~.------~------------ JOURNAL OF RESEARCH of the National Bureau of Standards - C. Engineering and Instrumentation Vol. 70C, No.4, October- December 1966 Notes on the Use of Propagation of Error Formulas H. H. Ku Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 20234 (May 27, 1966) The " la w of propagation o[ error" is a tool th at physical scientists have convenie ntly and frequently used in th eir work for many years, yet an adequate refere nce is diffi c ult to find. In this paper an exposi­ tory rev ie w of this topi c is presented, partic ularly in the li ght of c urre nt practi ces a nd interpretations. Examples on the accuracy of the approximations are given. The reporting of the uncertainties of final results is discussed. Key Words : Approximation, e rror, formula, imprecision, law of error, prod uc ts, propagati on of error, random, ratio, syste mati c, sum. Introduction The " law of propagation of error" is a tool that physical scientists have conveniently and frequently In the December 1939, issue of the American used in their work for many years. No claim is Physics Teacher, Raymond T. Birge wrote an ex­ made here that it is the only tool or even a suitable pository paper on "The Propagation of Errors." tool for all occasions. "Data analysis" is an ever­ In the introductory paragraph of his paper, Birge expanding field and other methods, existing or new, re marked: are probably available for the analysis and inter­ "The questi on of what constitutes the most reliable valu e to pretation fo r each particular set of data. Never­ be assigned as the uncertainty of any given measured quantity is theless, under certain assumptions give n in detail one that has been discussed [or many decades and, presumably, in th e following sections, the a pproxim ati ons res ulting will continue to be discussed. It is a questi on that involves many considerations a nd by its very nature has no unique answer. The from the use of these formulas are useful in giving subject of the propagation of errors, on the contrary, is a purely an estimate of the uncertainty of a reported value. mathematical matter, with very definite and easily ascertained T he uncertainty computed from the use of these conclusions. Although the general s ubject of the present article formulas, however, is probably somewhat less that is by no means new,' many scie ntists still fail to avail themselves of the enlighte ning conclu sions that may often thus be reached, the actual in the sense that no func ti on form is know n while others frequently use the theory incorrectly and thus arrive exactly and the number of variables considered usually at quite mi sleading conclusions." does not represent fully the contributors of errors Birge's remark 27 years ago still sounds fitting today. that affect the final result. For a number of years, the need for an expository paper on this topi c has been felt by the staff of the 1. Sta tist ica l Tolerancing Versus Imprecision Statistical Engineering Laboratory at the National of a Derived Quantity Bureau of Standards. Frequent inquiries have to be answered, yet a diligent search in current litera· 1.1. Propagation of error formulas are frequently ture and textbooks failed to produce a suitable ref· used by engineers in the type of problem yalled '~S ta ­ erence that treats the subject matter adequately. tis tical tolerancing." In such problems, we are The present manuscript was written to fill this need. concerned with the behavior of the characteristic In section 1, we consider the two distinct situations W of a system as related to the behavior of a charac­ under which the propagation of error formulas can teristic X of its component. For instance, an engi­ be used. The mathematical manipulations are the neer may have designed a circuit. A property W same, yet the interpretations of the results are en­ of the circuit may be related to the value X of the tirely different. In section 2 the notations are de­ resistance used. As the value of X is changed, " , fined and the general formulas given. Frequently W changes and the relationship can be expressed used special formulas are li sted at the end of the by a mathematical function section for conveni ent reference. In section 3 the accuracies of the approximations are discussed, W=F(X) together with suggesti ons on the use of the errors propagated. Section 4 contains s uggestions on the within a certain range of the values of X. reporting of final results. Suppose our engineer decides on W = Wo to be the desired property of the circuit, and specifies X = Xo I See. for instance. M. Merriman, Method of Least Squares, pp. 75- 79 (ed. 8, 1910). for this purpose. He realizes, however, that there 263 I /' will be vanatlOns among the large lot of resistors he dF]2 S2 ordered, no matter how tight his specifications are. var (11)= [dX n (1.2) Let x denote the value of anyone of the resistors in the lot, then some of the time x will be below Xo, while at other times x will be above Xo. In other words, x has a distribution of values somewhat clus­ tered about Xo. As x varies with each resistor, so Often he assumes that 11 is distributed at least approxi­ does w with each circuit manufactured. mately in accordance with the normal law of error and If our engineer knows the mean and standard gives probability limits to the statistical uncertainty of deviation (or variance) of x, based on data from the his estimate 11 based on the standard deviation calcu­ history of their manufacture, then he can calculate lated (o-w) and this assumption. the approximate mean and variance of w by the Cramer [1946] has shown that under very general propagation of error formulas: conditions, functions of sample moments are asymp­ totically normal, with mean and variance given by mean (w) == F(mean x), and the respective propagation of error formulas. 2 Since xn is the first sample moment, the estimate 11 will be approximately normitlly distributed for large n. Hence variance (w) == [dXdF]2 var (x), (1.1) our physicist is interested in the variance (or the standard deviation) of the normal distribution which the distribution of F(xn) approximates as n increases. where the square brackets signify that the derivatives (Note that both estimators tv and var (11) are functions within the brackets are to be evaluated at the mean of n.) For n large, the distribution of tv can be as­ of x. The approximations computed refer to the mean sumed to be approximately normal and probability and variance of an individual unit in the collection of statements can be made about 11. circuits that will be manufactured from the lot of 1.3 Hence, we have the two cases: resistors. The distribution of values of w, however, (1) The problem of determining the mean and vari­ is still far from being determined since it depends ance (or standard deviation) of the actual distribu­ entirely on the functional form of the relation between tion of a given function F(x) of a particular random Wand X, as mathematical variables, and the distri­ variable x, and bution of x itself, as a random variable. This type of (2) The problem of estimating the mean and vari­ approach has been used frequently in preliminary ance (or standard deviation) of the normal distri­ examinations of the reliability of performance of a bution to which the distribution of F(xlI) tends asymp­ system, where X may be considered as a multidimen­ totically. sional variable. As examples of problems studied under the first 1.2 Let us consider now the second situation under case, we can cite Fieller [1932] on the ratio of two which propagation of error formulas are used. This normally distributed random variables, and Craig situation is the one considered in Birge's paper, and [1937] and Goodman [1962] on the product of two is the one that will be discussed in the main part of or more random variables. Tukey, in three Princeton this paper. University reports, extended the classical formulas A physicist may wish to determine the "true" value through the fourth order terms for the mean and Wo of interest, for example, the atomic weight of silver. variance, and propagated the skewness and elon­ He makes n independent measurements on some re­ gation of the distribution of F(x) as well. These lated quantity x and calculates reports present perhaps the most exhausti ve treat­ ment of statistical tolerancing to date. _ 1 From now on we shall be concerned in this paper Xn = - (Xl + X2 +. + Xn) n with the second case only, i.e., the problem of esti­ mating the mean and variance, or standard deviation, as an estimate of the true value Xo and of the normal distribution to which the distribution of F(xn) tends as n increases indefinitely, and hence also the problem of using approximations to the 1 1/ S2=_- (X-X)2 mean and variance computed from a finite number of l L In n- I measurements. Since the mean and standard deviation are the parameters that specify a particular normal distribution, our problem is by its very nature less complicated than that of statistical tolerancing as an index of dispersion of his measured values. The where the actual distribution of the function may physicist is mainly concerned in obtaining an estimate have to be specified.