What exactly is the Horwitz equation?
The Horwitz function 1, also known as the Horwitz “Trumpet”, is an empirical relationship that has, thus far, not been derivable from fundamental principles. As noted by Thompson 2, and as shown below in Fig. 1 (from 2), the relative standard deviation of reproducibility (RSDR) increases as analyte concentration decreases:



Horwitz formulated his expression after noticing a pattern in the relative standard deviations reported in many (thousands at this point) collaborative trial results. The following figure (a screenshot from Hanley [3, p. 12037]) provides the essential equations and background information:



As Hanley notes, the Horwitz function appears to be in the Tweedie family of exponential dispersion fluctuation scaling relationships (https://en.wikipedia.org/wiki/Tweedie_distribution ). Specifically, the Horwitz function may be a compound Poisson-Gamma process, since the exponent, $\alpha $ in equation 2, is approximately 0.85. (More accurately, Thompson 1 notes that it is $1 - (log2)/2$).

Answer to OP’s first question

Assume measurements are Gaussian distributed, with $\mu $ as the population mean and $\sigma $ as the population standard deviation, and assume both parameters are unknown. Let $X_{lab} $ be a single measurement, i.e., a random sample from the Gaussian distribution. Then $z = (X_{lab} - \mu )/\sigma $. So z, which is the so-called ‘z score’, is simply the signed difference parsed in units of $\sigma $. A small z score means the measured result is close to the true value, i.e., $\mu $, assuming no systematic error. A large z score means the measurement has relatively poor accuracy.

But both $\mu $ and $\sigma $ are unknown, so $X_{cons}$ is used as an estimate of $\mu $ and $S_{target} $ is used as an estimate of $\sigma $. The result is a rough, and possibly biased, estimate of z. Frankly, this is too crude for me.

Answer to OP’s second question

Also as noted by Hanley, attempts have been made to derive the Horwitz function from “the binomial distribution, Zipf’s law, and a heuristic approach”: see 4 and 5 below. However, these have not been generally been accepted and it is fair to say that the final word has not been written on the origin of the Horwitz function. As well, the Horwitz function potentially has a problem at zero analyte concentration, as noted by Holland et al. [6]. This is problematic because analytical blanks do not have infinite standard deviations.

References:


W. Horwitz, “Evaluation of analytical methods used for regulation of foods and drugs”, Anal. Chem. 54 (1982) 67-76.
M. Thompson, “The amazing Horwitz function”, AMC Technical Brief No. 17, July, 2004 Royal Society of Chemistry. (2 pages).
Q.S. Hanley, “Chemical Measurement and Fluctuation Scaling”, Anal. Chem., 88 (2016) 12036-12042 (Open Access).
R. Albert, W. Horwitz, “A Heuristic Derivation of the Horwitz Curve”, Anal. Chem., 69 (1997) 789−790.
P. Hall, B. Selinger, “A statistical justification to relating interlaboratory coefficients of variation with concentration levels”, Anal. Chem., 61 (1989) 1465−1466.
R. Holland, R. Rebmann, C.D. Williams, Q.S. Hanley, “Fluctuation Scaling, the Calibration of Dispersion, and the Detection of Differences”, Anal. Chem., 89 (2017) 11568-11575.