applicatiow and analogiw Accuracy of Measurements and the U.S. Census George Gorln Oklahoma State University Stillwater. OK 74078
A Census of the U.S. Population is taken every 10 years, as is prescribed by the Constitution. Taking the Census is a complex operation and some aspects of it can help students to understand the problem of measurement error and the use of "significant figures" in properly describing the measurements. For the Census taken in 1980, the official count of the (resident) was But this datum ~ , U- S . oo~ulation ~ ~ ~ 226,545,805. ~ ~ is not accurate.'~n-order to understand why, one must keep in mind that the accuracy of any measurement depends on the procedure used in ohtaining it. The 1980 enumeration was done for the most part by mailing out forms and tallying the information on the forms which were returned. The Census Bureau itself admits that this procedure does not give accurate results, hut i t is clear, on the other hand, that t o do even marginally better would require a lot more time and money. The sheer magnitude of the Census makes i t difficult to identify and analyze the sources of error. I t is therefore appropriate t o consider first a similar hut much smaller operation, i.e., taking attendance in a moderately large class. Let us suppose that a student is asked to perform that task and that he or she reports a count of 203. Let us further suppose that the count is repeated, independently, by two other students, and that they get 202 and 201, respectively. From such results one can deduce, for certain, that the procedure of having a single person do the count, one time, does not give an accurate result. However, one can infer that the average of the three results, 202, is more likely to be exact than either of the other two values. This inference can he tested, if desired, by making still more measurements. With respect t o the Census, it is out of the question to make the same measurement more than once, and therefore one cannot use that approach to evaluate and improve the accuracy. But check determinations can he, and have been,
936
Journal of Chemical Education
edited by: RON DELORENZO Middle Georgia College Cochran. Georgia 31014
made on smaller and more controllable subsets of the population. In this way it has been estimated that the uncertainty is of the order of 1-2%. An important consequence of the Census is that i t determines how many representatives should he sent to the Congress from each State; as a result of the 1980 Census, for example, the State of New York lost five representatives, and Florida gained four. Not surprisingly, the losers were not h a ~ ~and v . a number of leeal proceedings have taken place, ch&iging the validity of h e aspect orthe operation. But, in eeneral. the courts have rejected the challenges, on the that the Census prockdures and results, albeit not exact, are more nearly accurate than anything else available. In science, critical~measurementsare m a d repeatedly, if possible, and the procedures used are analyzed 10 determine the range of uncertainty, which is given explicitly. Moreover, it is customary to express the values by the so-called "scientific notation", in which the order of magnitude is denoted by apower of 10. For example, the best value of the Avogadro constant, which was recommended in 1986, is 6.022 1367 f 0.0000036 X 1023m01-~.~ The uncertainty is about 1.5 parts in 107, or 105 times smaller than that in the 1980 Census. Many scientific measurements are reported without an estimated uncertaintv ranee, however. In that case, the following convention is generally used: the number of digits in the reported value is limited, so that only the last digit need he regarded as "doubtful". The Census is not a scientific measurement, strictly speaking, hut i t may he used as an example. An uncertainty of 1.5% corresponds t o 3,400,000 persons. I t is therefore evident that t h e last six digits in the official count are not significant and that the third digit is douhtful. If the population were expressed like a scientific measurement, i t would he given as 2.27 X lo8. In 1990, there will he another Census, and anew value will be obtained. But it, too, will be to some extent inaccurate.
' Cohen. E. R.; Taylor. B. N. J. Res. Nat. Bur. Stand. (U.S.)1987, 92,
85-95.
