Error Analysis and Significant Figures

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.

Systematic errors

Systematic errors arise from
a flaw in the measurement scheme which is repeated each time
a measurement is made. If you do the same thing wrong each
time you make the measurement, your measurement will differ
systematically (that is, in the same direction each time) from
the correct result. Some sources of systematic error are:

Errors in the calibration of the measuring instruments.
Incorrect
measuring technique: For example, one might make an
incorrect scale reading because of parallax error.
Bias of the experimenter. The experimenter might consistently
read an instrument incorrectly, or might let knowledge
of the expected value of a result influence the measurements.

It is clear that systematic
errors do not average to zero if you average many measurements.
If a systematic error is discovered, a correction can be made
to the data for this error. If you measure a voltage with a
meter that later turns out to have a 0.2 V offset, you can
correct the originally determined voltages by this amount and
eliminate the error. Although random errors can be handled
more or less routinely, there is no prescribed way to find
systematic errors. One must simply sit down and think about
all of the possible sources of error in a given measurement,
and then do small experiments to see if these sources are active.
The goal of a good experiment is to reduce the systematic errors
to a value smaller than the random errors. For example a meter
stick should have been manufactured such that the millimeter
markings are positioned much more accurately than one millimeter.

Random errors

Random errors arise from the
fluctuations that are most easily observed by making multiple
trials of a given measurement. For example, if you were to
measure the period of a pendulum many times with a stop watch,
you would find that your measurements were not always the same.
The main source of these fluctuations would probably be the
difficulty of judging exactly when the pendulum came to a given
point in its motion, and in starting and stopping the stop
watch at the time that you judge. Since you would not get the
same value of the period each time that you try to measure
it, your result is obviously uncertain. There are several common
sources of such random uncertainties in the type of experiments
that you are likely to perform:

Uncontrollable fluctuations in initial conditions in the
measurements. Such fluctuations are the main reason why,
no matter how skilled the player, no individual can toss
a basketball from the free throw line through the hoop each
and every time, guaranteed. Small variations in launch
conditions or air motion cause the trajectory to vary and
the ball misses the hoop.
Limitations imposed by the precision of your measuring
apparatus, and the uncertainty in interpolating between the
smallest divisions. The precision simply means the smallest
amount that can be measured directly. A typical meter stick
is subdivided into millimeters and its precision is thus
one millimeter.
Lack of precise definition of the quantity being measured.
The length of a table in the laboratory is not well defined
after it has suffered years of use. You would find different
lengths if you measured at different points on the table.
Another possibility is that the quantity being measured also
depends on an uncontrolled variable. (The temperature of
the object for example).
Sometimes the quantity you measure is well defined but
is subject to inherent random fluctuations. Such fluctuations
may be of a quantum nature or arise from the fact that
the values of the quantity being measured are determined
by the statistical behavior of a large number of particles.
Another example is AC noise causing the needle of a voltmeter
to fluctuate.

No matter what the source of
the uncertainty, to be labeled "random" an uncertainty
must have the property that the fluctuations from some "true" value
are equally likely to be positive or negative. This fact gives
us a key for understanding what to do about random errors.
You could make a large number of measurements, and average
the result. If the uncertainties are really equally likely
to be positive or negative, you would expect that the average
of a large number of measurements would be very near to the
correct value of the quantity measured, since positive and
negative fluctuations would tend to cancel each other.

Estimating random errors

There are several ways to make
a reasonable estimate of the random error in a particular measurement.
The best way is to make a series of measurements of a given
quantity (say, x) and calculate the mean ,
and the standard deviation from
this data. The mean is defined as

where x_i is the result of the i^th
measurement and N is the number of measurements. The
standard deviation is given by

If a measurement (which is subject only to random fluctuations)
is repeated many times, approximately 68% of the measured valves
will fall in the range .

We become more certain that
, is an
accurate representation of the true value of the quantity x the
more we repeat the measurement. A useful quantity is therefore
the standard deviation of the mean defined
as . The
quantity is
a good estimate of our uncertainty in .
Notice that the measurement precision increases in proportion
to as
we increase the number of measurements. Not only have you made
a more accurate determination of the value, you also have a
set of data that will allow you to estimate the uncertainty
in your measurement.

The following example will
clarify these ideas. Assume you made the following five measurements
of a length:

	Length (mm)	Deviation from the mean
	22.8	0.0
	23.1	0.3
	22.7	0.1
	22.6	0.2
	23.0	0.2
sum	114.2	0.18	sum of the squared deviations
	divide by 5	divide by 5 and take the square root	(N = number data points = 5)
mean	22.8	0.19	standard deviation
		divide by
		0.08	standard deviation of the mean

Thus the result is 22.84 ± .08 mm. (Notice the use
of significant figures).

In some cases, it is scarcely
worthwhile to repeat a measurement several times. In such situations,
you often can estimate the error by taking account of the least
count or smallest division of the measuring device. For example,
when using a meter stick, one can measure to perhaps a half
or sometimes even a fifth of a millimeter. So the absolute
error would be estimated to be 0.5 mm or 0.2 mm.

In principle, you should by
one means or another estimate the uncertainty in each measurement
that you make. But don't make a big production out of it. The
essential idea is this: Is the measurement good to about 10%
or to about 5% or 1%, or even 0.1%? When you have estimated
the error, you will know how many significant figures to use
in reporting your result.

Propagation of errors

Once you have some experimental
measurements, you usually combine them according to some formula
to arrive at a desired quantity. To find the estimated error
(uncertainty) for a calculated result one must know how to
combine the errors in the input quantities. The simplest procedure
would be to add the errors. This would be a conservative assumption,
but it overestimates the uncertainty in the result. Clearly,
if the errors in the inputs are random, they will cancel each
other at least some of the time. If the errors in the measured
quantities are random and if they are independent (that is,
if one quantity is measured as being, say, larger than it really
is, another quantity is still just as likely to be smaller
or larger) then error theory shows that the uncertainty in
a calculated result (the propagated error) can be obtained
from a few simple rules, some of which are listed in Table
1. For example if two or more numbers are to be added (Table
1, #2) then the absolute error in the result is the square
root of the sum of the squares of the absolute errors of the
inputs, i.e.

then

In this and the following expressions, and are the absolute random errors in x and y and
is the propagated uncertainty in z. The
formulas do not apply to systematic errors.

The general formula, for your
information, is the following;

It is discussed in detail in many texts on the theory of errors
and the analysis of experimental data. For now, the collection
of formulae in table 1 will suffice.

Table 1: Propagated errors in z due to errors in x and y.
The errors in a, b and c are assumed
to be negligible in the following formulae.