Defining Relative Error
Date: 03/11/2004 at 11:48:45 From: Mark Subject: Relative error v. Relative error? Dear Dr. Math, I am a Mathematics Editor at a small publishing company in NJ working on a 9th grade standards-based book and am having trouble discerning what precisely is relative error. I have searched the Internet and our own archives and have come up with two different definitions. At most sites I have seen this definition: Relative error is the absolute error divided by the "true" measurement. This is the definition with which I am the most familiar, and what I felt relative error was. Then I came across a different definition at http://www.glencoe.com/sec/math/mac/mac01/course3/pdf/m_31107.pdf : The relative error of a measurement is found by dividing the greatest possible error by the measurement itself. It strikes me that these are two very different formulas for finding what seems like two very different types of relative error. One is more of a comparison of your error to the true value. Meanwhile I'm not sure what to make of the other. Possibly it is a comparison between the degree of error and the actual value. Perhaps it should be called the Greatest Possible Relative Error? Any and all help discerning which definition is correct, and what perhaps one might call the latter definition (if not the relative error) would be greatly appreciated. Thanks! - Mark
Date: 03/12/2004 at 12:14:21 From: Doctor Peterson Subject: Re: Relative error v. Relative error? Hi, Mark. As you suggest, they really are two different things entirely. Your first definition, absolute error divided by true value, is a measure of _accuracy_; it tells us how close a particular measurement is to the correct value. We can only determine this when we know the true value. Your second definition, "greatest possible error" divided by the measured value, is a measure of _precision_, which depends only on the measuring instrument itself, not on the actual value. If we assume the instrument is correctly calibrated, then it tells how far off we might be due simply to the number of digits of accuracy or the spacing of the marks. If it is not calibrated, then precision is irrelevant to accuracy, but still meaningful in its own right. So what you've found is that the term "relative error" is used in two very different (but easily confused) contexts. We might call them "actual relative error" and "possible relative error". One of the references I gave before makes the same point: Error in Measurements - Introduction to Chemical Sciences, James A. Plambeck, Univ. of Alberta http://www.ualberta.ca/~jplambec/che/p101/p01017.htm The amount of error associated with a particular measurement may be considered from the point of view of precision or the point of view of accuracy. The precision of a measurement expresses the error, or deviation, of the measurement from the average of a large number of measurements of the same quantity, while the accuracy of a measured value expresses the deviation of the measurement from the true value of the quantity. Error is considered from the point of view of accuracy when the true value is known, but when the true value of a quantity is not known precision must be used in place of accuracy. It is impossible to obtain accuracy if precision cannot be obtained, but precision does not guarantee accuracy. Any significant systematic error (an error which, for some systematic or determinate reason, influences the measurement in a known or knowable way) may give results which are very precise--and highly inaccurate. Scientists often obtain the precision of a measurement not by actually carrying out a large number of measurements but from knowledge of the limitations of the apparatus used to carry out the measurement procedure....these precisions can be obtained using proper measuring techniques and are a measure of the deviation expected in repetitive measurements. So precision is what you can determine from the measurements alone (how close they are to one another), or from the nature of the instrument itself (how close a ruler's marking are, for example); in this context, relative error indicates the spread of possible measurements. On the other hand, accuracy is based on the true value, and in that context, relative error indicates how far the measurement is from the true value. Again, this page distinguishes the two forms of "relative error" by using different terms, "relative error" and "relative deviation": Accuracy and Precision http://king.prps.k12.ca.us/prhs/pasohigh/classes/Fairbank/public.www/homepage/physics/accpre.HTM Accuracy is the degree to which a measurement agrees with an accepted value for those measurements. They can be evaluated in absolute or relative terms. The absolute error is the absolute value of the difference between the accepted value and the measurement. This can be written as an equation as shown below. Absolute error = Observed - Accepted value Ea = |O - A| This can be expressed as a percentage error also. The percentage error is the relative error. It is expressed by the following equation. Absolute error Ea Relative error = -------------- x 100% Er = -- x 100 Accepted value A Data can also be evaluated in terms of how many measurements that are made in the same manner deviate from one another. This is known as precision and is evaluated in terms of absolute and relative deviation. Absolute deviation is the absolute value of the difference between the mean or average value and the measured value. This is expressed below in the equation. Absolute deviation = Observed - Mean value Da = |O - M| Another way to express the deviation or precision is as a percentage. This is the relative deviation and is expressed as follows. Average absolute dev Da Relative deviation = -------------------- x 100% Dr = -- x 100 Mean value M Actually that "relative deviation" is not quite what you asked about, because it deals with a set of actual measurements rather than possible measurements; but it is closely related. I would call your second definition "relative uncertainty" if I had to give it its own name. Uncertainty is just a different type of "error". And that turns out to be just what the following nice glossary uses: I'll close with this: Definitions of Measurement Uncertainty Terms http://www.physics.unc.edu/~deardorf/uncertainty/definitions.html I won't quote from it, because you'll want to be sure to read through the whole page to see both careful definitions, and the conflicts the author found among different sources. As you can see, all terms and concepts tend to be rather flexible, adapting to different situations by changing their meaning slightly, while retaining the essential concept. That can be a little confusing, but it is the way language works, even in math! So textbooks will never quite be able to match exactly with real-world uses of terms, because you don't want to confuse kids with this reality. A little awareness of it may be good, however! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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