Random or statistical uncertainties arise from random fluctuations
in a measurement. These random fluctuations can occur in measuring
devices. For example, electronic noise and air currents lead
to a rapid but small fluctuation in motion detector readings. These
fluctuations occur, even when the motion detector is measuring the
distance to a stationary object. Random fluctuations can also be
a characteristic of the quantity being measured. For example, if we
use a meter stick to measure the landing positions of a series of
projectiles shot from a springloaded launcher, we see significant
random variations which clearly do not arise from the limitations of
the meter stick. Instead, we suspect that the launch velocity given to
projectiles by the launcher is subject to small random variations.
Truly random fluctuations average to zero, and so the way to remove
them is to average a large number of measurements,

(1) 
The average value approaches the ``true value'' as the
number of measurements in the average approaches infinity. Finding the
``true value'' is impractical, so we settle for the ``best value''
given by the average. The average value is also called the mean value.
Random fluctuations are described by the normal distribution, or
Gaussian distribution, or the ``bell curve.'' The uncertainty in the
``best value'' of a large collection of normally distributed
measurements can be calculated using the standard deviation

(2) 
which describes the width of the distribution. More precisely, about
68% of a normal distribution falls within of the average
value. The standard deviation is the uncertainty in a single
measurement in the distribution. Rather than doing this calculation
``by hand,'' I recommend using the STDEV() function of your
spreadsheet.
The uncertainty in the average of a large number of measurements is
less than . This follows from the idea that the more
measurements we make, the closer the average value comes to the ``true
value.'' The standard deviation of the mean is given by

(3) 
We report this as the uncertainty in .
See the sample writeup in Appendix A for an example of an analysis of
normally distributed data.
Systematic errors are are due to a defect in the equipment or methods
used to make measurements. For example, a motion sensor can be
poorly calibrated so that it gives distance readings which are only 90%
of the true values. It has a systematic uncertainty (10%) that is
much greater in magnitude than the statistical uncertainty in its
readings. Systematic errors are often difficult to detect, because
they do not show up as fluctuations in the results of repeated
measurements. It is important to think about possible sources of
systematic errors and to try to correct them or rule them out, for
example by
 checking calibrations
 comparing results with accepted values
 comparing results obtained via independent means
We often do not have the luxury of a large collection of normally
distributed measurements to analyze. Instead, we must somehow estimate
the uncertainty of a single measurement. This is necessarily somewhat
subjective. If only a few measurements are available, it is more
reasonable to use the entire range covered by the measurements to
define the uncertainty instead of calculating the standard deviation
of the mean. If only one measurement is available the resolution of
the device and the variation in the quantity measured are important
guides. For example, the resolution of a meter stick is 1 mm. If it is
used to measure the length of a rectangular steel plate, an
uncertainty of 1 mm, or perhaps even 0.5 mm, is reasonable. If I use
the same meter stick to measure the height of a small child, issues
of variable posture and how I line up the stick lead to an uncertainty
of as much as 1 cm. Be conservative with your estimates. That is, when
in doubt, it is a good policy to report a larger uncertainty.
Frequently, calculations involve one or more measured quantity, and we
need to determine how the uncertainties in input quantities translate
into the uncertainty in the result. The guidelines below cover all
of the possibilities. Always check that the result and its
corresponding uncertainty have the same units. If they do not,
something went wrong.
When adding or subtracting, add absolute uncertainties in
quadrature.
For example, if , then

(4) 
When multiplying or dividing, add relative (percentage)
uncertainties in quadrature.
For example, if
, then

(5) 
When raising a value to a power, multiply its relative error by the
power. For example, if

(6) 
Use first derivatives to determine the approximate
variation of the result due to the uncertainty in each measured
quantity.
If a quantity is a function of the measured quantities , then

(7) 
When calculating a result which depends on measured
input quantities, determine the variations in the result due to each
input quantity, and add the variations in quadrature. In some cases,
upper and lower uncertainties differ.
For example, if , the individual variances are
and the upper and lower uncertainties are
This kind of analysis is a good job for a spreadsheet.
Results with uncertainties are typically reported in the form

(10) 
Units are always included, and are usually given after the
result and its uncertainty. It is common practice to round
uncertainties to one significant figure. Results should be rounded off
to the decimal place of the corresponding uncertainties. For example,
if an analysis of several measurements of my height reveals an average
of
m with a standard deviation of the mean of
m, I report my height as
m. The form

(11) 
is also sometimes used, where the uncertainty is given as a single
digit. In this form, my height is m. The uncertainty is
assumed to be in the last reported digit of the result. With
asymmetric uncertainties, one uses the form

(12) 
Copyright © 20022004, Lewis A. Riley

Updated Mon Jan 19 13:29:10 2004

This work is licensed under a Creative Commons License.