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are ill considered. We did not see the losses of sulfur in (NH4)&304and HzS04seen at BYU, but did see an 18 f 5% loss of sulfur in highly hydrated ferrous sulfate samples under special focused beam conditions. We feel that the much higher temperature of the BYU PIXE beam is implicated, while not supporting the theory of additional chemical loss mechanisms in realistic samples of particulate matter. While we feel that the samples and conditions at BYU are not representative of most laboratories,the effects they reported should be carefully studied and evaluated a t any laboratory analyzing sulfur, halogens, or labile species.
LITERATURE CITED
(6) Camp, D. C.; Van Lehn, A. H.; Loo, B. W. “Intercomparison of Samplers Used In The Determinatlon of Aerosol Composltlon”; Interagency Energy/Environment R and D EPA-600/7-78-118. (7) Macias, E. S.; Zwicker, J. 0.; Ouimette, J. R.; Hering, S. V.; Friedlander, S. K.; Cahill, T. A.; Kuhimey, G. A,; Rlchards, L. W. Atmos. Environ., in press. (8) Ouimette, J. R. Ph.D. Thesis Envlronmental Quality Laboratory, California Instltute of Technology, Pasadena, CA, 1980. (9) Cahili, T. A., Flocchlni, R. G., Feeney, P. J., Shadoan, D. J . Nucl. Instrum. Method 1974, 120, 193-195. (10) Thibeau, H. J.; Stadel, J.; Cline, W.; Cahill, T. A. Nucl. Instrum. Methods 1973, 111, 615-617.
Thomas A. Cahill* Department of Physics and Crocker Nuclear Laboratory University of California, Davis Davis, California 95616 Robert G . Flocchini Department of Land, Air 8.1 Water Resources University of California, Davis Davis, California 95616
(1) Hansen, L. D.; Ryder, J. F.; Mangelson, N. F.; Hill, M. W.; Faucette, K. J.; Eatough. D. J. Anal. Chem. 1980, 52, 821-824. (2) Shaw, R. W.; Stevens, R. K.; Courtney, W. J. Anal. Chem. 1980, 52, 2217-2218. (3) Hegg, D. A.; Hobbs, P. V. Anal. Chem. 1980, 52, 2218. (4) Hansen, L. D.; Mangelson, N. F.; Eatough, E. J. Anal. Chem. 1980, 52, 2219. (5) Camp, D. C.; Van Lehn, A. L.; Rhodes, J. R.; Pradzynski, A. H. X-Ray Spectrom. 1975, 4 , 123.
RECEIVED for review March 9,1981. Accepted May 17,1982.
Sir: We agree with Cahill and Flocchini that the question of the accuracy of XRF and PIXE determinations of sulfur requires better resolution. As stated in our paper, the effects of experimental parameters are largely unknown and until the relation between errors in sulfur determination and these parameters are better defined, no one can predict a priori when the errors will or will not occur. We welcome further assistance from Cahill et al. in resolving these problems. The PIXE system at BYU has, as has that at Davis, been shown to give accurate sulfur determinations on most samples; however, we have found some types of samples where the PIXE results are unquestionably low because of loss of volatile or labile compounds from the target. We agree with Cahill and Flocchini that sample heating is more serious under the conditions used a t BYU than those used at UCD. This was clearly illustrated in our paper by comparison of the results obtained on H2S04and (NH4)&304by proton PIXE at BYU and by a-PIXE at Davis. However, we disagree that our system or conditions are unique for PIXE equipment, if anything, the Davis system is the more unusual one. A recent tabulation indicates approximately 4 times as many laboratories using 1-4 MeV proton beams as are using a particle beams of various energies (I). Also, simple heating of the sample by the beam is not the only effect which must be considered. Cahill and Flocchini have missed the point made by the data on FeSO4.7Hz0. We used this compound to illustrate that charged particle beams can cause decomposition of chemical compounds, irrespective of the type of charged particle. We do not claim this particular compound to be present in airborne particles, but it would come as no surprise if iron sulfates, including FeS04-7H20,were found in emissions from steel mills and smelters. Other compounds which undergo decomposition in the charged particle beam may be present in samples of airborne particles from some sources. Our view of what may