1
Experimental Uncertainties
Measurements of any physical quantity can never be exact. One can only know its value with a
range of uncertainty. If an experimenter measures some quantity X, the measurement must be
written with an uncertainty: X ± ΔX. This expresses the experimenter’s judgment that the “true”
value of X lies somewhere between X - ΔX and X + ΔX.
I. Uncertainties of Measurements
1. Instrumental uncertainty
Every measuring instrument has an inherent uncertainty that is determined by the precision of the
instrument. This value is taken as a half of the smallest increment of the instrument scale. For
example, if the smallest mark on a ruler is every 1 mm, then 0.5 millimeter is the precision of a
ruler. What is the logic of this? Take a look at the two examples below:
A measurement of 73 mm
Where is the location of the end of the pencil on the ruler? If the smallest
unit on the ruler is 1 mm, then we would say that the end is at 73 mm. If
the end was further right than 73.5 mm, we would round up to 74 mm
since that would be the closest mark on the ruler
70 mm
80 mm
73.5 mm
A measurement of 74 mm
In this case the end of the pencil is past the middle in between 73 mm
and 74 mm. Thus when reading location of the end of the pencil on the
ruler, we would round our measurement up to 74 mm since it is the
closest mark on the ruler
70 mm
80 mm
73.5 mm
Originally authored by the Rutgers PER group. Adapted by D. Brookes and M. Kagan (2011).
2
What this means is that if somebody tells you that they measured the end of the pencil at 73 mm,
that means that the end could have been located anywhere between 72.5 mm and 73.5 mm. In
other words, their measurement (with accompanying uncertainty) was (73 ± 0.5) mm. This is
why you can always estimate instrumental uncertainty by taking the smallest unit of the
measuring instrument and dividing it by two. So for example, a stopwatch measures time in
hundredths of a second. If I measure a time of 10.04 seconds for somebody to run the 100m
sprint, then my uncertainty is ± 0.005 seconds, or more clearly written: the measured time is
(10.04 ± 0.005) seconds.
Instrumental uncertainties are the easiest ones to estimate, but unfortunately they are not the only
source of the uncertainty in your measured value. You must be a skillful and lucky
experimentalist to get rid of all other sources and to have your measurement uncertainty equal to
the instrumental one.
!
2. Random uncertainty
Sometimes when you measure the same quantity, you can get different results each time you
measure it. That happens because different uncontrollable factors affect your results randomly.
This type of uncertainty, random uncertainty, can be estimated only by repeating the same
measurement several times. For example, suppose your lab partner is dropping a ball from 2 m
above the floor. You are trying to measure the time for a ball to fall and hit the ground from the
moment your lab partner released it using a stopwatch that measures to a precision of 0.01
seconds. You repeat the measurement three times and get the following results: 0.63 seconds,
0.71 seconds, 0.55 seconds. These results are quite possible since your reaction time is about 0.1
seconds. To estimate the random uncertainty you first find the average of your measurements:
t =(0.63 s + 0.71 s + 0.55 s) / 3 = 0.63 s
You then estimate approximately how much the values are spread with respect to this average –
in this case we have a spread of about Δt = 0.08 s. That is, our time measurement was
t = (0.63 ± 0.08) s . Notice something very interesting here: the random variation in your timing
(0.08 s) is much larger than the instrumental uncertainty of the stopwatch (0.005 s). This means
that the instrumental uncertainty of the stopwatch is irrelevant to estimating the uncertainty in
your measured value! You simply forget about instrumental uncertainty altogether in a case like
this. You uncertainty estimate is governed entirely by the random uncertainty inherent in your
ability to start and stop the stopwatch precisely. Through multiple trials of the same
measurement, you can find the average value and estimate the random uncertainty by looking at
the spread of your measurements about that average.
3. The Effect of Assumptions
Assumptions inherent in your model may also contribute in uncertainty. For example, an uneven
surface may change the speed of a moving car, or energy loss during a calorimetry experiment
may cause the measured temperature to drift. Repeating the measurement will not let you get rid
of such effects. This type of uncertainty is not easy to recognize and to evaluate. First of all, you
have to determine the sign of the effect, i.e. whether the assumption increases the measured
value, decreases it, or affects it randomly. Then you have to try to estimate the size of the effect.
Originally authored by the Rutgers PER group. Adapted by D. Brookes and M. Kagan (2011).
3
It is difficult to give strict rules and
instructions on how to estimate
uncertainties in general. Each case is
unique and requires a thoughtful
approach. Be ingenious and
reasonable.
