Measurements & Uncertainties

Before you come to lab, make sure that you have read the “Graphing Tutorial” and the “Uncertainty &
Error Analysis Tutorial” at the beginning of the lab manual.

Introduction
Experimental measurements and theoretical formalism provide the foundation of scientific disciplines.
Experimental observations of the world around us motivate theorists to construct mathematical descriptions
that explain the observations. In turn, the validity of a theoretical model is tested by the ability to confirm
the mathematical predictions with experimental measurements. Experiment drives theory, and theory drives
experiment.

In order for this reciprocal relationship to function successfully, careful and reproducible measurements
must be made. Because even the best measurements have some degree of uncertainty associated with them,
experimental data are of little use unless they include a statement of the size of the uncertainties associated
with the measurements and of how the uncertainties were determined. This tells others how much
confidence they should have in the experimental measurements. This lab focuses on estimating and
propagating uncertainties associated with experimental measurements in the context of the introductory
physics laboratory.

Equipment
• Meter stick
• Cylindrical objects
• Electronic balance
• Ruler

Background
The accuracy of a measurement depends on many factors including the precision of the measurement tool,
the measurement method employed, and the person performing the measurement. Slight changes in how the
measurement is performed and/or who performs it can yield different results for the same measurement.
Even when the same person repeats a measurement multiple times using the same protocol, slight variations
in the measurements are likely to occur. Because there is always some degree of uncertainty in
measurements, no one measurement is precisely correct. Rather we report our best estimate for the
measurement.

In order for others to interpret the validity of our measurements, all measurements should include an
estimate of uncertainty. Say we are measuring the value of some parameter X - we would report our
measurement as

X = X best ± !X

where Xbest is our best estimate of the measured parameter, and ΔX is the uncertainty we associate with the
measurement of X. A statement of this nature indicates that we believe that any measurement of X is likely
to fall somewhere in the range of (Xbest - ΔX) and (Xbest + ΔX). If we were to repeat this measurement
multiple times, it is likely that our measurements would fall somewhere in this range.

The best estimate of a measurement, Xbest, is typically obtained by taking the average value from a set of
measurements that were repeated multiple times. For example, the data in Table 1 are from three
independent measurements of the length of a rectangle. The best estimate of the length of the rectangle is
the average value from the three measurements. When it is not possible to repeat measurements multiple
times, as will be the case in some of the labs for this course, the best estimate for the measurement is simply
the most careful measurement you can perform.

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Trial Length (cm)

1 2.20
2 2.10
3 2.15
l1 + l 2 + l 3 cm
l=
3 2.15

Table 1: Determining the best estimate for the length of a rectangle: l = 2.15 cm

Once the best estimate for a measurement is obtained, we need to estimate the uncertainty associated with
the measurement. One form of uncertainty in experimental measurements has to do with the precision of
the tool being used to perform the measurement. Consider, for example, measuring the height of a table
using a meter stick. The smallest divisions on a meter stick are 1 mm increments, so we could make a
precise measurement good to the nearest millimeter, 91.4 cm, for example.

Typically, however, we record a measurement to the precision of the measurement tool and then estimate
the next decimal place. Our measurement of the table height might then become 91.44 cm where the
hundredths place is an estimate. The meter stick permits us to measure the height of the table to the nearest
millimeter (0.1 cm). The uncertainty associated with this measurement due to the precision of the meter
stick is then less than this smallest division, on the order of 0.5 mm = 0.05 cm. We would report our
measurement of the table height as 91.44 ± 0.05 cm.

When a measurement is repeated multiple times and the errors are random, the standard deviation can be
used as a measure of the uncertainty; this approach is illustrated in Table 2. In this example, the height of
the table, h, is measured three different times. The best estimate of the table height is obtained from the
average of the three measurements

h=
( h1 + h2 + h3 )
3

The uncertainty in the measurement of the table height is obtained from the standard deviation

#
( ) + (h2 " h ) + (h3 " h ) &('
2 2 2
%$ h1 " h
!=
3" 1

We then report the height of the table to be 91.47 ± 0.08 cm.

Trial Height, h (cm)

1 91.45
2 91.40
3 91.55
Average 91.47
Standard Deviation 0.08

Table 2: Using repeated measurements to estimate the average uncertainty in the height of a table

Uncertainties estimated from repeated measurements are often better indicators of the actual uncertainty
associated with a measurement because they take into account more than just the precision of the
measurement tool.

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Most experimental parameters of interest are derived from one or more measured values, each of which
have an uncertainty associated with them. For example, the density of an object, ρ (the Greek letter “rho”),
is related to the mass, M, and volume, V, by

M
!=
V

Measuring the mass and volume of the object will permit us to calculate the density. Both M and V have
uncertainties associated with them, and these uncertainties will propagate through the calculation of the
density and influence its uncertainty. In order to determine the uncertainty of the calculated value for the
density, we need to address how uncertainties are propagated when measured parameters are combined
arithmetically or evaluated in functions. The ‘Uncertainty & Error Analysis Tutorial’ at the beginning of
the lab manual discusses this topic in detail. Read this tutorial carefully before proceeding. Appendix 1
(p. 7) summarizes some important rules for propagating uncertainties that were developed in that tutorial.

