Error Analysis and Significant Figures

 Errors using inadequate data are much less than those using no data at all --- C.
Babbage (1791 - 1871)
No measurement of a physical quantity can be entirely accurate. It is important to
know, therefore, just how much the measured value is likely to deviate from the
unknown, true, value of the quantity. The art of estimating these deviations should
probably be called uncertainty analysis, but for historical reasons is referred to as error
analysis. This document contains brief discussions about how errors are reported, the
kinds of errors that can occur, how to estimate random errors, and how to carry error
estimates into calculated results. We are not, and will not be, concerned with the “percent
error” exercises common in high school, where the student is content with calculating the
deviation from some allegedly authoritative number.
SIGNIFICANT FIGURES
Whenever you make a measurement, the number of meaningful digits that you write
down implies the error in the measurement. For example if you say that the length of an
object is 0.428 m, you imply an uncertainty of about 0.001 m. To record this
measurement as either 0.4 or 0.42819667 would imply that you only know it to 0.1 m in
the first case or to 0.00000001 m in the second. You should only report as many
significant figures as are consistent with the estimated error. The quantity 0.428 m is said
to have three significant figures, that is, three digits that make sense in terms of the
measurement. Notice that this has nothing to do with the "number of decimal places".
The same measurement in centimeters would be 42.8 cm and still be a three significant
figure number. The accepted convention is that only one uncertain digit is to be reported
for a measurement. In the example if the estimated error is 0.02 m you would report a
result of 0.43 ± 0.02 m, not 0.428 ± 0.02 m.
Students frequently are confused about when to count a zero as a significant figure.
The rule is: If the zero has a non-zero digit anywhere to its left, then the zero is
significant, otherwise it is not. For example 5.00 has 3 significant figures; the number
0.0005 has only one significant figure, and 1.0005 has 5 significant figures. A number
like 300 is not well defined. Rather one should write 3 x 102, one significant figure, or
3.00 x 102, 3 significant figures.
ABSOLUTE AND RELATIVE ERRORS
The absolute error in a measured quantity is the uncertainty in the quantity and has
the same units as the quantity itself. For example if you know a length is 0.428 m ± 0.002
m, the 0.002 m is an absolute error. The relative error (also called the fractional error) is
obtained by dividing the absolute error in the quantity by the quantity itself. The relative
error is usually more significant than the absolute error. For example a 1 mm error in the
diameter of a skate wheel is probably more serious than a 1 mm error in a truck tire. Note
that relative errors are dimensionless. When reporting relative errors it is usual to
multiply the fractional error by 100 and report it as a percentage.