Anal. Chem. 1989, 6 1 , 1465-1466
LITERATURE C I T E D
Table I. Sauare Wave Voltammetric Data for Oxidation of Ascorbic Acid
Anderson, J. E.; Tallman, D. E.; Chesney, D. J.; Anderson, J. L. Anal. Chem. 1978. 50, 1051-1058. Weisshaar, D. E.; Tallman, D. E. Anal. Chem. 1983, 55, 1146-1151. Dayton, M. A.; Brown, J. C.; Stutts, K. J.; Wlghtman, R. M. Anal. Chem. 1980, 52, 946-950. Ponchon, J. L.; Cespuglio, R.; Gonon, F.; Jouvet, M.; Pujol, J. F. Anal. Chem. 1979, 57,1483-1486. Wightman, R. M. Anal. Chem. 1981, 53, 1125A-1134A. Edmonds, T. E.; Guoliang, J. Anal. Chim. Acta 1983, 757,99-108. Lipka, S.M.; Cahen, G. L., Jr.; Stoner, G. E.; Scribner, L. L., Jr.; Gileadi, E. J . Electrochem. SOC.1988, 135, 368-372. Caudill, W. L.: Howell, J. 0.;Wightman, R . M. Anal. Chem. 1982, 54, 2532-2535. Subramanian, R. V.; Crasto, A. S. Polym. Compos. 1986, 7 , 20 1-218. Bell, J. P.; Chang, J.; Rhee, H. W.; Joseph, R. Polym. Compos. 1987, 8 , 46-52. Chang, J.; Bell, J. P.; Shkolnik, S. J . Appl. Polym. Sci. 1987, 3 4 , 2105-2124. Shaw, 6.R.; Creasy, K. E. J . Electroanal. Chem. Interfacial Electrochem. 1988, 243,209-217. Shaw, 8. R.; Creasy, K. E. Anal. Chem. 1988, 60, 1241-1244. Park, J.: Shaw, 6.R. Anal. Chem. 1989, 67, 848-852. Wang, J.; Golden, T.; Varughese, K.; El-Rayes, I. Anal. Chem. 1989, 61, 508-512. Creasy, K. E.; Wang, C. L.; Shaw, 6.R. Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, Atlanta, GA, March 6-10,1989. Shaw, 6.R.; Haight, G. P., Jr.; Faulkner, L. R . J . Nectroanal. Chem. Interfacial Electrochem. 1982, 140,147-153. Aso, C.; Kuhitake. T.; Nakashima, T. Makromol. Chem. 1989, 724, 232-240. Petersson, M. Anal. Chim. Acta. 1986, 787, 333-338. Cox, B. G.; Jedral, W.; Palou, J. J . Chem. Soc., Da/ton Trans. 1988, 733-740.
peak peak current potential, density, mV mA/cm2
conditionsn glassy carbon electrode; 2 mM ascorbic acid ring-modified poly(viny1ferrocene) composite electrode; blank electrolyte, pH 3.0 ring-modified poly(viny1ferrocene) composite electrode; 2 mM ascorbic acid, pH 3.0
529
0.2b
391
0.3”
362
6.4c
Other conditions are as in Figure 7. Current density was calculated by using geometric surface area. CCurrentdensity was calculated by using the combined surface areas of all fibers. Figure 8. Both charging and faradaic currents increased over the 1-2.5-h time periods shown. The changes over time were more dramatic a t lower pH’s because as the hydronium ion activity increased, swelling of the pyridine-containing electrode surface became more extensive and occurred more rapidly. The higher charging currents indicate that increased active surface area was created as time passed. Since the faradaic current rose much more rapidly than the charging current, it is evident that the polymer, positively charged due to pyridine protonation, concentrated the highly charged ferricyanide ions. ACKNOWLEDGMENT We thank Jongman Park for advice on vinylpyridine copolymer matrices, Chia Lin Wang for data on the single-fiber ring-modified electrode, Andrew Garton for assistance with FTIR spectroscopy, and James P. Bell, Angel Wimolkiatasak, and Jemei Chang, Department of Chemical Engineering and Institute of Materials Science, for useful discussions related to electropolymerized interlayers in composites. Expertise in scanning electron microscopy offered by Carol Blouin, Institute of Materials Science, is acknowledged gratefully.
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*
To whom correspondence should be addressed.
Kenneth E. Creasy B r e n d a R. S h a w * Department of Chemistry U-60 University of Connecticut 215 Glenbrook Road Storrs, Connecticut 06269-3060 RECEIVED for review July 5, 1988. Revised March 24, 1989. Accepted March 27,1989. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, to the Research Corporation, and to support by the National Science Foundation under Grant NO. CHE-8707973.