For example, you measure the
diameter of the baseball assuming it is
a perfect sphere. However, the real
size of the ball may differ by 1 or 2
mm if you measure in different
dimensions. This difference will
determine the uncertainty of your
measurement.
II. Comparing uncertainties
What we have done so far is to discuss two methods by which you could estimate the uncertainty
in a measurement. (The assumptions method is a third method, if you like.) If you (lab group A)
measure your ball drop time as t A = (0.63 ± 0.08) s , what you have a number that tells you the
uncertainty in your measurement (namely 0.08 s). We will call this the absolute uncertainty in
your time measurement. Suppose another lab group (lab group B) measured t B = (0.82 ± 0.09) s
in their experiment because they dropped their ball from 2.5 m as opposed to 2m in your
!
experiment. By looking at the absolute uncertainty for two quantities (ΔtA = 0.08 s and ΔtB =
0.09 s) you cannot immediately decide which quantity is more uncertain. This is because the
!
magnitudes of the measured quantities are different. How can we decide
which quantity has a
larger uncertainty? We need to compare their relative uncertainties. The relative uncertainty of a
measurement is the ratio of the absolute uncertainty and the quantity itself. In other words, the
relative uncertainty in tA is δtA = ΔtA/tA = 0.127. This may be expressed as a fraction, or as a
percentage by multiplying the ratio by 100% (12.7%). Likewise the relative uncertainty in tB is
δtB = ΔtB/tB = 0.11, or 11%. Thus, even though tB has a larger absolute uncertainty it is a more
precise measurement than tA because it has a smaller relative uncertainty.
To help you understand this rather strange idea, consider the two circles in the figure to the right.
Each of them has exactly the same
fuzzy edge (9 units of blur) but the
larger circle looks sharper. Why?
The diameter D of the larger circle
is about 90 units (arbitrary units)
and on the same scale the smaller
circle has a diameter of about 30units. The absolute uncertainty
ΔD is the same for each circle,
about 9 units. However, the
relative uncertainty δD = ΔD/D is
about 10% for the large circle and
about 30% for the small one. This
Originally authored by the Rutgers PER group. Adapted by D. Brookes and M. Kagan (2011).
4
should give you some sense of why relative uncertainty is a better indicator of the uncertainty of
the measured quantity than its absolute uncertainty.
Note: Common sense and good judgment must be used in representing the uncertainties 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% uncertainty in
measuring the boiling point of water (100 degrees) but a whopping 10% 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 relative uncertainty to
insignificance!). However in most calorimetry tasks, the value of interest is not temperature itself
but only the change of the temperature or the temperature difference.
III. Reducing Uncertainties
The example with the circles shows a way to reduce relative uncertainty in a measurement. The
same absolute uncertainty yields a smaller relative uncertainty if the measured value is larger.
Suppose you have a block attached to a spring and want to measure the time interval for it to
oscillate up and down, back to its starting position. If you use a watch that displays time in
seconds to measure the time interval, the absolute uncertainty of the measurement is about 0.5 s.
If you now measure a single time interval of 5 s, you get a relative uncertainty of 10%:
[(0.5 s/5 s)*100]. Suppose you measure the time interval for 5 oscillations instead and you
measure 25 s. The instrumental uncertainty is still 0.5 s! The relative uncertainty in your
measurement of the time interval is now:
time interval relative uncertainty = (0.5 s/25 s) *100% =2%
By measuring a longer time interval (five oscillations instead of one), you have reduced the
uncertainty in your time interval measurement by a factor of 5!
Of course you should not forget about the obvious way of reducing relative uncertainties by
minimizing absolute uncertainty with a better design, decreasing the effect of assumptions, or
increasing the accuracy of instrument if it is possible.
IV Finding the Uncertainty in a Final Calculated Value
1. Why do you need to know uncertainty?
How can you tell if a value that you calculated from experimental measurements agrees with a
predicted value? How can you tell if the data you have collected fits a physical model? If you
found the same quantity two different ways by two separate experiments, how can you tell if the
two measured values agree with each other? You cannot answer these questions without
considering the uncertainties of your measurements and how those uncertainties affect a final
value that you calculate using those measurements. Indeed, are the values of two quantities the
same if the difference between them is smaller than the uncertainty in their measurements?
Thus, to make judgment about two values X and Y, you have to find the ranges where these
values lie. If the ranges X ± ΔX and Y ± ΔY overlap, you can claim that the values X and Y
agree within your experimental uncertainty.
