Error Analysis in Experimental Physical Science

§1 - Introduction

"To err is human; to describe the error properly is sublime."
-- Cliff Swartz, Physics Today 37 (1999), 388.

As you may know, most fields in the physical sciences are bifurcated into two branches: theory and experiment. In general, the theoretical aspect is taught in lectures, tutorials and by doing problems, while the experimental aspect is learned in the laboratory.

The way these two branches handle numerical data are significantly different. For example, here is a problem from the end of a chapter of a well-known first year University physics textbook:

A particle falling under the influence of gravity is subject to a constant acceleration g of 9.8 m/s2. If …

Although this fragment is perfectly acceptable for doing problems, i.e. for learning theoretical Physics, in an experimental situation it 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 measurement such as of 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 in an experimental context we must say 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/s2.

This series of documents and exercises is intended to discuss how an experimentalist in the physical sciences determines the errors in a measurement, i.e. the numbers that appear to the right of the ± symbols in the above statement. The level is appropriate for beginning University students in the sciences.

We should emphasise right now that a correct experiment is one that has been correctly performed. Thus:

The error in an experimentally measured quantity is never found by comparing it to some number found in a book or web page.

Also, although we will be exploring mathematical and statistical procedures that are used to determine the error in an experimentally measured quantity, as you will see these are often just "rules of thumb" and sometimes a good experimentalist uses his or her intuition and common sense to simply guess.

Although these notes are delivered via the web, many people find that reading the type of material discussed here is more effective on paper than on a computer screen. If you are one of these people you may wish to print out these notes and read them in hardcopy.

However, because of different browsers and choices of default fonts printing this version of the document which you are now reading may not "layout" very cleanly on the page . Thus another version of this document has been prepared in the Portable Document Format (pdf) with page breaks controlled for a better layout. Using pdf files requires that you have the Acrobat Reader software installed, which is available free from http://www.adobe.com.

The file size of the pdf version is 290k and the hardcopy will be 37 pages. It will appear in a separate window. You may access the pdf version of this document here.

At the University of Toronto, some studeents answer all the questions that appear in these notes; this includes all the questions that appear in the exercises. These answers are then collected and marked. The maximum mark on the assignment is 100. Each question is marked out of 3 points except for the following, which are marked out of 4 points:

• Exercise 3.2: Questions 1, 2, and 3 for a total of 12 possible points for this Exercise.
• Exercise 3.3: Questions 1, 2, and 3 for a total of 12 possible points for this Exercise.
• Question 9.2

University of Toronto students who are required to do the assignment must turn in the assignment using a form that is available as a pdf document here.

These notes and the underlying code were written by David M. Harrison, Department of Physics, University of Toronto in January 2001. You may email the author at mailto:harrison@physics.utoronto.ca

The main pages of these notes are pure html with no JavaScript or similar enhancements. The simulations in some of the Exercises are written in Perl using the CGI, GD and GDGraph modules.

The quincunx applet accessed in §4 was written by David Krider when he was at RAND, and was slightly modified by David M. Harrison.