exist in “realistic samples of particulate matter” is obviously more complex than that of Cahill and Flocchini. Special sampling in indoor work environments or from specific industrial sources will lead to the encounter of chemical species not found in typical ambient aerosols. It was just such a sampling program in a smelter which led to the discovery of the problems reported in our paper. For example, we have consistently found S(1V) compounds, both inorganic and organic, in such sources as smelter plumes, plumes from combustion of fossil fuels, and urban atmospheres. Such compounds are missed in analyses by ion chromatography (2-4) as well as by PIXE. For example, recent results (4)
indicate the organic S(1V) present in atmospheric samples is not eluted as an anion in the ion chromatograph. The comparisons of ion chromatographic determinations of sulfate with PIXE determinations of total sulfur such as cited by Cahill and Flocchini thus raise another interesting question, Le., why are the PIXE total sulfur values never larger than the sulfate values? We believe it is because all of the samples used in the intercomparisonscontained only sulfate salts and, further, only those sulfate salts which are stable in charged particle beams. The sulfate salts in the samples used in the intercomparison studies probably were ammonium, sodium, and/or calcium sulfate which are the salts one would expect to find in atmospheric particulate samples collected at sites reasonably distant from industrial and urban activities. Since nearly all of the PIXE data taken by the group at Davis have been on such samples, we would expect the results to be accurate, and we do not, in general, question the results of PIXE determinations of sulfur in typical background ambient atmospheric particulate samples, especially when a group can show accurate results for sulfur determination in ammonium sulfate. Citing more intercomparisons in which PIXE and XRF are shown to agree with other methods proves only that the methods are in agreement on those samples. It does not disprove the fact that errors in the PIXE and XRF methods do or may exist for some samples. The disagreements reported in our papers do prove that PIXE and XRF methods are subject to the same types of cautions with respect to interferences and matrix effects as any other method for elemental analysis, e.g., atomic absorption. Unlike more fully developed methods, however, the effects of the variable experimental parameters on errors in PIXE methods have not ye3 been thoroughly investigated. For example, how do beam intensity, particle energy, and target composition effect sample heating? How do storage time and conditions affect stability of nonsulfate sulfur compounds? Furthermore, intercomparison results such as 0.96 f 0.12 (PIXE/ion chromatography) and 0.98 f 0.35 (PIXE/flash volatilization) with large standard deviations could hide a multitude of problems in either method. When we see agreement in intercomparisons of XRF or PIXE and other methods for sulfur determinations on samples containing sulfides, sulfites, mercaptans, thiosulfates, sulfonic acids, and elemental sulfur, we will be convinced to accept XRF and PIXE as “nondestructive and general” methods for sulfur determination. We are not the first to raise the issue of negative errors in XRF caused by loss of materials from the sample in the X-ray
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beam. As discussed in the introduction of our paper, the loss of C1 and S in XRF analyses has been reported previous to our work.
LITERATURE CITED (1) Cahlll, T. A. Ann. Rev. Nucl. Part. Scl. 1980, 3 0 , 211-252. (2) Hansen, L. D.; Richter, El. E.; Rollins, D. K.; Lamb, J. D.; Eatough, D. J. Anal. Chem. 1979, 5 1 , 633-637. (3) Eatough, D. J.; Eatough. N. L.; Hill, M. W.; Mangeison, N. F.; Ryder, J.; Hansen, L. D.; Meisenhisirner, R. G.; Fischer, J. W. Atmos. Envlron. 1979, 13. 489-508. (4) Eatough, D. J.; Hansen, L. D. “Organic and Inorganic S(1V) Compounds in Airborne Partloulate Matter”; Schwartz, S.E., Ed.; Advances
in Envlronmentai Science and Technology: press; Vol. 12.