Originally authored by the Rutgers PER group. Adapted by D. Brookes and M. Kagan (2011).
5
X-!X
X
[
X+!X
Y
[ ]
]
Y-!Y
Y+!Y
If the two values X and Y do not agree to within their experimental uncertainties, we say they are
different.
X-!X
[
X
X+!X
Y-!Y
]
[
Y
Y+!Y
]
2. Measured and Calculated Quantities
Suppose you want to determine the uncertainty in the final value of a quantity that is calculated
from several measured quantities. The uncertainties in these measured quantities propagate
through the calculation to produce uncertainty in the final result. Consider the following
example. Suppose you know the average mass of one apple m with the uncertainty Δm. If you
want to calculate the mass M of the basket of 100 apples, you will get the value
M ± ΔM = 100 m ± 100 Δm.
The relative uncertainty of calculated value of M remains the same as the relative uncertainty of
the single measured value for m
ΔM / M = Δm / m.
If you have more than one measured quantity, estimating uncertainty becomes a bit more
complicated. The way we will handle it is with the weakest link rule.
3. Weakest link rule
The percent uncertainty in the calculated value of some quantity is at least as great as the greatest
percentage uncertainty of the values used to make calculation. Thus to estimate uncertainty in
you calculated value, you have to:
1. Estimate the absolute uncertainty in each measured quantity used to find the calculated
quantity.
2. Calculate the relative uncertainty in each measured quantity.
3. Pick the largest relative uncertainty. We call this largest relative uncertainty the weakest link.
4. We say that the relative uncertainty in our calculated value is equal to the weakest link (the
largest relative uncertainty in our measured values). We can then apply the relative uncertainty
of the weakest link to the calculated quantity to determine its absolute uncertainty.
Here’s an example: You’ve been asked to estimate the volume of your laptop computer. First,
you measure the length, width, and thickness with a meter stick (which has an absolute
uncertainty of 0.05cm)
Measurement
Value (with absolute uncertainty)
Length
(39.4 ± 0.05) cm
Width
(28.7 ± 0.05) cm
Thickness
(4.3 ± 0.05) cm
!
Relative uncertainty
0.05 cm
= 1.27 "10#3 = 0.127%
39.4 cm
0.05 cm
= 1.74 "10#3 = 0.174%
28.7 cm
0.05 cm
= 11.6 "10#3 = 1.16%
4.3 cm
!
Originally authored by the Rutgers PER group. Adapted by D. Brookes and M. Kagan (2011).
!
6
From this table, you can see that the thickness has by far the largest relative uncertainty - the
thickness measurement is our weakest link!
The volume of the laptop is
V = LWT = (39.4 cm)(28.7 cm)(4.3 cm) = 4862 cm3
Since the thickness measurement has the largest relative uncertainty (1.16%) we say this is the
relative uncertainty in our final calculated volume V. To determine the absolute uncertainty of
our calculated volume, we multiply the volume by the relative uncertainty of the weakest link:
!
!V = (4862 cm 3 )(1.16 "10 #2 ) = 56 cm 3
So, the final estimate for the volume of the laptop is
V = (4862 ± 56) cm3 .
3. Comparable uncertainties
If a final calculated value depends on several measured quantities that each has comparable
relative uncertainties, then the rules are more complicated: they depend on the type of the
! use for the calculation. Keep in mind, however, that the overall
mathematical relationship you
relative uncertainty cannot be less than the relative uncertainties of the independent measured
quantities. In other words, no matter what the relationship is, the relative uncertainty can only
increase when doing calculations.
Consider a simple example first: Let a calculated value C be the produce of two measured
quantities, A and B. In other words: C=AB. Now let’s rewrite A and B to include their
" !A %
uncertainties: A ± !A = A $1±
' = A(1± ! A) . Likewise, B ± !B = B(1± ! B) . Now we can
#
A &
estimate the product of A and B as AB(1± ! A)(1± ! B) = AB(1± ! A ± ! B ± ! A! B) . If we assume
that the relative uncertainties δA and δB are much smaller than 1, then we can neglect the δAδB
term and the following statement is approximately true:
C(1± !C) = AB(1± ! A ± ! B)
Thus the relative uncertainty δC in the measured quantity C is approximately:
!C = ! A + ! B .
The relative uncertainty of a product of measured values is approximately equal to the sum of the
individual relative uncertainties if those relative uncertainties are roughly equal. A useful
consequence of this is that if C=A2=A.A, then:
!C = ! A + ! A = 2! A
For instance, if you measure the sides of a square with an uncertainty of 3%, the resulting
uncertainty in its area would be 6%.
