Propagation of Significant Figures Lowell M. Schwartz University of Massachusetts, Boston; MA 02125 Not long ago the following'question arose with regard t o a eeneral chemistrv exercise. Students were asked to weieh several pennies on a balance and to calculate the mean we& with oroner regard to sienificant firmres. One student recorded five mensureients: 2.i4, 2.45, 2.%, 2.50, and 2.47 g. In calculating the mean he summed the five weights finding a total of 12.36 g, divided by five on a calculator and obtained 2.4720000 e. In order to write this number with moper sienificant figures, he reasoned correctly as we had taught. -
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1) When numbers are added or subtracted, write the result with the same number of decimal places as the number having the fewest decimal places. 2 ) When numbers are multiplied or divided, write the result with the same number of significant figures as that number having the fewest significant figures. 3 ) Numbers that are exact are treated as if they had infinitely many significant figures. Since all his weighings were to two decimal places, the student followed rule 1 and wrote the sum with two decimal places, as 12.36 g. Then, understanding that the denominator 5 in the division 12.3615 was exact because the count of five weighings was exact, he reasoned that the mean weight was limited by the four significant figures in the sum and so reported his result with four significant figures, as 2.472 g. Meanwhile, we bad explained the meaning of significant figures in terms of those uncertainties which unavoidably creep into most numbers derived from ex~eriment.The more certain or accurate a number, the more significant figures that must be used to convev that level of accuracv. The last significant digit in a given number has some uncertainty in &i value; previous significant digits in that number should be firm. This student saw the inconsistency and asked, "When I weighed each penny, I was sure of the first two digits in the weight, but thethirddigit in the second decimal place was in doubt so the weighings are good to three significant figures. But the average of these weighings seems to be good to four significant figures. How can the average of five numbers each with uncertaintv in the hundredths nlace have uncertaintv in the thousandths place?" Good question. Somewhat he& tantlv and unhaonilv . . . I answered that the rules for combinine significant figures areonly approximateand sometimes lead tu sienificant fieure counts that arr slirhtlv off. I hestitated because I also knew another possible a n s w e r h a t the precision of the mean was meater than that of the individual measurements, and if thestatistical uncertainty of the mean turned out to be less than 0.01 g, the mean should, indeed, be written with three decimal places. However, I later calculated the standard error of the mean as 0.012 g. This confirmed that the statistical uncertainty of the mean as expressed as the standard error estimate was within the hundredths decimal place. Therefore, in this particular case the correct expression for the mean seems to be 2.47, not 2.472 e. This episode demonstrated to me tLt the subject material on significant figures, which I have been teaching for many yeari, has not been entirely correct. T o no a v a i l j searched through general and analytical chemsitry textbooks for some insight into the prohlem. All of the 20 or so books included some discussion of significant figures and all presented virtually the same information: that the number of significant
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figures represent the accuracy or uncertainty of a written number: how to count sienificant firmres in a eiven number: and how t o propagate iignificant ?igure or 2ecimal place counts throueh s i m ~ l earithmetic o~erations.However. no text that I sa; and "oneof several a r h s p u b l i s h e d inTHls JOUHNAI.~ mentions the limitations of the propagation procedures. Recause this subject appears in virtually all general chemistrv texts. it amears to br of sufficient interest to merit a better Aders&dKg than what currently exists. I have done some numerical exploration towards this end and offer the results in this article. It is hoped that this will stimulate others to do so also. I shall focus on the propagation of significant figures through those arithmetic operations which are of interest in general chemistry: the ubiquitous addition, subtraction, multiplication, division, and also the taking of common logarithms and antilogarithms, which come up in pH, solution equilibria, Nernst equation, and chemical kinetics calculations. Because the word "number" is needed in too many different contexts, I shall adopt the practice of using the word "result" for a number which is calculated from other numbers. These other numbers I will call "data" (or the singular "datum"). Thus "results" are calculated from "data." The rules for addition, subtraction, multiplication, and division are well known and these are set off as rules 1 and 2, respectively, in the opening paragraph. The rules for logarithms and antilogarithms are less well known but are given in several texts. These are: 4 ) When taking common logarithms, if L is the logarithm of A , ( L = log A ) , the number of decimal places in L is the same as the number of significant figures in A . Conversely, if A is the antilogarithm of L , ( A = antilog L or A = l o L ) ,the number of significant figures in A is the same as the number of decimal places in L . The procedures which are numbered 1,2, and 4 above are often called "rules of thumb," a descriptor that implies not only that the procedures are simple t o use but also that they are crude in the sense that they do not always work. The particular ,problems that I wish to explore are ( 1 ) how to propagate significant figures properly under any circumstances and ( 2 ) the specification of those conditions under which the rules of thumb break down. Both these problems require an understanding of the relationship between the uncertainty of a number and the significant figure representation of that number. The specification of how to write a number with significant figures appropriate to its uncertainty has been stated many alternative wavs but mv oreference is: The uncertaintv in the value of a n u m k r shouidbe within the decimal place occupied bv the riehtmost dieit onlv. For examnle. if the uncertaintv in the number 1.006 is f6.05, the uncertainty is within thk hundredths decimal lace and so the number is ~ r o ~ e r l v written 1.00, the rightmost written digit being in ihehuidredtbs place. Or, if the uncertainty in 6398.7 is f25, theuncertainty reaches to the tens place. The units digit 8 and the tenths digit 7 are not significant. Thus the number is properly
' F a example. Hurley. F. H.. J. CHEM.€OK.. 17, 334 (1940);Pinkerton. R., and Gleit. C. €.. J. CHM. €OK.. 44,232 (1967);Anderlik. 6.. J. CHEM.EOUC.. 57. 591 (1980l. Volume 82 Number 8 August 1985
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to he significant. The analytical concentration of the acid C = 0.072412 M, and a t this low concentration the fraction of the acid which dissociates is substantial. We solve the quadratic equation K. = x2/(C - x), where x is the hydronium ion concentration, and find x = 0.042212 M and the pH is 1.3746. Next, in order to find the proper significant figure count, we must calculate maximum and minimum pH values corresponding to extremes in the uncertainty ranges of the four data: 500 mL, 3.26 g, 90.04 g mol-', and K. = 0.059. The datum 500 mL presents an obstacle. The significant figure count of this number is ambieuous and we cannot oroceed without additional information. Having noted that the particular flask used was marked "Class A" by the manufacturer, we consult a supply-house catalog and find that 500-mL volumetric flasks so marked have a tolerance o f f 0.2 mL. Thus the volume datum is good to four significant figures and should have been written 500.0 mL. The next obstacle is to decide how to combine data uncertainties to actually achieve a maximum in the DH. Usina algebraic or chemical reasonink-, we conclude that
