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/