On a side-note, we can observe something very interesting if we do the same analysis in terms of
absolute uncertainties instead of relative uncertainties:
!C " C # !C = AB(! A + ! B) " B # !A + A # !B
In other words,
!(AB) = B " !A + A " !B ,
which very much resembles the product rule for derivatives!
Here is a trickier example. Suppose you measured the period of a pendulum with an uncertainty
3% and want to calculate the pendulum’s frequency. What would be its percent uncertainty?
First recall that f=1/T. Now as the period is T(1±δT), the estimate for the frequency will be
Originally authored by the Rutgers PER group. Adapted by D. Brookes and M. Kagan (2011).
7
1
1
≈ (1  δT ) = f (1  δT )
T (1 ± δT ) T
,
as you might remember from calculus. Although the sign of the percent uncertainty is reversed
on the righthand side, it is practically the same relative uncertainty! (δf = δT) Thus the frequency
will also be determined with an uncertainty of 3%. In general, the percent uncertainty of a
measured quantity equals the percent uncertainty of its calculated inverse.
IV Summary
When you are doing a lab and measuring some quantities to determine an unknown quantity:
• Decide which factors affect your result the most.
• Wherever possible, try to reduce the effects of these factors that cause uncertainty.
• Wherever possible, try to reduce uncertainties by measuring longer distances or time
intervals, etc.
• Decide the absolute uncertainty of each measurement.
• Then, find the relative uncertainty of each measurement.
• If one relative uncertainty is much larger than the others, you can ignore all other sources
and use this uncertainty to write the value of the relative uncertainty of the quantity that
you are calculating.
• If relative uncertainties of several measured quantities are comparable, the overall
relative uncertainty can be computed as follows:
o Multiplication by a constant does not affect the relative uncertainty
o When two or more measured quantities are multiplied (or divided), the relative
uncertainties should be added up. In particular, when raising a measured quantity
to a power (e.g., n=3), the percent uncertainty should be multiplied by the
corresponding factor (e.g., 3).
o When two or more measured quantities are added/subtracted, their absolute
uncertainties add up.
• Find the range where your calculated quantity lies. Make a judgment about your results
taking into the account this uncertainty.
Originally authored by the Rutgers PER group. Adapted by D. Brookes and M. Kagan (2011).
8
Exercises
1. Is the last result (δf = δT) consistent in terms of units? Explain
2. What is the relative uncertainty δD of the product of three independent measured quantities:
D=A.B.C?
3. What is the relative uncertainty δC of a ratio of two independent measured quantities C=A/B?
4. If the uncertainty of the baseball’s radius is 2%, what is the percent uncertainty in its volume?
5. What is the percent uncertainty in the length measurement 2.35 ± 0.25 m?
6. Suppose that after a hike in the mountains, a friend asks how fast you walked. You recall that
the trail was about 6 miles long and that it took between two and three hours. What is your
average speed if you assume the average time of 2.5 h?
Assume that absolute uncertainty in your distance measurement is 0.1 mile. Estimate the relative
(percentage) uncertainty in your distance measurement?
What is an absolute uncertainty in your time measurement? What is its relative uncertainty?
Compare the relative uncertainties in time and distance. Which measurements are more accurate?
Determine whether you can use the weakest link rule. Determine the relative and absolute
uncertainties in the speed estimation.
7. Suppose you want to measure time for the ball falling from a height of 1 m. You took three
measurements of the time interval and obtained 0.5 s, 0.6 s, and 0.4 seconds. What is average
time of the fall? What is an absolute value of the random uncertainty in the time measurement?
What is the relative uncertainty?
8. Suppose now that you measured the time interval for the ball to fall from a 10 m height and
got 1.5s, 1.7s, and 1.6s. Estimate the relative uncertainty assuming the absolute value is the same
as in the previous task. Compare the relative uncertainties in tasks 3 and 4. Make a conclusion.
9. You drive along highway and want to estimate your average speed. You notice a sign
indicating that it is 260 miles to Boston. In 45 min you pass another sign indicating 210 miles to
Boston. Make reasonable assumptions for the absolute uncertainties of your time and distance
measurements. Estimate relative (percentage) uncertainties. State your average speed with the
uncertainty.
10. You want to know how fast your coffee is cooling in your mug. For this you measure
temperature with a thermometer. Your first measurement is 76±1 ºC (you use the usual
thermometer with the smallest increment 1ºC). In 15 min temperature is 68±1 ºC. What is the
temperature drop (state the uncertainty range)? What is the relative uncertainty in your
measurement?
