The winner of the best measurement challenge (published in volume 415 issue 16) is:

Cristhian Paredes, Instituto Nacional de Metrología de Colombia, Bogotá, D.C., Colombia.

The award entitles the winner to select a Springer book of their choice up to a value of €100.

Our Congratulations!

The best measurement challenge [1] is about the isotope pattern of bromine-containing molecules, which has been the topic of a previous Analytical Challenge [2, 3]. The mass spectrum of tribromobenzene features the familiar a:b:c:d = 1:3:3:1 pattern. Despite the fact that the two signal ratios b:a and c:a appear identical, turns out that they give different quality results for the isotopic abundance of bromine-81. To understand this observation, it is useful to establish data-generative measurement model.

The isotopic pattern of tribromobenzene can be modeled, as a first approximation, by taking into account only the bromine atoms. This can be accomplished using Pascal’s triangle [4] and it involves a single variable—the isotopic abundance of bromine-81:

$${A}_{312}:{A}_{314}: {A}_{316}: {A}_{318} \approx {(1-{x}_{81})}^{3} :{3x}_{81}{\left(1-{x}_{81}\right)}^{2} :{{3x}_{81}}^{2}(1-{x}_{81}) :{{x}_{81}}^{3}$$

From here, we can derive the model for the two isotope ratios, R314/312 and R316/312:

$${R}_{314/312}=3{x}_{81}/\left(1-{x}_{81}\right)$$
(1)
$${R}_{316/312}=3{x}_{81}^{2}/{\left(1-{x}_{81}\right)}^{2}$$
(2)

The inversion of these expressions provides us with the explicit measurement models for the quantity of interest, x81:

$${x}_{81}=\frac{{R}_{314/312}}{{R}_{314/312}+3}$$
(3)
$${x}_{81}=\frac{{R}_{316/312}-\sqrt{3{R}_{316/312}}}{{R}_{316/312}-3}$$
(4)

Now, we can perform sensitivity analysis to determine how these two estimates of x81 are affected by measurement uncertainty associated with isotope ratios R314/312 or R316/312. This can be done by formal computation of the first-order derivatives, dx81/dR314/312 and dx81/dR316/312, or by simply calculating the values of x81 from several values of R314/312 or R316/312, as shown in Fig. 1.

Fig. 1
figure 1

Isotopic abundance estimates of bromine-81 (x81) from isotope ratios R314/312 and R316/312 of tribromobenzene (ignoring the contributions of carbon and hydrogen isotopes)

Full size image

Figure 1 shows that the estimates of bromine-81 abundance from the 316/312 ratio are twice less affected by the small variability (measurement uncertainty) of isotope ratios than those from the 314/312 ratio. This can be seen from the slopes of linear equations that approximate the relationships between x81 and R316/213 or R314/312 in the vicinity of R = 2.90. Put differently, ± 1% variations in R314/312 will lead to ± 0.50% variations in x81 whereas ± 1% variations in R316/312 will lead to twice smaller (± 0.25%) variations in x81.

Thus, from statistical considerations, one should choose the 316/312 ratio. Of course, real-life measurements often involve more than just statistical considerations. For example, some ions may have spectral interferences. Nevertheless, this Analytical Challenge offers an example where statistical considerations can differentiate between two seemingly similar measurements.