Notes Qn Propagation of Errors For
Fhysies {8Sn ZB0, & ZBS Labs at Golden West Goltege
by
Konrad M. Stein
Error analysis is the study and evaluation sf uneertainty in measurement.
Experience has shown that no measurement, however eaiefully made, ean be
completely free of uncedainties. Sinee the whole structure and application of
science depends on measurements, it is therefore crucially important to be able
to evaluate these uncertainties, and keep them to a minimum.
L Ercrs as Uncertaintieg
ln science, the word "error" does not carry the usual connotatioR of "mistake"
or "blunder"" "Error" in a scientific measurement means the inevitable
uneertainty that is a part of all measurements. As such, errors are not mistakes;
you caRRot avoid them by being very careful. The best you can hope to do is to
eRsure that errors are as small as reasonably possible, and have some reliable
estimate of how large they are.
ll.,Feportinq Uncertainlies
The correct way to state the result of any measurcment is to give the best
estimate of the quantity concerned and the rangc within whicF the experimenter
is confident it lies. For example, suppose we measure the period of a simple
pendulum and express the result as,
T =2.4
t 0.1 sec.
This means that the best estimate we can make is that the period is 2,4 see, but
it could be as low as 2.3 sec, or as high as 2.5 sec. We are confident that the
true period is somewhere between 2.3 and 2.S sec, and to the best of our
knowledge, it is 2.4 sec. Thus, in general, we write,
(measured value of X) = Xuesr * dX
lll. $iqnifieant Fisure-s
Several basic rules for stating uncertainties are worth emphasizing. First, since
the quantity 6X is an estimate of an uncertainty, it should not be stated with too
much precision. As a basic rule for stating uncertainties,
Experimental uncertainties should usually
bc rounded off to one significant figure.
For example, if we measure the aceeleration of gravity to be g.BZ m/secz, and if
a calculation of the uneertainty yields 0.02385 rn/secz, then the result should be
1en9$ed.ast,9 = (9.8? -0.02) m/secz" There is only one significant exception
to this rule: lf the leading digit
in the uncertainty 6X is a 1, then it may be better to
keeg two significant figures in 6X. A second basic rule for stating un-certainties is
the following:
*
The last signifieant figure in any stated answer
should usually be sf the same order of magnitude
(in the same decimal position) ae the uncertainty"
For example, the an$wer 92.81 with an uncertainty of 0.3 should be rounded off
ts 92.8 0.3 . However, numbers to be used in caleulations should generally
be kept with one more significant figure than is finally justified. This wiii reduce
the inaccuracies introduced by rounding numbers, At the end of the calculation,
the final answer should be rounded to remove this extra (and insignificant) figure.
r
lV. So,.mparison of measured aqd acccpted values
There is very little point to performing an experiment if one does not draw some
sort of a conelusion from the experiment. The vast majority of experiments lead
to quantitative eonclusions, that is, to state numerical results. lt is therefsre
important to recognize that the statement of a single number is uninteresting. An
interesting eonclusion must compare two or more numbers: a measuremenfwith
the aecepted value, a measurement with a theoretically predicted value, or
several measurements to show that they are related to one another in
aceordance with some physical law. lt is in such a comparison of numbers that
error analysis is so important. Suppose for example that we measure the speed
of ssund in air and deduce the following result:
measured speed = 329
* 5 m/s.
Sjnce the accepted speed of sound is 331 m/s, the experimenter would say that
his measurement was satisfactory, since the accepted speed lies within the
estimated range of the measured speed.
v. Fraetional Unegq.tgjntigq
The uncertainty 6X in a measurement
(measured X) = X0",, +" 6X
indieates the reliability or precision of the measurement. However, the
uncertainty dX by itself does not tell the whole story. An uneertainty of one inch
in a distance of one mile would indicate an unusually pr"ecise measurement,
whereas an uneertainty of one inch in a distanee of three inches would indicate a
rather crude estimate. Obviously the quality of a measurement is indicated not
just by the uncertainty dX, but also by the ratis of dX to the best value of X; this
leads us to consider the fraetionqlgncertaintv, defined by
fractional uncertainty =
ffi
ln most serious me€surcments, the uneertainty 6X is much smalter that the
measured best value sf X, so the fractional uncertainty therefore is usually a
small number. lt is often convenient to multiply it by 100 and quote it as a
percentage error. For example,
length L = 50
t 1 cm can be written as L = 50 cm *. Zo/o.
