In the Classroom
There Seems To Be Uncertainty about the Use of Significant Figures in Reporting Uncertainties of Results Julio F. Caballero Argentine Consultants in Education, P. O. Box 66321, Houston, TX 77266 Delphia F. Harris Department of Chemistry and Physics, University of St. Thomas, Houston, TX 77006-4694
Error is inherently associated with any experimental determination. It is clear that when a numerical result is reported that the uncertainty should be estimated. A survey of the literature indicates that chemists consistently provide estimates of uncertainties for reported results. There are circumstances under which considerable effort is required to estimate uncertainties. Some computer programs that provide linear or curve-fitting routines to determine values also provide uncertainties associated with the computed values. Computers, like calculators, fill the available places with digits unless programmed to round to a certain number of digits. In whatever way uncertainties are estimated, another step is required before determining the final form of the reported result. The relative magnitude of the uncertainty compared with the result should dictate the number of significant figures with which the result should be reported. In order to provide as much information as possible, the last digit of the result is assumed to contain error and may be largely uncertain (1, 2). The next to the last digit may conTable 1. Examples of Inappropriate and Appropriate Rounding Published Value 771 ± 135
Ref
Appropriate Rounding
Uncer-taintya
3
(7.7 ± 1.4) × 102
14
65.9 ± 10.1
3
(6.6 ± 1.0) × 10
10
e᎑ (646 ± 69) / T
4
e᎑(650 ± 70)/ T
1.79 ± 0.78
5
1.8 ± 0.8
e(8450 ± 850)/ T
6
1
e
(8400 ± 800)/ T b
7 8 8
(1.4488 ± 0.0290) × 10 ᎑46 7
(1.449 ± 0.029) × 10᎑46
29
᎑19.9998 SE 0.100842
᎑20.00 SE 0.10
10
aThis
8
column shows that each uncertainty has a value between 3 and 30, whatever the place value. bThis example illustrates laboratory rounding. When rounding involves 5 exactly, round to the even number.
996
tain some uncertainty. Common guidelines taught to undergraduates (1, 2) indicate that uncertainty should be reported with one or two significant figures with a value between 3 and 30 whatever the place value. Reports of uncertainties that exceed 30 appear regularly in the literature. Table 1 illustrates just a few examples that were obtained by a very cursory review of recent issues of two journals. A number of them include uncertainties with three or more significant figures. These examples are taken from both research and chemical education articles, and are the reason for this note. Scientists who take such care in designing and carrying out complex experiments need to take more care in deciding on the appropriate form for reporting their results. Faculty who teach students not to write down all the digits that their calculator generates should not report in a published article all the digits that a computer fit generates. Referees and reviewers of manuscripts also need to give careful attention to how results and their uncertainties are reported. Let’s practice what we teach. Literature Cited 1. Shoemaker, D. P.; Garland, C. W.; Nibler, J. W. Experiments in Physical Chemistry, 4th ed.; McGraw-Hill: New York, 1981; p 52. 2. Skoog, D. A.; West, D. M.; Holler, F. J. Fundamentals of Analytical Chemistry, 6th ed.; Sauders: New York, 1992; pp 31, 32. 3. Wallington, T. J.; Ball, J. C. J. Phys. Chem. 1995, 99, 3201. 4. Wu, F.; Carr, R. W. J. Phys. Chem. 1995, 99, 3128. 5. Horowitz, A.; Crowley, J. N.; Moortgat, G. K. J. Phys. Chem. 1994, 98, 11924. 6. As cited in ref 5, DeMore, W. B.; Sander, S. P.; Golden, D. M.; Hampson, R. F.; Kurylo, M. J.; Howard, C. J.; Ravishankara, A. R.; Kolb, C. E.; Molina, M. J. JPL Publication 92-20; Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, 1992. 7. Mina-Camilde, N.; Manzanares, I. C.; Caballero, J. F. J. Chem. Educ. 1996, 73, 804. 8. David, C. W. J. Chem. Educ. 1996, 73, 46.
Journal of Chemical Education • Vol. 75 No. 8 August 1998 • JChemEd.chem.wisc.edu
