Horwitz trumpet

It is a well-known fact that the precision in trace analysis decreases with diminishing concentration in a similar way as it does with decreasing sample weight (Sect. 2.1). The dependency of the repeatability and reproducibility standard deviation on the concentration of analytes has been investigated systematically at first by Horwitz et al. [1980] on the basis of thousands of pieces of interlaboratory data (mostly from food analysis). The result of the study has been represented in form of the well-known Horwitz trumpet which is represented in Fig. 7.3. 

Fig. 7.3. The so-called Horwitz trumpet Dependency of the relative standard deviation, rsd(x) on the concentration (x)...

They further simplify the Horwitz trumpet relationship in two forms as ... 

Coefficient of variation of interlaboratory results as a function of sample concentration (expressed as g analyte/g sample). The shaded region has been referred to as the Horwitz trumpet" because Of the way it flares open. [From W. Horwitz, Evaluation of Analytical Methods Used for Regulation of Foods and Drugs." Anal. Chem. 1982, 54.67A]... 

The target value for the standard deviation, o, should be circulated in advance to the PT participants along with a summary of the method by which it has been established. It will vary with anal)4e concentration, and one approach to estimating it is to use a functional relationship between concentration and standard deviation. The best-known relationship is the Horwitz trumpet, dating from 1982, so called because of its shape. Using many results from collaborative trials, Horwitz showed that the relative standard deviation of a method varied with the concentration, c (e.g. mg g" ), according to the approximate and empirical equation ... 

Note again the appearance of the 2 in equation (4.17), because two sample materials are studied. Here it is a simple matter to calculate that the estimate of si is (3.615). The mean of all the measurements is 49.33/2 = 24.665, so the relative standard deviation is (100 x 5.296)/24.665 = 21.47%. This seems to be a high value, but the Horwitz trumpet relationship would predict an even higher value of ca. 28% at this concentration level. It should be noted that possible outliers are not considered in the Youden procedure, so the possibility of rejecting the results from laboratory 1 does not arise. 

Limitations in Analytical Accuracy Part 1 - Horwitz s Trumpet... 

Workman, J. and Mark, H., Chemometrics in Spectroscopy Limitations in Analytical Accuracy - Part 1 Horwitz s Trumpet, Spectroscopy 21(9), 18-24 (2006). 

Figure 71-1 Relationship of Laboratory CV (as %) with analyte concentration as powers of 10 f. (For example, 6 on the abscissa represents a concentration of 10 or 1 ppm.) Note the shape of the curves has been referred to as Horwitz s trumpet.

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