For a detailed discussion of propagation of uncertainties, see ‘An Introduction to Error Analysis: The Study
of Uncertainties in Physical Measurements’ by J.R. Taylor.

Procedure
In this lab we will investigate three different approaches for handling uncertainty in experimental
measurements. In the first set of measurements, we will consider propagation of uncertainties when the
measurement tools are the limiting factors in the precision of our measurements. In the second exercise, we
will acquire a data set that has some inherent variability due to the person performing the measurement and
the actual measurement technique. In this case, we will appeal to repeated measurements to estimate the
uncertainty in our measurements. In the third example, we will investigate how graphical analysis of a data
set can eliminate systematic errors in measurements.

Measuring the Density of a Metal Cylinder

The density of an object, ρ , can be determined from the relationship

M
!=
V

given that the mass, M, and the volume, V, are measurable quantities. The volume of a cylinder can be
computed from

!d 2 l
V=
4

where d is the diameter, and l is the length of the cylinder.

• Use the digital balance and ruler to measure the mass, diameter, and length of the cylinder. Based
on the precision of the measurement tools, include an estimate of the uncertainty for each of your
measurements.
• Compute the volume and density of the cylinder based on your measurements.
• Determine an algebraic expression for the fractional uncertainty of ρ in terms of the fractional
uncertainty of M and V.
• Determine an algebraic expression for the fractional uncertainty of V in terms of the fractional
uncertainty of d and l.
• Calculate numerical values for the absolute uncertainty of the volume and density, and then report
the values for the volume and density of the cylinder in the form of Xbest ± ΔX

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Measuring Human Reaction Time

We will now consider an experiment to estimate human reaction time. These data will exhibit more
variability from measurement to measurement than those associated with measuring the density of the metal
cylinder because many factors associated with how the experiment is performed may change slightly from
trial to trial. Because of this variability in the measurements, we will use repeated measurements to obtain
average values for the reaction time and the associated uncertainty.

In order to estimate human reaction time, we will consider how long it takes one person to catch a meter
stick that is dropped through their hands by a second person. One person should hold the meter stick
vertically at the 100 cm end. A second person, whose reaction time is to be determined, should place their
hand around, but not quite touching, the meter stick at the 40 cm mark. The person catching the meter stick
should watch the meter stick near their hand (and not the person holding the stick). As soon as the person
catching the stick sees the meter stick start to move, he/she should grab the meter stick.

We are interested in measuring the distance that the meter stick drops relative to the 40 cm starting point

d = y final ! y initial
d = y final ! 40cm

where yfinal is the final position of the person’s hand and yinitial = 40 cm is the initial position of the person’s
hand. For each person in your group, perform 10 trials and record the final position in a data table in your
notebook. Table 3 illustrates a representative data table for the reaction time experiment.

Trial yfinal (cm) d (cm)

1
2
!
10
Average
Standard Deviation

Table 3: Sample data table for human reaction time measurements

For an object falling freely from rest under the influence of gravity, the distance, d, that it travels in some
time interval, t, is described by

gt 2
d=
2

Rearranging the previous equation, we can solve for the reaction time in terms of our experimentally
measured distance

2d ! 2 $ 0.5
t= =# &d
g #" g &%

As you will see when you perform multiple measurements, your reaction time will vary slightly from trial to
trial.

• Compute the average value of d and its standard deviation (σ d) for your set of ten measurements.
• Compute the best estimate of your reaction time.
• Determine an algebraic expression for the fractional uncertainty of your reaction time in terms of
the fractional uncertainty of d.

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• Report your reaction time in the form t = tbest ± Δt.

Eliminating Systematic Error through Graphical Analysis

In the two previous examples, we addressed how to propagate uncertainties for single and repeated
measurements. If a systematic error in our measurements existed (for example, a balance was consistently
reading 0.5 g less than the actual masses of objects), both of these earlier approaches would be influenced
by this systematic error. Even if we had taken great care in performing the measurements, and had
correspondingly small uncertainties in our final results, our answers could be offset from the actual value of
the parameter, and we would have no way of knowing it. Before proceeding, carefully read the Graphing
Tutorial.

One way of avoiding systematic errors is to appeal to graphical analysis. Let us consider the following
example to illustrate how graphing a data set can eliminate systematic errors.

Hooke’s law states that the extension of a spring is proportional to the force causing the extension.

F =k! x

where F is the magnitude of the force applied to obtain the extension of the spring, x, and k is the spring
constant that describes the stiffness of the spring.

