QUANTITATIVE ANALYSIS and the THEORY of MEASUREMENT

FRANK H. HURLEY

The Rice Institute. Houston. Texas

T HE importance of measurement is so great to a science that i t can be argued whether a science without measnrement is a science at all. The

rise of chemistry out of alchemy is largely due to the fact that the purely descriptive material of alchemy was translated into precise mathematical terms through measurement; or, as Ida Freund said, "Chemistry only took rank as a science when i t made quantitative work its basis." The nature of measurement and of calculation involving measured quantities is of such fundamental importance to the science that a failure to direct the attention of a student of chemistry to the stndy of measurement and calculation is to keep from him the essence of the science itself, and to deprive him of a basis of judgment regarding the measurements which he will make if he continues in chemistry, and also those of other scientists which will come before him for evaluation. There are many interesting philo- sophical ramifications to the theory of measnrement which bear directly on the validity of the scientific method itself, but if it were only this which gave the stndy of measurement its importance, the pursuit of such studies would be of doubtful value for most chem- ists. The fact is, however, that a stndy of measure- ment is of considerable practical importance, and a chemist with some knowledge of the subject is often in a position to save himself both time and money in actual experimental work by designizig experiments to fit his needs accurately, instead of procee&ng blindly in the hope that if enough measurements are made or if only the most expensive equipment is used, everything will turn out aU right.

The most logical and appropriate place for the study of measurement and calculation in the chemistry cur- riculum is in quantitative analysis, for it is a t this stage that the student is, or should be, introduced to meas- uring instruments worthy of the name, and is taught to use them in an exact fashion, which is usually very dierent from the hurried and sometimes slipshod fashion which characterizes most of the measurements of the freshman or high-school course. Moreover, in quantitative analysis the object of the work is gener- ally to make measurements for their own sake, without any special emphasis on their application to some theory or set of conditions which might be under investigation. Under these circumstances i t would seem that the course in analysis offers the best opportunity for a study of the so-called "calculus of observations."

Uufortunatelv. of the manv textbooks on quantita-

tive analysis which are in common use, very few have recognized this opportunity by the inclusion of ade- quate discussions of the problem of measnrement There are two noteworthy exceptions to this generali- zation, namely the texts of Fales (2) and of Kolthoff and SandeU (3), both of which contain separate chap- ters dealing with the subject. In the main, however, the topic is treated by a few very brief statements or rules with no clue to their basis except (in the better books) a footnote referring to more complete treat- ments. In some texts even the inadequate discussion is confused or erroneous. For example, the following quotations taken from the 1929 edition of a well-known text contain statements which are totally incorrect.

"Thus suppose that the mean of four runs such as buret readings is 30.255 cc. and that the average devia- tion of a single reading is 0.005; then 0.005/<4 = 0.0025 is the probable error [sic] in the above buret readings. The result is then expressed as 30.255 +

0.0025 and means that an error of 0.0025 cc. is intro- duced in the buret reading. .

"It is to be noted that close$ agreeing checks are not a necessary 'indication of a good precision because a large error, of the same kind, may have been intro- duced in all the determinations."

No references to sources of correct information ac- company these erroneous statements.

No one would expect a chemistwho is not specifically interested in precise measurements to have a complete knowledge of the mathematical theory of statistics and probability, but it is not unreasonable to expect him to understand how measurement and calculation are related to probability theory, to know exactly what is meant by expressions such as "probable error" and "standard deviation," and, most especially, to under- stand the precise significance of his own measurements. Benedetti-Pichler (1) has recently drawn attention to the importance and value of statistics in quantitative analysis in an interesting paper with which all students of analysis might be expected to be familiar.

The nature and scope of a discussion of measure- ment and calculation to be included in a course in quantitative analysis will, of course, be subject to limitations depending on the time available, but the following outline may serve as the basis of such a dis- cussion.

I. Experimental Errors (A) Indeterminate Errors

(1) Gauss' Error Curve (a) The "best representative value"

(2) The arithmetical mean and the num- ber of measurements

(a) Indeterminate error of a single observation

1. The average deviation 2. The standard deviation 3. The probable error

(b) Indeterminate error of the mean (c) The practical number of meas-

urements (3) The trustworthiness of individual

measurement (a) Rejection of measurements

(B) Determinate Errors (1) Classification

(C) Absolute and Relative Error 11. Accuracy and Precision

(A) Definitions (B) Determination of Precision (C) Determination of Accuracy

(1) The problem of the "true value" (D) Accuracy and Precision in Quantitative

Analysis 111. Significant Figures

(A) Definition and Usages (1) The purpose of significant figures

(B) Semi-exponential Notation (C) Pure Numbers

IV. Error and Calculation (A) Propagation of Error

(1) Addition and subtraction (2) Multiplication and division

(23) Rules of Computation (C) Methods of Computation

(1) Slide rule (2) Logarithms (3) Shortened arithmetic

C

There are several sources, in addition to those al- ready mentioned, from which such a discussion can he drawn (4, 5, 6a), but it may not be amiss to note here, certain points on which there is some disagreement or difficulty. At the outset it is necessary to define deter- minate and indeterminate errors, and the following definitions of these terms are suggested.

Indeterminate errors are errors of measurement which have signs and magnitudes determined solely by chance, and therefore cannot be controlled by the observer.

Determinate errors are errors of measurement which have signs and magnitudes determined by laws relating them to their causes, and therefore can be controlled or evaluated, provided the laws and causes can be discovered. It is believed that these definitions are more ac-

curate and informative than many which have pre- viously been given. The terms "vectorial" and "non- vectorial" are very satisfactory substitutes for "deter-

minate" and "indeterminate," since in many instances it is easier to "determine" the indeterminate error than the determinate.

