In mathematics, error analysis is the study of kind and quantity of error, or uncertainty, that may be present in the solution to a problem. This issue is particularly prominent in applied areas such as numerical analysis and statistics

The Purpose of Error Analysis

For students who only attend lectures and read textbooks in the sciences, it is easy to get the incorrect impression that the physical sciences are concerned with manipulating precise and perfect numbers. Lectures and textbooks often contain phrases like:

A particle falling under the influence of gravity is subject to a constant acceleration of 9.8 m/. If …

For an experimental scientist this specification is incomplete. Does it mean that the acceleration is closer to 9.8 than to 9.9 or 9.7? Does it mean that the acceleration is closer to 9.80000 than to 9.80001 or 9.79999? Often the answer depends on the context. If a carpenter says a length is “just 8 inches” that probably means the length is closer to 8 0/16 in. than to 8 1/16 in. or 7 15/16 in. If a machinist says a length is “just 200 millimeters” that probably means it is closer to 200.00 mm than to 200.05 mm or 199.95 mm.

We all know that the acceleration due to gravity varies from place to place on the earth’s surface. It also varies with the height above the surface, and gravity meters capable of measuring the variation from the floor to a tabletop are readily available. Further, any physical measure such as g can only be determined by means of an experiment, and since a perfect experimental apparatus does not exist, it is impossible even in principle to ever know g perfectly. Thus, the specification of g given above is useful only as a possible exercise for a student. In order to give it some meaning it must be changed to something like:

A 5 g ball bearing falling under the influence of gravity in Room 126 of McLennan Physical Laboratories of the University of Toronto on March 13, 1995 at a distance of 1.0 ± 0.1 m above the floor was measured to be subject to a constant acceleration of 9.81 ± 0.03 m/.

Two questions arise about the measurement. First, is it “accurate,” in other words, did the experiment work properly and were all the necessary factors taken into account? The answer to this depends on the skill of the experimenter in identifying and eliminating all systematic errors. These are discussed in Section 3.4.

The second question regards the “precision” of the experiment. In this case the precision of the result is given: the experimenter claims the precision of the result is within 0.03 m/s. The next two sections go into some detail about how the precision of a measurement is determined. However, the following points are important:

1. The person who did the measurement probably had some “gut feeling” for the precision and “hung” an error on the result primarily to communicate this feeling to other people. Common sense should always take precedence over mathematical manipulations.

2. In complicated experiments, error analysis can identify dominant errors and hence provide a guide as to where more effort is needed to improve an experiment.

3. There is virtually no case in the experimental physical sciences where the correct error analysis is to compare the result with a number in some book. A correct experiment is one that is performed correctly, not one that gives a result in agreement with other measurements.

4. The best precision possible for a given experiment is always limited by the apparatus. Polarization measurements in high-energy physics require tens of thousands of person-hours and cost hundreds of thousand of dollars to perform, and a good measurement is within a factor of two. Electrodynamics experiments are considerably cheaper, and often give results to 8 or more significant figures. In both cases, the experimenter must struggle with the equipment to get the most precise and accurate measurement possible.

Different Types of Errors

As mentioned above, there are two types of errors associated with an experimental result: the “precision” and the “accuracy”. One well-known text explains the difference this way:

The word precision” will be related to the random error distribution associated with a particular experiment or even with a particular type of experiment. The word accuracy” shall be related to the existence of systematic errors—differences between laboratories, for instance. For example, one could perform very precise but inaccurate timing with a high-quality pendulum clock that had the pendulum set at not quite the right length. E.M. Pugh and G.H. Winslow, p. 6.

The object of a good experiment is to minimize both the errors of precision and the errors of accuracy.

Usually, a given experiment has one or the other type of error dominant, and the experimenter devotes the most effort toward reducing that one. For example, in measuring the height of a sample of geraniums to determine an average value, the random variations within the sample of plants are probably going to be much larger than any possible inaccuracy in the ruler being used. Similarly for many experiments in the biological and life sciences, the experimenter worries most about increasing the precision of his/her measurements. Of course, some experiments in the biological and life sciences are dominated by errors of accuracy.

On the other hand, in titrating a sample of HCl acid with NaOH base using a phenolphthalein indicator, the major error in the determination of the original concentration of the acid is likely to be one of the following: (1) the accuracy of the markings on the side of the burette; (2) the transition range of the phenolphthalein indicator; or (3) the skill of the experimenter in splitting the last drop of NaOH. Thus, the accuracy of the determination is likely to be much worse than the precision. This is often the case for experiments in chemistry, but certainly not all.

Question: Most experiments use theoretical formulas, and usually those formulas are approximations. Is the error of approximation one of precision or of accuracy?




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