ERRORS OF' MEASUREMENT WITH THE SLIDE RULE MADISON L. MARSHALL Texas State College for Women, Denton, Texas
STu~~iw ins the physical sciences who have completed their undergraduate work usually have a broad foundation in mathematics, physics, and chemistry. Courses in physics and chemistry rely heavily upon the experimental method in that they test in the laboratory as many of the theories and laws as time will permit. The student is frequently called upon in these courses to measure the evidence of certain phenomena quantitatively. In many Gases, the quantitative efficiency of a given method of procedure determines its relative value among other methods. In spite of the fact that the student is called upon to carry out many operations involving quantitative procedure, it is not always certain that he has standards provided which enable him to measure his degree of perfection in that operation. The student's understanding of the standard may be entirely superficial and routine. Courses in physical chemistry' recognize these facts when they provide a discussion for the study of errors of measurement. The actual laboratory work, if any is provided, frequently consists in reading a thermometer or barometer or other simple instruments a large number of times and then examining the readings obtained. Such a simple and monotonous experiment is quite likely to create in the mind of the experimenter an antipathy for the experiment before it is hardly begun. In order to provide the student with an experiment which will offer him an opportunity togxamine the degree of his own skill and a t the same tlme provide him with an interesting device for doing so, the author provided for his students in physical chemistry an experiment with the slide rule. This experiment has been employed for several years and hss been performed by approximately 50 different students. Approximately 20 per cent of this group a t the time of experiment had already attained some degree of facility with the rule. This situation introduced no difficulty, however, for those students who could already use the rule were required to do a larger number or a greater variety of measurements. The standard experimental procedure consisted in carrying out five different calculations on a standard twelve-inch slide rule, two involving multiplication, ' DANIEL^, F., J. R. MATHEWS,AND J. W. WILLIAMS,"EXperimental Physical Chemistry," McGraw-Hill Book Company, J. J.. "Laboratom Methods of Inc.. New York. 1941: JASPER. ~hykiealchem&tq?," oughto on Mflin Co., ~ o s t i nMassachu, A,, "Practical Physioal Chemistry," setts, 1938; FINDLAY, Longmans. Green and Co.. New York, 1935.
two involving division, and one involving a combination of multiplication and division. Multiplication and division problems were chosen not only because they provide a rudimentary hut essential practice on the rule, but also because these operations are the ones most frequently encountered in physical chemistry. It was felt that inclusion of other manipulations in the exercise would confuse the beginner and allow h i to lose sight of his general problem, the study of errors. For the same reason, t,he decimal point was located after the first digit in each number, and the student could easily show by mental analysis that the decimal point would fall after the first digit in each of the answers. Multiplicands, multipliers, divisors, and the dividends were given with either three or four significant figures depending upon the portion of the slide rule on which the number occurred. In most cases, setting the cursor or index of the slide rule required that the student estimate the position on the rule for the last digit. Table 1 No. 1 2 3
4
5
Omration
Mukiplieation Multiplmation Dmision niviviin Multiplication and division
2.345 X 3.172 1.200 X 4.350 X 0.891 8 6 4 s 41fi4 (5.05 i 0.855 L(3.240 s 2.695) x 8.961 s 9.10
4.420)
In addition to computing the answers for each of the above problems a definite number of times (five usually were required), the student was also required to snbmit a written report presenting briefly the theory of probabilities as it dealt with types of errors, precision, accuracy, error measurement, significant figures, propagation of errors, and also to include a critical examination of the calculated results in the light of the theory. This exercise lent itself admirably to the execution of these studies. The absolute values of the operations can be determined easily by long multiplication and division. In contrast to this, absolute values in most physical measurements are not known and, consequently, a comparison between the measured and exact value cannot he made. In other words, in most physical measurements the precision can be measured but not the accuracy. In order to prevent any preconceived notions concerning the final value of a given computation influencing the student, the exact value
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JULY, 1948
was not calculated until the conlputations had been completed on the rule. In order to illustrate the difference between determinant and indeterminant errors, different quality slide rules were employed. In addition to a modern, well-calibrated slide rule, a cheaper rule purchased a t the variety store was employed. It was easy to demonstrate with the inferior rule the larger error incurred when an instrument is poorly calibrated. The error measurement calculations usually required were average deviation, mean square error, and probable error. The calculations of each error function were required for a single observation and for the mean of five,calculations. Computed values were in good agreement with probability theory in that the accuracy of n determinations were times greater than a single determination. As interesting as the theory itself was the analysis of the final results in relation to the utility of the slide rule. An analysis of a portion of the calculation for the eight students is given in Table 2 which includes only the probable error function. Since the other error functions measure essentially the same thing as the probable error, the inclusion of them in the above data is not essential. Referring to Table 2, the largest probable error column contains a selection of the largest probable error incurred by the eight students for each opera$ion. In a like manner, the second column represents the smallest error incurred by the eight students for. the stated operation. In all the data the probable error function war, calculated to the base unity. The average probable error includes the largest and smallest probable error, and represents 40 calculations per operation and a total of 200 actual calculations for the exercise. The data do not include large errors of manipulation due to slipping of the index or cursor or to other mechanical trouble which the student became aware of as he carried out the calculation. Deviations were expressed as 'deviations from the mathematically correct value rather *than from the mean of the five calculations. No selection was exercised in the choice of the 8 students. In t,hose cases where arithmetical errors were made in computing the error functions, these error functions were recalculated. All error functions were computed to the base unity and expressed to the fourth decimal. This is equivalent to expressing them as parts per 10,000. This was done a t first with some hesitation since a twelve-inch slide rule is not usually considered a precision instrument. However, an examination of the data presented in Table 2 shows satisfactory agree-
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Table 2 Probable Error of a Single Calculation Expreaszd a s Part p=r Unity .
Oper* tion
no. 1 2 3
Largest orobable krror of eight students 0 0038 0.0015 O.ODfi2
Smallest ombable 'emor o f
Average
Auemge orobable error o f
0.0013
Probable Error of the Mean of Nva Calculations Expressed a s P a ~ t per s Unity
4
Operabion no. 1
2 3 4
5
Largest probable ewer of eight students 0.0017 0.0007 0.0028 0.0005 0.0038
Smallest probable error of eight students 0.0003
