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withdrawn, cooled and weighed. The heating is again continued for another interval, another weighing made and so on until constant weight is attained. The percentage of moisture is always figured on the original weight of sample, and in stating the result found the temperature of drying should always be mentioned.

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CHAPTER IV

PRECISION

50. Before undertaking his laboratory work, the student will find it very profitable to become acquainted with the general principles of precision;' not only in order to know how carefully he should make his measurements but also to know how reliably he can count upon his results.

It has perhaps been too generally taken for granted that quantitative methods are on a thoroughly satisfactory basis, and that as a consequence the student has only to run duplicate determinations until exact checks are obtained in order to insure accuracy.

Such an idea is very harmful because it implies that determinations are free from all errors except those arising from some fault of the student. Thus the student misses the very essential point that, besides his own mistakes, there are also the errors which lurk in every experimental method even when conducted with the most extreme care. It must be continually borne in mind that every determination is an approximation which is attended with more or less error and that in no case is it possible to reduce the error to zero.2 Hence we are confronted with the question as to what are allowable limits of error in order to know in any given case whether we may consider our results as possessing the necessary degree of precision.3

1 The following notes on precision are intended to be only such a brief treatment of the subject as will be of interest to the student of analytical chemistry. For a comprehensive treatment of the subject of precision, the reader is referred to any of the following works: H. M. Goodwin, Elements of the Precision of Measurements and Graphical Methods, Massachusetts Institute of Technology, Boston, 1908, 2nd ed. L. D. Weld, Theory of Errors and Least Squares, The Macmillan Co., New York, 1916. T. W. Wright and J. F. Hayford, Adjustment of Observations by Least Squares, D. Van Nostrand Co., New York, 1906, 2nd. ed., 298 pp.

2 As already stated in §5 the smallest limit that we can practicably reduce our error to in quantitative analysis is about one part per thousand.

3 By the precision of a result is meant the narrowness of the limits within which the true value may be assumed to lie with respect to the measured. These limits it is possible to get an estimate of, as explained in the following paragraphs of this chapter.

It is to be pointed out that the term "accuracy" is often used carelessly in place of the term "precision" in connection with the results of physical or chemical measurements.

By the accuracy of a result would be meant the concordance between it and the true value

To deal with this question we notice that the final values which are set down for the results of determinations are derived measurements in that they are obtained by means of calculation from one or more directly measured quantities. Consequently the final values will be affected by the errors in the primary measurements, or, in other words, the ultimate error with which we are concerned in any given determination is a function of the errors which exist in the primary measurements. Let us now discuss the general nature of errors, and then later show the effect which errors in the primary measurements have upon the final result. 51. Classification of Errors. When any quantity is measured to the full precision of which the instrument or method employed is capable, it will, in general, be found that the result of repeated measurements do not exactly agree. This is true not only of results obtained by different observers using different instruments and methods, but also of results obtained by the same observer under similar conditions. The cause of these discrepancies lies in various sources of error to which all experimental data are subject. When we examine the nature of these sources, we find that errors may be conveniently grouped into two classes:

Indeterminate
Determinate

Indeterminate, or as they are often called, Accidental Errors are those errors which manifest themselves by the slight variations that occur in the case of successive observations made by the same observer under as nearly identical conditions as possible. They are due to causes over which the observer has no control and which in general are so intangible as to be incapable of analysis, as, for example, slight differences in the judgment of the observer, or in the behavior of the instruments, or in the relative amounts of the substances which have reacted, as we proceed from one determination to another.

of the quantity measured. This concordance, however, it is impossible to ascertain because the true value of any physical quantity is unobtainable; the value that we get as a result of our measurements, after correcting for all known sources of error, being only a more or less close approximation to the true value.

Hence to employ proper terms, we must speak of the precision of a measured result, and not of the accuracy.

Determinate, or as they are often called Constant Errors, are errors of such a nature that they persist in a constant way from one determination to another; consequently their value can be more or less closely determined and their effect on the result thereby largely reduced. They arise mainly from the three following sources:

1. Instrumental Errors, due to poor construction or faulty graduation of an instrument, as, for example, the inequality in the length of the arms of a balance, the discrepancies between the indicated and the actual values of a set of weights, the inaccuracy in the graduation of volumetric apparatus.

2. Personal Errors, due to certain peculiarities of constitution or susceptibility on the part of individual observers, as, for instance, the habitual recording of end points too late on account of the inability to judge color changes sharply.

3. Errors of Method, due to causes such as: incorrect sampling; incompleteness of precipitation; contamination of precipitate by impurities; the so-called incompleteness of oxidation-reduction reactions, namely, the difference between the equilibrium point of a reaction and the equivalent point as represented by the stoichiometrical equation; the excess of standard solution necessary to establish the end point in a volumetric determination, etc.

52. Indeterminate Errors. It has been shown that the magnitude and sign of errors of this kind follow a perfectly definite law, namely, the law of chance, the equation for which is,1

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where y is the frequency of the occurrence of an error of the magnitude x, and h is a constant, the value of which depends upon the character of the observations and which affords a measure of their precision. The curve represented by this equation is called the Curve of Error or the Probability Curve. See Fig. 6.

4 This equation is due to Gauss. See, "Theoria Motus Corporum Coelestium," Lib. II, Sec. III, Art. 178 (1809); also "Bestimmung der Genauigkeit der Beobachtung," Zeit. f. Ast. Mar., 1816.

An inspection of this curve shows that

1. Small errors occur more frequently than large ones.

2. Very large errors are unlikely to occur.

3. Positive and negative errors of the same numerical magnitude are equally likely to occur.

The fact that positive and negative errors of the same numerical magnitude are equally probable leads to the conclusion that the

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best value to select for a series of observed readings is such a value that the differences between the observed readings and it shall balance each other. This value we find to be the Arithmetical Mean.

It is easy to verify this choice of the arithmetical mean. Let the several measured results be designated by a1, a2, and their arithmetical mean by m. Then

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'It must be remembered that this generalization represents a limiting case corresponding to a very great number of observations; therefore deductions from it with respect to a small number of observations apply less rigidly the fewer the number of observations.

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