Perhaps the most difficult part of the discussion of measurement is that relating to the trustworthiness of individual measurements. There are many criteria for the rejection of aberrant observations and it is dif- ficult, if not impossible, to adopt any one which will he universally applicable and which will actually effect the rejection of untrustworthy results. No matter what criterion is adopted, the rejection must finally be subjected to the scrutiny of common sense, a criterion which is probably no worse (nor any better) than the mathematical formulas. The convention most fre- quently adopted in quantitative analysis (and the one recommended by Benedetti-Pichler) is the arhi- trary rejection of a measurement with a deviation greater than four times the average deviation. This is a variant of Wright's criterion (fib), based on the probable error. It is worth mentioning that the figure given in most analysis texts for the probability on which this rule is based is incorrect. The figure usu- ally given, 993/1000, refers to four times the probable error instead of the average deviation. The correct probability is 999/1000 (64.

The recommendation that relative errors in quan- titative analysis be given in parts per thousand rather than per cent. to avoid confusion with analytical results, which are usually expressed in per cent., has been pretty widely adopted. The use of the symbol, O/m. for parts per thousand (1) also merits adoption, and the terms "per mil" and "permillage error" might also be convenient.

The subject of Significant figures is one which most students find rather confusing and the source of the confusion is not hard to find. They are given a set of rules and conventions which are stated to cover the subject and then must reconcile these with flagrant violations on the part of the instructor and the text- book. The important point about signscant figures, and the point which is frequently omitted in the huny to get to the rules, is that the purpose of giving a measurement its correct number of significant figures is to indicate its approximate precision. Thus, it is quiteimpossihle to write a measurement with its correct number of figures unless there is some knowledge as to its reproducibility. To relate significant figures to the estimations involved in reading a measuring instru- ment is one thing, and to relate them to the real re- producibility of the measurement is another, a dis- tinction carefully pointed out by Benedetti-Picbler.

The rules for the number of significant figures to be used in the expression of a measured quantity diier rather widely among authors. If the mean figure 21.23 per cent. is obtained as the result of an analysis, most writers on the mathematical theory of measure- ments would say that i t indicates that the correct value lies between 21.225 per cent. and 21.235 per cent., or has a precision of *0.005 unit. Others would say that the indicated precision is *0.01 unit. Most

chemists do not actually use such a narrow criterion errors of its antecedent measurements is important in and would feel perfectly correct in writing the figure as the consideration of a completed analytical result, but given if it were obtained as the mean of the two meas- an even more important use of such knowledge is its urements 21.20 per cent. and 21.26 per cent. There is application to projected measurements. Unless a peneral aneement that in writ in^ a number repre- student has some roundi in^ in the theorv of nronara-

figure. The source of the conflicting conventions is in thislastfigure, and some convention as to how "doubt- ful" thelastfiguremay be must be adopted before gen- eral agreement can be obtained. Perhaps the most satis- factory way to treat the last significant figure is to give a rough indication of its doubtfulness by the use of sub- scripts. Using this convention, a measurement written as 21.23 would indicate a precision (average deviation of a single observation) of *O.Ol unit. If the average deviation is greater than this, the number would be written as 21.23. The adoption of this convention re- duces all rules for the use of significant figures to the following. In writing a number which represents a measured quantity, use the number of significant figures which comes nearest to expressing the real preci- sion of the measurement, using a subscript if the last figure is less precise than * 1.

In making calculations with numbers representing measurements, i t is unusual to k d a student who re- gards these figures other than as pure numbers. A study of the way in which errors of measurement are propagated in the ordinary arithmetical operations is not ordinarily a part of his previous mathematical training, and a discussion of this topic should be in- cluded in the course in quantitative analysis. The knowledge of the way a final answer is affected by the

urement and to make arrangements to perform i t by methods which will satisfy his requirements of accuracy and precision. This is especially true of analytical determinations which are obtained by computation from several antecedents. Most quantitative analyses are of this character, and in every analysis a preliminary calculation is required to decide what size sample of the material to be analyzed should be taken to fit the conditions of the analysis. In most textbooks these calculations have already been performed for the student, and the directions for a determination usually begin, "Weigh out 0.5 g. of . . . ." This is very convenient for the instructor, but i t is apt to lead the student to believe that 0.5 g. is the proper size sample to be used no matter bow much or how little of the constituent being determined is present. The failure to indicate how the sample sizes, the quantities of reagents, concentrations of solutions, and so forth, were decided upon is a gap in the instruction which should not be left open. Decisions about these quantities are always partially, and sometimes wholly, based on the propagation of experimental error in calculation. A student who has been shown how these decisions are made will be in a position to make them for himself when the need arises.

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LITERATURE CITED

(1) BENEDETTI-PICHLER. "The application of statistics to quan- (4) MELLOR, "Higher mathematics fa. students of chemistry titative analysis," Ind. Eng. Chem., Anal. Ed., 8, 373 and physics." Longmans, Green and Co., London, 1916, (Sept., 1936). Chap. IX, pp. 498-566.

(2) PALES, "Inorganic quantitative analmis," The Century Co., (5) PALMER. "The theory of measurements," The McGraw-Hill

New York City, 1925, Chap. IV, pp. 5&73. Book Co., Inc., New York City, 1912,248 pp.

(6) (a) TurnE AND SATTERLY, "The theory of mearurements," (3) K o m ~ o a a am SANDELL, "Texthwk of quantitative inor- Longmans, Green and Co., London, 1925,333 pp.

ganic analysis," The Mamillan Ca.. New York City, (b) Ibid., p. 212. 1937, Chap. XV, pp. 250-78. (c) Ibid., p. 194.