If we hang a spring vertically and attach masses to the end of it, the restoring force of the spring balances
the weight of the masses, so
mg = k ! x

In this experiment our independent variable is the mass (m) that we hang on the spring. The resulting
extension of the spring (|x|) depends on the mass and is, thus, our dependent variable. Solving for the
dependent variable, we find that

!g $
x = # &'m
"k %

If we were to plot |x| vs. m we would obtain a straight line with a slope equal to the ratio of the acceleration
of gravity to the spring constant. From a linear fit to this data set, we could determine the spring constant
from the slope of our line,

!g$ ! g $
slope = # & ' k = # &
"k% " slope %

If the hanging masses were not carefully calibrated and had a systematic error such that what we perceived
to be a mass m was really (m + merror), then the previous equation would become

!g $
x = # & ' (m + merror )
"k %
!g$ !g$
x = # & ' m + # & ' m error
"!
k% "!
k%
slope intercept

Inspection of the above equation shows that we could still estimate the spring constant from the slope of a
line fit to our data without the systematic error of the balance affecting our results. What we would observe
in our graph is that our line would have the same slope as before, but the systematic error introduces a
vertical offset into the data. This would show up in the y-intercept of the graph.

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The data in Table 4 are measurements of the period of a pendulum, T, when the pendulum length, L, is
varied. Mathematically, we expect these two parameters to be related by

L
T = 2!
g

• Manipulate the equation for the period of a simple pendulum so that it becomes linear in the
independent variable L.
• Create a plot1 of the pendulum data in Table 4 that is linear in L.
• Add a best-fit line to the plot, and determine the acceleration of gravity (g) from the fit to the data.

Length, L (m) Period, T (s)

0.29 1.09
0.38 1.24
0.66 1.65
0.80 1.79
1.19 2.19
1.64 2.56
1.72 2.66
2.50 3.03
3.38 3.68

Table 4: Period measurements for a pendulum with varying lengths

1
Refer to the ‘Graphing Tutorial’ for tips on graphing.

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Appendix 1: Rules for Propagation of Uncertainties

• ADDITION & SUBTRACTION: If several quantities x,…,w are measured with uncertainties
Δx,…,Δw, and the measured values are used to compute

q = x + … + z − (u + … + w),

then the uncertainty in the computed value of q is the quadratic sum of the original uncertainties:
!q = (!x) 2 + ...+ (!z) 2 + (!u) 2 + ...+ (!w) 2

• MULTIPLICATION & DIVISION: If several quantities x,…,w are measured with uncertainties
Δx,…,Δw, and the measured values are used to compute
x !…! z
q= ,
u !…! w

then the uncertainty in the computed value of q is the quadratic sum of the fractional uncertainties in
x,…,w:
!q " !x %2 " !z %2 " !u %2 " !w %2
= $ ' + ...+ $ ' + $ ' + ...+ $ '
q # x & # z & # u & # w &

• FUNCTIONS OF ONE VARIABLE: If the quantity x is measured with uncertainty Δx, and the
measured value is used to compute q(x), then the uncertainty in the value of q(x) is given by

dq
!q = " !x
dx

• FUNCTIONS OF MORE THAN ONE VARIABLE: If several quantities x,…,w are measured with
uncertainties Δx,…,Δw, and the measured values are used to compute q(x,…,w), then the uncertainty
in the computed value of q is the quadratic sum

$ "q '2 $ "q '2

!q = & # !x ) + ...+ & # !w )
% "x ( % "w (

where ∂q/∂i is the partial derivative of q with respect to the ith measured variable

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Concluding Questions
When responding to the questions/exercises below, your responses need to be complete and coherent. Full
credit will only be awarded for correct answers that are accompanied by an explanation and/or justification.
Include enough of the question/exercise in your response that it is clear to your teaching assistant to which
problem you are responding.

1. When calculating the density of the cylinder in the first experiment, did the fractional uncertainty
in M or V contribute the most to the fractional uncertainty of the density? Use your own data to
support your response.
2. Describe some possible sources of error that contributed to the uncertainty associated with
measuring your reaction time.
3. Using the error propagation rule for functions of a single variable, derive a general expression for
the fractional error, Δq/q, where q(x) = xn and n is an integer. Explain your answer in terms of n, x,
and Δx.
4. You have a ruler and a ream (500 sheets) of paper, and you are asked to measure the thickness of a
single piece of paper. Realizing that the paper thickness is much less than the precision of your
ruler, you decide to measure the thickness of the entire ream of paper to estimate the paper
thickness. If you measure the ream to be 5.20 ± 0.05 cm thick, what is the thickness of one piece of
paper? Write an algebraic expression for the uncertainty in the sheet thickness in terms of the
uncertainty of the ream thickness. Then determine a numeric value for the uncertainty in the sheet
thickness. Discuss why this is a good method for determining the thickness of a single sheet of
paper rather than trying to measure the thickness of a single sheet directly.