Originally authored by the Rutgers PER group. Adapted by D. Brookes and M. Kagan (2011).
9
Solutions to Exercises
1. Relative uncertainties δf and δT are both dimensionless quantities since they are expressed as
!T
!f
ratios: ! f =
and !T =
. So yes, units are indeed consistent since there aren’t any.
T
f
2. ! D = ! A + ! B + !C .
3. !C = ! A + ! B .
4
4. V = ! R 3 . Therefore δV=3δR=6%.
3
5. The percent uncertainty, also called ‘relative uncertainty,’ is
0.25 m
= 0.106 = 10.6% .
2.35 m
6 miles
= 2.4 miles per hour . The relative uncertainty in the distance
2.5 hours
!
0.1 miles
measurement is
= 0.0167 = 1.67% . The absolute uncertainty in the time measurement
6miles
is 0.5 hours since the hike might have taken as short as 2 hours or as long as 3 hours. The relative
!
0.5 hours
uncertainty in the time measurement is
= 0.2 = 20% . The distance measurement is
2.5 hours
!
more accurate
because its relative uncertainty is smaller. Because it is significantly smaller the
weakest link may be used to estimate the uncertainty in the speed. The weakest link rule says to
use the largest relative uncertainty from the data as the relative uncertainty of anything you
! relative uncertainty in the speed is 20%. The absolute uncertainty
calculate from the data. So, the
in the speed is then 2.4 miles per hour " 0.2 = 0.48 miles per hour .
6. Your average speed is
0.5 s + 0.6 s + 0.4 s
7. The average time of the fall is
0.5 s . We can estimate the absolute
3
! by looking at the range of measured values. The absolute uncertainty is 0.1 s since all
uncertainty
0.1 s
the trials lie within 0.1s of the average. The relative uncertainty is
= 0.2 = 20% .
0.5 s
!
0.1 s
8. The average is 1.6 s and the relative uncertainty is
= 0.0625 = 6.25% . Conclusion: The
1.6 s
! than the time of fall from 1m since
measurement of the time of fall from 10 m is more accurate
the relative uncertainty is lower.
!
9. This is a little trickier. Each distance measurement
could reasonably have an absolute
uncertainty of 0.5 miles. That is: (210 ± 0.5) miles and (260 ± 0.5) miles . The distance
measured is 260 miles - 210 miles = 50 miles. However, because of the uncertainty in each
measurement, the interval could be as large as 260.5 miles " 209.5 miles = 51 miles , and could
be as small as 259.5 miles " 210.5 miles = 49 miles . Thus the distance interval measured is
! uncertainty is 1 mile.
! The relative uncertainty is
(50 ± 1) miles . The absolute
1 mile
estimate of the absolute uncertainty in the time
= 0.02 = 2% . A reasonable !
50 miles
!
!
!
Originally authored by the Rutgers PER group. Adapted by D. Brookes and M. Kagan (2011).
10
0.5 min
= 0.0111 = 1.11% . The average speed
45 min
is the distance traveled (50 miles) divided by the time it took (45 min = 0.75 hours)
50 miles
v avg =
= 66.7 miles/hour . What is the uncertainty in our estimate of the average
0.75 hours
!
speed? Because the two relative uncertainties
in the distance and time measurements are
comparable to each other we should add them to find the relative uncertainty in the calculated
average speed. The relative uncertainty in the speed is 1.11%+2%=3.11%. So the absolute
uncertainty is 66.7 miles/hour x 0.0311 = 2.1 miles/hour. The average speed is then
(66.7 ± 2.1) miles/hour .
measurement is 0.5 min. Its relative uncertainty is
!
!
!
10. The temperature drop is 76°C - 68°C = 8.0°C. The process we need to use here is similar to
problem 5. Remember, our measurement is the drop: 8.0°C, but to find the uncertainty in this
measurement we need to know the uncertainty in each temperature, 76°C and 68°C. We know
that the temperature measurements are (76 ± 1)°C and (68 ± 1)°C . Thus our measurement of the
temperature drop could have been as large as 77°C " 67°C = 10°C and as small as
75°C " 69°C = 6°C . This makes the absolute uncertainty of the temperature drop is 2°C . Based
on the above analysis the temperature drop is (8 ± 2)°C .
!
!
!
!
!
Originally authored by the Rutgers PER group. Adapted by D. Brookes and M. Kagan (2011).