolve a covariance-axis ambiguity.

Let $Z=sm+ϵ$ with sign $s\in \{-1,+1\}$, known $m>0$ and symmetric noise $ϵ$ having density $f(x)=f(-x)$. If the recorded value is $Y=|Z|$, then for $y>0$Symmetry gives $g(y\mid +1)=f(y-m)+f(y+m)=g(y\mid -1)$. Independent repetitions multiply identical densities, so their likelihood ratio remains one. With equal sign priors and equal decision costs, no decision rule based solely on these magnitudes can have average sign-error probability below1/2.

For normal distribution noise with standard deviation $\sigma$, the magnitude has a folded normal distribution:where $\varphi$ is the standard normal density. The formula depends on $|m|$, not its sign. The pinned numerical reference provides the equivalent standardized folded-normal density and its scale transformation.^[ref]

For the original calibration scale $m=0.20\mathrm{V}$ and $\sigma =0.050\mathrm{V}$, recording a signed value can distinguish the two models strongly, but recording only its magnitude cannot do so under the symmetric model. This is an information loss in the measurement map, rather than a shortage of averaging trials.

The symmetry assumption is essential. As an original counterexample, let noise equal−1 with probability2/3 and+2 with probability1/3, giving zero mean but an asymmetric distribution. With $m=1$, the positive-sign magnitudes are0 or3, while the negative-sign magnitudes are2 or1. Their disjoint supports permit sign recovery if this asymmetric noise model is independently established. The calibrated asymmetry itself supplies sign-reference information absent from the symmetric case.

Similarly, a genuinely signed projection or a known phase reference can break the original ambiguity. An unverified readout polarity or unknown amplitude sign can preserve it. Specify which signed information the apparatus actually records before interpreting a magnitude fit as an oriented calibration.

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.

L. S. Nelson is a coauthor, with F. C. Leone and R. B. Nottingham, of the 1961 paper The Folded Normal Distribution by F. C. Leone and others. It develops estimation of the underlying normal distribution's mean and standard deviation from measurements reported as absolute values, with tables and an example to support the procedure.^[ref]

The distinction between the observed mean and the signed mean is important. For $Y=|X|$ with $X\sim N(\mu ,{\sigma }^2)$, $Y^2=X^2$, soA second moment alone does not separate offset from spread. For example, $(\mu ,\sigma )=(0,1)$ and $(0.6,0.8)$ both give $\mathbb{E}{Y}^2=1$, although their full folded normal distributions differ. Additional distributional information or moments are needed to distinguish these cases.

When $\mu =0$, the observed mean is $\sigma \sqrt{2/\pi }$, about $0.798\sigma$, rather than zero. Interpreting the average absolute deviation as the original signed mean would therefore create an apparent positive offset even for a centered normal distribution. These elementary consequences explain the measurement problem addressed by the joint paper; they are illustrative derivations rather than an attribution of a particular calculation to Nelson alone.

The verified source prints the author as L. S. Nelson. A shared surname and initials are insufficient to attach a separate biography or portrait without a source linking the records.

R. B. Nottingham coauthored The Folded Normal Distribution by F. C. Leone and others with F. C. Leone and L. S. Nelson, published in Technometrics in 1961. Their paper addressed measurements whose algebraic signs were not recorded. Even if a signed quantity follows a normal distribution, its recorded magnitude follows a different probability distribution.^[ref]

The practical issue is information loss during observation. If $X\sim N(\mu ,{\sigma }^2)$ and the instrument reports $Y=|X|$, then $\mu$ and $-\mu$ produce the same folded normal distribution. Thus magnitudes alone cannot distinguish an offset to one side of a reference from an equally large offset to the other side. This conclusion follows from the observation rule, not from the quality of a numerical fitting method.

An illustrative inspection records distances from a reference line but omits which side each point lies on. A pair of signed readings $-2$ and $+2$ becomes two identical magnitudes $2$. Retaining the original signs would preserve a distinction that no later analysis of those two magnitudes can reconstruct. The paper's contribution is to treat this altered observation distribution explicitly rather than fit an ordinary signed normal distribution to magnitude data.

The cited work is jointly authored; the primary byline does not assign individual responsibility for particular formulas or tables. The printed initials are retained here without attaching unrelated biographical records to this author.