REPORT FOR ANALYSTS the scale in which the measurement is expressed. In such cases, the usual measures of reproducibility, such as standard deviation, average deviation, or coefficient of variation, are unsuited for evaluating test methods. For ex ample, the scales on both the Williams plasticity and the Mooney viscosity instruments are graduated in arbitrary units. If one of these instruments had a larger coefficient of variation than the other, it would suffice to make an appro priate nonlinear change in scale (such as logarithmic exponential, or raising to an appropriate power) to reverse the order of superiority of the two instru ments from the viewpoint of coefficient of variation. Sensitivity, on the other hand, is invariant, no matter what trans formation is made in the scale of meas urement. Speetrophotometrie ApplicaHons Sensitivity has the further advantage of focusing attention on the importance x>f the behavior of the experimental error in any type of measurement. For ex ample, in speetrophotometrie work, when Beer's law is applicable, it is customary to select concentrations i\ the neighborhood of 37% transmittance for the measurement of transmittance or absorbance. The reason for this choice is the fact 1 dc that the quantity - X —, where c is conc dl centration and I is transmitted intens ity, is smallest when — = e~l = 0.37, •10
provided that the equation expressing Beer's law, — = e~kc, holds. When this line of reasoning is used, the underlying assumption—viz., the constancy of the error of transmittance measurements over the entire range of transmittance values—is not immediately apparent. The sensitivity criterion can be applied to this problem in the following manner: In the above formulation of Beer's law, the concentration in the cell c re sults generally from dilution of the original unknown concentration c0, by a dilution factor, D. Thus, c = —. The problem is to select D in such a way that the sensitivity of the measurement, y, is maximized.
Since = e-Kc = e"
y The Westphal balance is useful in making density determinations
The sensitivity is therefore: Κ _ X C -Î - e D
D
cri/h
The maximum of this function, with respect to D, is obtained when the numerator is the maximum, provided that the denominator is constant. I t is easily found that this maximum occurs when D = kco—i.e., when - = e _ 1 = Jo 0.37. But if the behavior of the error term is different—i.e., if the standard deviation of transmittance varies systematically from one end of the scale to the other—the maximum value of the sensitivity no longer occurs a t the transmittance value of 37%. The sensitivity of a method applies tc a single determination. The average of η determinations has a sensitivity in creased by the factor \/n. Thus, if method A has a sensitivity twice that of method B, a single determination b y method A is equivalent to four deter minations by method B . Such informa tion is especially valuable in the over-all evaluation of alternative methods of measurement, which should be based not only on their technical merit but also on economic considerations. The over-all cost to obtain analytical infor mation with a desired degree of con fidence is equal to the cost per determi nation multiplied by the number of de terminations necessary to attain this confidence. Thus, sensitivity provides an objective means of making rational decisions in the choice of methods of measurement.
the slope of — with respect to c0 (neglect-
Literature Cited
ing the sign) is:
(1) Mandel, J., Stiehler, R. D., / . Re search Natl. Bur. Standards S3, 155 (1954), R P 2527; Natl. Bur. Stand ards Tech. News Bull. 40,139(1956).
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VOL. 29, NO. 4, APRIL 1957 ·
19 A
