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The Vocabulary of Error Analysis
The tutorial pages following this one will detail practical methods for determining the size and nature of errors in an experiment. As preparation for these pages, you should read through these definitions first. You can also come back here to look words up when you get confused.
An error in a measurement is the difference between the result of the measurement and the true value of whatever you were trying to determine. An error can be expressed in two ways:
An absolute error is an error expressed in physical units. For
example, if we measure the acceleration due to gravity in the lab room
to be 9.7 m s
,
the absolute error is -0.1 m s.
Absolute errors should always have the physical units indicated.
A relative error, or fractional error, is an error expressed as a fraction of the value measured or the true value (if the error is small, it makes little difference). In the above example, the error was 0.1/9.8 = 1% relative to the true value. If the true value is not known, relative errors are given with respect to the measurement. Relative errors should always be displayed as a percentage, to avoid confusion with absolute errors.
An uncertainty is a range, estimated by the experimenter, that is likely to contain the true value of whatever is being measured. For example, if you measure a distance with a meter stick you usually assign an uncertainty of ± 1mm to the result. Uncertainties can be expressed in absolute terms or relative terms, just as errors can. People often say "error" when they mean uncertainty, just because it doesn't take as long, but what is meant can usually be figured out from the context. I'll try not to do it myself in these pages.
A confidence level is the probability that the true value of your experiment falls within a given range of uncertainty. Confidence levels can be exactly defined if you have a good understanding of the nature of your errors.
A systematic error is not a way of expressing an error, but a breed of problem which often plagues experiments. A systematic error biases your measurements in some predictable way, although you may not know how to predict it. A simple example would be a voltmeter which only displays 90% of the true voltage, so the the size of the error changes depending on what you're measuring. Systematic errors can get very complex, but once understood you can modify your results to remove them.
A random error is an error which is present every time you take the measurement, but which varies unpredictably in size and direction. Random errors are always present, but fortunately follow well-behaved statistical rules. The effects of random errors can be reduced only by repeating the measurement as often as possible.
An illegitimate error is not one born out of wedlock, but a one-time mistake in the procedure which produces a bizzare value. If you know you made a mistake (for example, kicking the equipment in frustration) then you can just throw out that measurement. Usually the mistake is more subtle, for example misreading a display or an unexpected power surge in the equipment. In this case, people usually use some statistical criteria to throw out data which are well outside the normal range of possibility.
The precision of a measurement is the total amount of random error present. A very precise measurement has small random errors, but just because a measurement is precise doesn't mean that it's accurate (see below); undiscovered systematic errors might skew your results drastically.
The accuracy of a measurement is a way of talking about the total error in your final result. An accurate measurement is very close to the true value. Just because a measurement is accurate doesn't mean it's precise; an accurate value with a wide possible range isn't very useful.
This is enough to establish a common ground. There is much more to learn, of course, but other new words will be expained as they come up.
Created by Ben Mathiesen
Last Updated by Andy Pawl
Sept. 2003
apawl@umich.edu