INDETERMINATE ERROR ESTIMATES FROM SMALL GROUPS OF REPLICATES
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RALPH A. JOHNSON University of Illinois, Urbana, Illinois
THE
discussion of errors has become a regular part of cept of it and, second, the discipline of statistics to deinstruction in auantitative analvsis. Two cateeories fine it quantitatively and supply the mechanism for its of errors are generally given: determinate and indeter- interpretation and application. There is a growing recminate. Discussion of the determinate errors fits ognition of the indeterminate error as an attribute rrnoothly into the discnssion of rhrrniml st~l~jrvt matter; which can be measured and clearly described. Alfor cxamnlv. . ,the nrini~inlwof muilibrinrn a11dstuirhiwn- though experienced experimentalists intuitively apetry are applied in estimating the titration and solubil- praise their indeterminate error estimates quite accuity errors. Further, the determinate error concept is a rately, the statistical approach gives uniformity and sysconcrete one. It requires little imagination to realize tem to this highly important operation. Statements of that the true value may not be the nominal value of a standard deviation or fidurial limits and the number of weight or buret graduation and that calibrations correct measurements contributing to these estimates are rethis error. The discussion of indeterminate, or variable placing such vague statements as, "The method is good error, is not so well favored. Its meaning is not simply to two per cent," or, "Results agree within three parts established through an extension of familiar chemical per thousand." With definite statements of the indecalculations. It cannot be "pinned down" and clearly terminate error at hand, more realistic evaluations of and simply viewed as a physical defect. Its inevitabil- and comparisons between the methods and instruments ity and unpredictability are philosophically incompat- of chemical measurement can be made. Application of the reproducibility1 estimate is often ible with the scientific exactness implicit in discussing elementary chemical theory and making the related illustrated by describing it to be a measuring stick for laying off probability regions under the normal districhemical calculations. bution curve as in Figure 1. For example, 95 per cent of the measurements deviate from the mean by less than 2 o and a result deviating by more than 2.5 r d l occur by chance only once in approximately one hundred times. Although statistics is now more widely and more correctly used than ever before, there is still some misunderstanding of the subject. The budding chemist will not venture far into the world of his profession before being faced with the proposition that "statistics are not applicable to small groups of data." If the statement is understood to be general, statistics may easily come to be regarded as irrelevant to most chemical analyses or as a form of applied witchcraft used specially to dignify poor data. A more thorough analysis of the situation recognizes that classical statistics fails in certain respects when applied t o small groups of replicates. Fisure 1. The Normal Distribution Shoring Curnulatin Fsquencie. I Function of Standard Dniation However, it rarefullv defines the nature of the failure and reveals that procedures are available which take For the indeterminate error, the learning situation is proper account of the number of replicates available. created in the laboratory. Having found results differ- ~t is the peneral Dumose of this Dauer t," +,his .--.. ing among chenlical analysts in spite of their applying analvsis. I t is its saecific . Durnose to evaluate the stathe utmost in care and skill, the initiate to chemical used to the indeterminate error. ~t asanalysis is eager to accept a category of errors that are sumes knowledge of classical statistics of normal disalways present and are simply a matter of chance. The tributions as described in elementary textbooks of quantemptation now comes to relieve one's self of all re- titative analvsis. sponsibility, t o proceed deliberately through the ritual The term reproducibility is preferred to precision in this sense of the and leave the outcome to chance' A because the former ht~spreatw semantic value and is less ambigusatisfactory approach to this kind of error requires, OU,. cf. definition of precision in "Webster's Collegiate Dicfirst, imagination to achieve a realistic, qualitative con- tionary," G. and C. Merriam Co.
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JOURNAL OF CHEMICAL EDUCATION
466
The behavior of reproducibility estimates may be illustrated by an imaginary experiment. I t is supposed that a million chloride analyses are carried out on a given sample and that the variation of the results follows a normal distribution. From this very large group of data, a standard deviation is calculated. If another large set of replications is done, the new stand-
ard deviation virtually coincides with its counterpart in the first group. It is important to recognize that the estimate of reproducibility, i. e., the standard deviation, is itself so reproducible when derived from so large a population as a million that it may be regarded as a single-valued attribute of the analysis. I t is accordingly termed the universe standard deviation and designated as u. A standard deviation so obtained is a reliable measuring stick for laying off probability regions. It is this situation that is treated in classical statistics. Because of time limitations, most measurements in chemistry are limited t o two t o ten replications. How do these standard deviation estimates compare in magnitude with the universe standard deviation? How do the small-group estimates agree among themselves? For the answers to these questions a new model is set up: the data from the chloride analyses previously carried out are randomly separated into groups of ten, and standard deviations are calculated for each group. The chemist now inspects the resulting array of standard deviations in search of an index to the indeterminate error of his method. His first general impression is that standard deviations from small groups of replicates are not exact; they are hardly well defined measuring sticks for use in probability estimations. To distinTABLE 1 Factors for Bias Correction for Standard Deviation Estimates
N
=
no. of replicates; c = ?'
guish them as approximations of their single-valued counterpart, u, they are termed standard deviation estimates and are symbolized by s. Upon further analysis, he realizes as significant two characteristics of the behavior (or misbehavior) of the small sample statistics: (1) the estimates are not single-valued. They vary on either side of the universe value, u, as shown in Figure 2. Further, as the number of replicates decreases, the proportion of very high and very low results becomes greater. (2) The deviation estimates tend to be low and the discrepancy increases as fewer replicates are involved. This tendency is manifest in two ways: (a) low values appear more frequently than high values, and (b) the over-all average is less than u. The concentarationof values on the low side is indicated in Figure 2. A tendency of a statistic to deviate more in one direction than another is called bias. (Note that this bias is not a subjective matter.) For standard deviation estimates, the bias is negative. With this general survey and analysis of the behavior of reproducibility estimates a t hand, the chemist is prepared to face a small group of replicates encountered in one of his everyday analytical problems and make some specific inferences about the indeterminate error therein represented. I n the remainder of this paper, three such inferences are given along with some detailed back
