INTRODUCTION

These web documents contain supplementary material for laboratory work in the introductory (first year) physics course. They provide an emphasis and point of view which some laboratory manuals lack.

A large part of this material commonly goes by the name "error analysis," a name that often strikes fear into the tender heart of freshmen. Some physics teachers also fear to tackle the subject; and laboratory manuals vary greatly in the extent and character of their treatments. Some ignore it entirely. Some include it in the introduction, then never refer to it again. Some manuals include exhaustive treatments of error theory far beyond the needs of the freshman laboratory.

In the first year physics laboratory one seldom takes sufficient data to mathematically justify the use of the standard deviation. The student who takes upper level physics courses will encounter laboratory work that requires careful statistical treatment of errors; the freshman does not.

The first year physics course has a very full academic menu, so we must use care and good judgment in deciding what essential material to include and what material to omit.

To totally omit a concern for errors does a disservice to the student and leaves the false (and harmful) notion that "getting the right answer" represents the major objective of laboratory work. If students measure the "goodness" of the experiment by how close they come to the "textbook value," then we have perpetuated an attitude that runs counter to good scientific method.

I propose a shift of emphasis. To talk about sophisticated statistical measures of error has little value when one hasn't taken enough data to even know what kind of error distribution the measurements have. One might as well use the crude "maximum error" measure, even with its limitations.

We can, and should, emphasize propagation of errors, and the error propagation equation. No matter what the error distribution, or what fancy, or crude, error measures one uses, error propagation equations describe the effect that errors have on results. The error equations have a far greater importance than merely calculating the error in a result. When a student uses the error equation to optimize the experimental procedure and thereby minimize errors in results, this improves the experiment and its results no matter what error measure one happens to use.

This approach has value even if the student has never heard of "standard deviation" or "kurtosis." As engineer Hilbert Schenck, Jr. has said, "The problem is not...that a model or method may be used clumsily or in the wrong place. Far more serious in the present state of engineering experimentation is that the statistical model will not be used at all." [1]

The error propagation equation serves to guide the experimental strategy, identifying those variables that most affect the error, and it shows what must be done to attain a desired precision in the result. The error equation may sometimes guide the experimenter in the choice of the sizes of variables to produce the best results.

Once the student learns the technique of error propagation analysis, and its use becomes habitual, a solid foundation has been built for later work, and for conceptually understanding the more sophisticated mathematical analyses of error.

Attention to error propagation encourages the student to think, in a critical manner, about the entire experiment, from overall strategy to the minutest detail. This may represent the most valuable benefit from concern for error analysis in the elementary laboratory.

Donald E. Simanek, September 2, 1988

1. Schenck, Hilbert, Jr. Theories of Engineering Experimentation, 2nd Ed. 1968. McGraw-Hill.

Preface to the 1996 revision:

This printing results from a thorough reworking of the text from start to finish, with particular emphasis on tightening the prose, eliminating flabby passages, improving clarity and consistency, and improving readability. Hardly a page escaped numerous changes. Certain parts, particularly the introduction and chapter 9, now conform to the standards of E-prime, by avoiding all forms of the verb "be." Such strict adherence to E-prime does not seem to me appropriate for material which must include physical and mathematical definitions. However, attention has been paid throughout the text to avoid misleading and ambiguous statements resulting from "be", "is", etc.

Preface to the 2004 revision:

This online version is a project that will extend through late 2004 and into 2005. Some additional material and edits have been made, but the material is substantially unchanged.

Preface to a future revision, 2018.

This manual was written before personal computers came on the scene. That was the slide-rule era, where three significant figures was the norm for calculations in elementary physics courses and laboratories. Most of the contents of this manual are still valid, but some laboratory apparatus is now seldom seen, replaced by motion sensors, digital readouts, and other coveniences. One suspects that lab equipment may evolve to the point that you can press a "start" button and the robotic equipment will do all the work, log the data, process it, and spit out a handsomly printed completed laboratory report worthy of an "A".

Computer software can be helpful not only to decrease drudgery, but to improve resuslts, if used intelligently. In the past experimenters had to consider how many significant figures were enough carry through intermediate calculations to avoid compromising the quality of results. Now almost all computational software carries far more significant and insignificant figures (15 is typical) than are necessary, and the result can be rounded off.

Unforuntately this processs doesn't tell us how many of those figures in the result are significant. One method, seldom used in the past, now becomes practical. It is the "crank three times" (C3T) method. Calculate the result once the normal way. Then do it again with each variable larger by the amount of its estimated error, and save the result. Then do it again with each variable smaller by the amount of its estimated error, and save the result. Now see how much the three results vary. This gives you the range of error in the result.

This sounds reasonable, and is easy enough to do with computer devices. But it has a flaw. Sometimes the worst result obtains when some data quantities produce the greatest error in the result when some are smaller and some larger than their average. (Consider the error in division: R=X/Y.) We'd have to consider all combinations of larger and smaller in a complex calculation to be certain we have explored the full range of possible results.

Back in the slide rule era, some students did use the C3T method for dealing with error propagation.

The C3T method is the Monte Carlo Method Lite.

The C3T method, unfortunately, doesn't tell us which variable's error affects the answer the most and which the least? We'd like to know this in order to improve the experimental strategy. Which variables require the most attention to their improvment, and which can we dismiss as insignificant. The standard methods described in this manual do that quite well, and whatever method you use to express error in the result, it would be wise to develop the determinate error equation anyway—in advance of even performing the experiment. All of the simple methods have an achiles heel. One must establish the error distribution of each variable to be sure they are near enough to Gaussian to even justify the standard deviation (or even the average deviation) as a measure of its error. This requires taking quite a large number of data points for each data variable. If the error distribution of any one deviates significiantly from a Gaussian curve, most standard error propagation methods are inappropriate.

We see that every step in this process requires careful thought. Perhaps this is why no software is yet available that can carry it out in all situations reliably.

A final caution: The chapter on significant figures is an historic relic describing a method for calculating error in results that it is a blunt instrument, totally inadequate, and in some cases gives worthless results. It was "taught" in many high schools, and some colleges, and in some textbooks it was the only method given. It deserves to be consigned to history.