Note that the fractional unceftainty is a dimensionless quantity. The fraetional
unceftainty is an approximate indication of the quality of a measurement,
whatever the size of the quantity measured. Fraetional uncertainties of 10% or
so are usually characteristic of rough measurements. Fraetional uneertainties of
1o1o $ Za/o a'r€ characteristic of fairly accurate measurements, and are about the
best one can hope for in many experiments in the introductory physics
laboratory.
Vl. Fropagatiqn of Uneertaintiee
Most physical quantities cannot be measured in a single direct measurement, but
are instead found in two distinct steps. First one measures one or more
quantities X, Y, .... , that can be measured directly and from which the quantity of
interest can be calculated. $econd, using the measured values of x, y, ... , one
calculates the quantities of interest. For example to find the area of a rectangle,
one first measurcs the length I and then the width w. Then, the area is
caleulated from the formula A = lw. When a measurement involves these two
steps, the estimation of experimental uncertainty also involves two steps. One
must first estimate the uncertainties in the quantities that are measured directly,
and then find out how these uncertainties "propagate" through the calculations
directly to produce the uncertainty in the final result. To this end, the following
rules are given here, without justifieation, to be used in your error calculations for
your lab repofts.
(a) $u$s and Differenees. $uppose that x, y, and z are measured with
uncedainties dx , dy , and 6z , and the measured values are used to compute a
quantity q = (x + y - z). lf the uncertainties in x, y, and z are known to be
independent and random, then the uncertainty in q is the quadratic sum,
oq=m
(b) Frodu_clp and Quotients, Suppose that x, y, and z are measured with
uncertainties 6x , 6y , and 6z , and the measured values are used to compute a
quantity q = (xy/z). lf the uncertainties in x, y, aRd z are known to be
independent and random, then the uncertainty in q is expressed as a fractional
uncertainty given by,
ffi=ry
(c) UFeertaintv in a Power. lf x is measured with an uneertainty 6x and is
used to calculate the power g x" (where n is a fixed, known numLer), then the
:
fractional uneertainty in q is lnl times that in x:
ffi = l"lffi
(d) UFcgrtatnlyln Anv Fqnetiqn of One Variable. lf x is measured with an
uneertainty dx and is used to calculate the function q(x), then the uneertainty 6q
is given by,
dq =
l#l*
(e) Unqert?inly ln.A Funetipn of Eeveial Variables" Suppose that x, y, and
z are measured with uncertainties 6x , 6y , and dz , and the measured values are
used ts compute the function q(x, y, z). lf the unceriainties in x, y, and z are
independent and random, then the uncertainty in q is:
oo=
Excreiseq. These exercises must all be done correcfly in order to be
successfully completed. Pay particular attention to significant figures, and use
the rules given above for the propagation of errors. These exercises will be
graded by the instructor and will be your first laboratory exercise for the
semester.
1. $lppose that a student measures g, the aeeeleration of gravity, by measuring
the time t for a stone to fall from a height h fqom rest above the ground. The
formula that applies in this situation is, [ =
]St'. After making s-everal timings
the student concludes that t = (1.6 + 0.1) sec, and measures the height h to be,
6 = (46.2
0.3) ft. Calculate the acceleration of gravity in ft/sz and express the
result along with the uncertainty.
*
2. suppose that we have measured an angle d as, d = (20 * 3) degrees. Find
cos 0 along with its uncertainty. warnins: Be sure to express the angle in
radians when taking the derivative.
3. Suppose that a student wishes to determine g, the acceleration of gravity, by
using a simple pendulum. The length of the pendulum is measureo ts bc
1= (92.95 0.1) cm while the period is determined to be
T = (1.936
.004) sec. What value of g would the student report in his lab
notebook? Recall that for a simple pendulum
= Z"1f b .
*
*
,f
4. Consider the function q(x, y, z, d) = x[ y - z(sin f )].
Supposex=2.34 +0.06cm, y=7.5 *.3cm,z=9.82 * 0.05cm,and
d = 10 * 1 degree. Find q and its uncertainty.
5. Calculate the gravitational foree of attraction, using Newton's law of universal
gravitation, between two masses, ffi = (19.7 + 0.2) kg and 1y = (9.4
0.2) kg
separated by a distanee r = (0.641 + 0.009) m. Assume that the gravitationalconstant G = 6.67 x 10-11 N . mz / kgz is exact to three significant figure. Express
your result as F
(percentage). Note that Newton's law of universal gravitation
*
*
is,
lPl = G#
This completes these notes on error propagation.