A Statistical Justification Relating Interlaboratory Coefficients of Variation with Concentration Levels Sir: One of the most intriguing empirical relationships in modem analytical chemistry was published by Horwitz in 1980 (1, 2); see for example Figure 1 of ref 2. The graph results from an examination of over 50 interlaboratory collaborative studies conducted by the Association of Official Analytical Chemists on various commodities for numerous analytes over the last 100 years. Individual methods are tested by a t least half a dozen laboratories on a series of blind samples. The results are analyzed for bias and interlaboratory variability. The graph relates the interlaboratory coefficients of variation (CV) found during proficiency testing with the concentration levels at which those particular analyses need to be carried out. The smooth relationship accomplishes this with no respect for the quite different methodologies and instrumentation used for the various analyses. We have taken the liberty of labeling the graph as the “Horwitz Trumpet”. The trumpet has profound implications for the level of detection and the precision that can be ex-
pected in setting legal controls in health-related legislation. The statistical techniques were developed to detect significant deviations from predetermined quality requirements and to provide a warning when this was no longer being fulfilled. The classical expectation of quality assurance is the production of identical interchangeable articles. However when an analytical laboratory produces results on a foodstuff, the results are not a series of identical measurements. The foodstuff varies in composition and there is no predetermined, absolute reference point from which to measure. Frequently a consensus value is established by adding a constituent and then recovering it. This somewhat artificial process is the closest that is obtained to a reference. We now present a very simple theory that accounts very well for the experimental observations. Let us postulate the following origin of the “trumpet curve”. Suppose each laboratory result were the summation of many simple yes/no binomial components, each estimating the
0003-2700/89/036 1-1465$01.50/0 0 1989 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 13, JULY 1, 1989
concentration p . Then each point on the curve would be computed from a binomial distribution with parameters n and p , for some n. Each recorded sample result may be thought of as the mean of n birtomial components. Of course we are not saying that all these experiments are carried out individually-only that the statistical accuracy of the results is such that the same coefficient of variation could hypothetically be achieved by carrying out n individual "yes-no" experiments. The yes-no experiments may be thought of as the individual quanta of which a large, sophisticated analytical scheme is composed. The variance of the mean of a binomial value is n-'p(l p ) , while the expected value is p . Thus the coefficient of variation is the approximation being valid for small p . For fmed n,a graph of (np)-1/2 against p or -log p , for p decreasing from 1to zero, has the Horwitz trumpet shape. However, the fit of the calculated curve to the experimental curve at the center shows a positive deviation at high p and a negative deviation a t low P. To obtain a good fit, the number of "effective binomial components" that constitute a result must increase as p decreases. That is, at lower levels of detection as the experiment becomes more difficult, we postulate an increase in the number of "effective components" involved. The increased difficulty at lower concentrations includes the increasing importance of extraction, cleanup, and contamination, as well as the increasing demands on the instrumentation and the operator. The question now arises as to what dependence of p on n to assume. Zipf s law or the principle of least effort (3)argues that a geometric relation exists for the extra "effort'! (in our case increase in n log n) needed to cope with greater difficulty (in our case smaller p ) for given cost of the experiment. In more detail, note that the effort or negative entropy associated with an n-component experiment is proportional to n log n. The difficulty of each component is proportional to log p , and so the total difficulty of the experiment is proportional to n log p. The cost of an n-component experiment is proportional to n. Minimization of the effort for a given degree of difficulty and a given cost may be shown mathematically to produce n = pp-*, where a and 3( are constants whose values are determined by the given difficulty and cost. (Use the method of Lagrange multipliers.) Zipfs law may also be derived from Fermat's principle of least time (4). Taking n ( p ) = &I-" in the formula CV@) = ( n ~ ) - ' we /~, obtain
CV(p) = (pp-"p)-"2 = (pp1-")-"2, 0 < p < 1
(1) Equivalently, log,, C V ( p )and log,, p-' admit the linear relationship
I
"
'
/
"
'
I
60%
b /*
Flgure 1. A comparison between the empirical Horwitz trumpet and the theoretical equation (2).
The fit is good. It supports the idea that the effort put into detecting lower concentrations increases in such a way that the "precision" changes as though sample size increased inversely with the concentration p . The actual relationship is as Therefore the precision keeps pace to some extent with increasing "intrinsic error", although a would have to equal 1.0 rather than 0.7 before the precision would be equal a t all concentrations. Conversely, one could argue that the effort put into measuring a t high concentrations is correspondingly reduced because the high precision in that region is unwarranted. The value of a is as approximate as the Horwitz trumpet curve itself. However it is the very resilience of the trumpet that is its attraction. We believe that the utter simplicity of our model and its good fit make it attractive in turn. The analytical chemist a t work appears to provide a beautiful example of Zipf s "Principle of Least Effort"-no offence meant!
LITERATURE CITED (1) Horwitz, W. Evaluation of Analytical Methods Used for Regulation of Food and Drugs. Anal. Chem. 1982, 54(1).67A. (2) Horwitz, W.; Laverne, R. K.; Boyer, W. K. Quality Assurance in the Analysis of Foods for Trace Constituents. J . Assoc. Off. Anal. Chem. 1980, 63(6),1344. (3) Zipf, G. H. Human Behaviour and the Principle of Least Effort; Hafner: New York, 1965. (4) Feynman, R. P. The Feynman Lectures on Physics; Addison Wesley Publishing Co.: Reading, MA, 1973; Vol 1, Part 2 [Optics 26-41,
Peter Hall Approximate values a = 0.7 and p = 2 500 are obtainable by nonlinear regression from data in ref 2. Equation 1 then becomes
CV(p)= ~ * . ' ~ / 5 0 (2) which is very close to the formula 100 CV(p) = 2 ~ ' - " log,, . ~ 2. To test the suitability of this formula, we interpolated from 10-4.8 f lo4 , Figure 1 at p = 10-9.0,and 10-lo.o, read off values of CV(p),and compared these with the values of CV(p)given by formula 2. The curve obtained from formula 2 lies almost directly on top of the trumpet curve; see Figure 1.
Department of Statistics The Australian National University G.P.O. Box 4 Canberra 2601, Australia
Ben Selinger* Department of Chemistry The Australian National University G.P.O. Box 4 Canberra 2601, Australia
RECEIVED for review October 17, 1988. Accepted March 16, 1989.
