F. STANDARD METHODS FOR EXPRESSING ERROR
1. Absolute Error.
Uncertainties may be expressed as absolute measures, giving the size of the a quantity's uncertainty in the same units in the quantity itself.
Example. A piece of metal is weighed a number of times, and the average value obtained is: M = 34.6 gm. By analysis of the scatter of the measurements, the uncertainty is determined to be m = 0.07 gm. This absolute uncertainty may be included with the measurement in this manner: M = 34.6 ± 0.07 gm.
The value 0.07 after the ± sign in this example is the estimated absolute error in the value 3.86.
2. Relative (or Fractional) Error.
Uncertainties may be expressed as relative measures, giving the ratio of the quantity's uncertainty to the quantity itself. In general:
(Equation 5)absolute error in a measurement
relative error =
size of the measurement
Example. In the previous example, the uncertainty in M = 34.6 gm was m = 0.07 gm. The relative uncertainty is therefore:
(Equation 6)m / 0.07 gm
= / = 0.002, or, if you wish, 0.2%
M / 34.6 gm
It is a matter of taste whether one chooses to express relative errors "as is" (as fractions), or as percents. I prefer to work with them as fractions in calculations, avoiding the necessity for continually multiplying by 100. Why do unnecessary work?
But when expressing final results, it is often meaningful to express the relative uncertainty as a percent. That's easily done, just multiply the relative uncertainty by 100. This one is 0.2%.
3. Absolute or relative form; which to use.
Common sense and good judgment must be used in choosing which form to use to represent the error when stating a result. Consider a temperature measurement with a thermometer known to be reliable to ± 0.5 degree Celsius. Would it make sense to say that this causes a 0.5% error in measuring the boiling point of water (100 degrees) but a whopping 10% error in the measurement of cold water at a temperature of 5 degrees? Of course not! [And what if the temperatures were expressed in degrees Kelvin? That would seem to reduce the percent errors to insignificance!] Errors and discrepancies expressed as percents are meaningless for some types of measurements. Sometimes this is due to the nature of the measuring instrument, sometimes to the nature of the measured quantity itself, or the way it is defined.
There are cases where absolute errors are inappropriate and therefore the errors should be expressed in relative form. There are also cases where the reverse is true.
Sometimes both absolute and relative error measures are necessary to completely characterize a measuring instrument's error. For example, if a plastic meter stick uniformly expanded, the effect could be expressed as a percent determinate error. If a one half millimeter were worn off the zero end of a stick, and this were not noticed or compensated for, this would best be expressed as an absolute determinate error. Clearly both errors might be present in a particular meter stick. The manufacturer of a voltmeter (or other electrical meter) usually gives its guaranteed limits of error as a constant determinate error plus a `percent' error.
Both relative and fractional forms of error may appear in the intermediate algebraic steps when deriving error equations. [This is discussed in section H below.] This is merely a computational artifact, and has no bearing on the question of which form is meaningful for communicating the size and nature of the error in data and results.