Rules for Propagation of Slgniflcant Figures To the Editor:
To the Editor:
Stieg asserts that the last significant figure in a number reflects the precision but not the accurac; of that number. Because I and other faculty members in my department write significant figures to conform to the total uncertainty; i.e., the random error uncertainty plus the systematic error uncertainty, I wonder really how much variation there is in this practice. Perhaps a survey of the readership of this Journal will give some idea. Such a survev would encouraee readers to giie some thought to the meaning of signific&t figures and, if no consensus exists, would help to establish a consensus. In order to focus properly on the difference between random error (precision) and systematic error (accuracy), I offer the following simple example and experiment that can he duplicated in any chemistry laboratory. A common Erlenmever flask is labeled 125 mL and also carries the notation f5%. By f5% the manufacturer means to note that when the flask is filled to the 125-mL line, the volume of liquid is anywhere from 119 to 131 mL. When I fill any one particular flask to the 125-mL line and report that I have 125 mL, I understand that I have a systematic error of f 6 mL in that volume of liquid. In order to find the random error associated with that same volume, however, I must perform a series of replicate experiments such as the one I report as follows. I weighed a dry 125-mL Erlenmeyer flask on a balance having much greater precision than the precision of the volumes to be measured. For this purpose Iused a single pan balance with f10 mgreadability. Then I filled the flask to the 125 mL line with water, recorded the temperature of the water, and reweighed. The net weight of the water divided by the density of water at the recorded temnerature gave me the volume of water. I repeated this procedure with the same flask 10 times. The mean of the 10 water volumes was 128.23mL, the standard error of this mean was 0.06 mL and the standard deviation of the 10 measurements was 0.17 mL. These results tell me (a) that the systematic error of the 125-mL line on this particular flask is about +3 mL and well within the manufacturer's tolerance, and (b) that the precision of a single volume measurement a t the 125-mL level is he about 0.2 mL. Now I take a different 125-mL- -flask - - ~ of - -~t.--. ~ same manufacture and fill with water to the 125-mL line. I am suite sure that I can fill this second flask with the same 0 . 2 - m ~precision, hut without performing a calibration experiment I must accept the manufacturer's f 6 mL tolerance for my systematic error estimate. How should I report the volume of water in the second flask? If I followed Stieg's practice of considering only the random error o f f 0.17 mL, I would report 125.0 mL. However, if I estimated the total uncertainty at roughly f 6 mL systematic error plus f0.17 mL random error, the f 6 mL uncertainty reaches to the units place and so I would report
I was surprised to see the recent attack on the sound rules of significant figure propagation ( I ) . The author fails to recognize that the last digit in a number impliesprecision or random error in the measurement result estimated by the number. Significant figures say nothing about accuracy. This implied random error is only an order of magnitude estimate, the magnitude being the base of the number system used. In the decimal system, then, this means that the rules will hold if this estimate is within 1000%,or f (order of magnitude) of any "better estimate" of random error. One such better estimate is the standard deviations in a number. If we let the implied random error in a number be estimated by s, then we can calculate (2)a better estimate of the resulting propagated error hy using the propagation of random error eqs 1and 2:
then
then
Eight example calculations support the rules (see table). The percent difference does not exceed 1000%for any of these eight examples. Let us not, then, rush to discard these useful rules of thumb. Perhaps also each of us should make sure what significance significant figures have.
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Literature Cited
Scott stieg Harvey Mudd College
Claremant, CA 91711
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Examples that Defend Rules for Propagation of Significant Figures
%WR
implied
Implied 5
A
B
Resuit
SA
34.9 1 0.023 101.0 0.24 0.24 0.0001 0.0001
62.7 2 10.1 99 0.48 0.48 42.91 42.91
5 = 97.6 D=-I S = 10.1 D =2 P = 0.12 Q = 0.50 P = 0.004 0 = 2 X lo-B
0.1 1 0001 0.1 0.01 0.01 0.0001 0.0001
0.1 1 0.1
I 0.01 0.01 0.01 0.01
% difference in $W,
$nu~t
calculated
implled
by eqs (1)or (2)
by
from
sig, fig.
~r.."~calculated
0.14 1.4 0.10 1.0 0.006 0.03 0.004 2 X lo-'
Volume 64
implied
0.1 1 0.1 1 0.01 0.01 0.001 1 X lo-@
Number
5
-28% -28% 0% 0% 67% -67% -400% -50%
May 1987
471