Wiley:
New York, in
L. D. Hansen* N. F. Mangelson M. W. Hill D. J. Eatough Departments of Chemistry and Physics, and the Thermochemical Institute Brigham younguniversity Provo, Utah 84602 RECEIVED for review April 19, 1982. Accepted May 3, 1982.
Inaccuracies in the Calculation of Standard Deviation with Electronic Calculators Sir: The correct use of statistical methodology,while always an integral component of the scientific method, is becoming increasingly apparent today as some journals begin to impose expert statistical review on manuscripts prior to their acceptance for publication (1). Likewise, the development of inexpensive electronic calculators over the last decade has led to their universal adoptiion by the scientific community and has greatly facilitated statistical calculations. Unfortunately, as electronic calculators have become more familiar and ubiquitous, there has been an increasing temptation to view them as infallible devices. The blind acceptance of the result in the calculator display by students hae been noted (2, 3), and it is very easy to be lulled into this trap. The usual advertisement of an internal accuracy for most keyboard operations of ten or more significant digits, regardless of display, over a range of 10*lWor greater, is effectively mesmerizing. Not all calculator functions are this accurate at all times, however, and occasional warnings of certain small errors can be found in owner’s manuals upon careful reading. [For example, inaccuracies of several percent are obtained upon executing si+ of specific, very small arguments(4).] The use of modern calculators to compute the statistical standard deviation of certain data sets can lead to appreciable and unsuspected errors. The problems arise with data sets consisting of tightly bunched arrays, and severe calculator round-off error is the cause. Although many investigators may be familiar with this issue from their own past experiences, it is not at all documented in the various owners’ manuals provided by calculator manufacturers. An awareness of this problem is important, however, because a lack of knowledge of its consequences can be very serious. FORMULAS AND CALCUL.ATIONS The usual form of the equation for the experimental or sample standard deviation is
The two formulas employed by today’s most popular electronic calculators, however, arle the algebraic identities of S
and s2
=
[
Ex; - n f 2 n-1
If the data in the set consist of many digits with variation only in the last several places, then round-off error is possible in the computation of S1and Szbecause their numerators will be small differencesbetween two large numbers. In calculators where the capacity is insufficient to carry the necessary number of digits required to propagate the true variation, results of zero or worse are obtained. While answers of zero are obvious nonsense and serve as flags of inaccuracy, severe errors not readily recognized are also possible. By way of example, the following results were obtained with Hewlett-Packard HP-97 and HP 9825A calculators. For the data set xi = (19.51408, 19.51383, 19.51388, 19.51386, 19.51385), in which variation begins in the fifth significant digit, SI = 0 for an error of 100% and Szis 2.2% smaller than S. We have found similar examples in arrays where the variation began as soon as the fourth significant digit. Behavior contrary to the normal statistical expectations for standard deviation is also possible. In general, as n becomes large the standard deviation becomes a more dependable expression of precision. When the three data x, = (16.8299, 16.8298,16.8297) are repeatedly averaged such that n = 3,6, 9, ..., however, the opposite trend is observed for S1. When nonzero, the calculated S1values are consistently higher than the corresponding S values; at n = 3 the error in S1 is 124%, at n = 15 the S1error is 216%, at n = 45 the S1error is 477%, and at n = 96 the S1 error is 1090%. CODING DATA A simple remedy to this problem is to code the original data array by the subtraction of an appropriate constant from the xi prior to calculating the standard deviation. An elementary example demonstrates the cause of round-off error as well as the advantageous consequences of a preliminary coding of the data. Let n = 2 and x i = (50000000 and 50000002). Then the numerator of eq 2 for S , becomes 5,000,000,200,000,004 -
10,000,000,400,000,004 2
In order to obtain this result in this fashion, a calculator must be capable of carrying at least 17 digits internally. Otherwise, an incorrect answer of zero is obtained. If these x i are coded 0003-2700/82/0354-1877$01.25/0
0 1982 American Chemical Society
