The Importance of Graphs in Undergraduate
Physics
Christopher Deacon
D
Christopher Deacon
received his Ph.D. from the
University of Birmingham,
England, in 1983 and is currently laboratory coordinator
in the physics department at
Memorial University of
Newfoundland. When not
supervising classes he likes
to find new ways of doing
experiments that are easy,
fun to do, and help students
learn that practical physics
is not as difficult as they
might have first thought.
Dept. of Physics and
Physical Oceanography
Memorial University of
Newfoundland
St. John’s, NF
Canada A1B 3X7
cdeacon@physics.mun.ca
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THE PHYSICS TEACHER
rawing a graph is a natural part of
doing physics, with applications in
all areas from a high-school physics
class to the presentation of advanced theoretical or experimental research results. Modern
computer packages are very good, but are of
limited use unless the user is fully aware of
their strengths and weaknesses. The student in
a laboratory class still has to decide whether
the results of a regression analysis are physically meaningful. This requires a thorough
understanding of the experiment being performed and a full appreciation of the reasons
for plotting a graph at all.
No one underestimates the convenience of
graphing software, but the ability to plot and
meaningfully interpret a graph is an essential
skill that physics students should learn first
while using pencil and paper, and develop
through the many experiments that involve
plotting a graph of one form or another.
Why plot graphs? Primarily, we plot a
graph to obtain a picture of the data. A clear
picture reveals several things that might not
be obvious from a table of data alone and, in
addition, provides answers to the following
questions:
1. How does a change in one variable lead to
a change in the other? Plainly, one variable will tend to either increase or
decrease relative to the other, and will not
necessarily show a straight-line trend. A
graph can show that the variables change
according to some physical law. Aberrant
data in a sequence of measurements,
whether they be single points or an entire
set, will usually show up in a graph more
readily than in a table of data. Thus if we
perform an experiment to measure the
magnetic field along the axis of a coil, we
intuitively expect the field strength to
decrease as we move away from the center of the coil. If the graph of field strength
versus distance shows any other behavior,
Vol. 37, May 1999
then a mistake has been made and should
be corrected. This is a good reason for
graphing as the data are being taken. To
discover that there is no correlation
between the plotted quantities may be an
important result in a research project. At
the introductory level, it might occur if a
student plots the wrong variables or even
performs the wrong experiment.
2. Do we have sufficient data? The graph
should show enough data points over the
range considered to obtain the complete
picture. In the laboratory I never answer
questions such as, “How many data points
should I take?” because I believe that students should be able to decide for themselves when there is enough data. Clearly
two points are not enough for most undergraduate experiments, a hundred is time
consuming and unnecessary.
3. Is there a region of interest that suggests
further analysis? If the shape of the graph
changes, more data should be taken to
confirm that the data are real, such as the
region around the resonant frequency of
an LCR circuit. This follows from point 2
above, since if there are too few data
points or if the range of data considered is
too small, there is a danger that students
will fail to see a change in slope, for
example, which might be the very reason
for performing a particular experiment at
all.
A picture of the data can give a qualitative
but concise description of the experiment
without the need to perform any further calculations. If the data are uncorrelated there is
clearly no relationship between the variables
and no further analysis is necessary. For a
more quantitative description we need to
establish whether a mathematical relationship
exists between the variables. Some functional
The Importance of Graphs in Undergraduate Physics
relationships can be identified straightaway. Equations of
the kind y = kxn occur regularly in physics. The exact
value of n may not be obvious from the data, but the student should be able to identify and draw graphs of the
function for n > 1, n < 1 and n = 1. The most common
functions encountered by students include y = kx2,
y = k兹x苶, and y = 1/x, as well as the straight line.
A mathematical relation may already be established by
theory and we can use the underlying theory or the results
of a curve-fitting procedure to extract some meaningful
physics from the data. For example, the position of a body
falling freely under gravity is described by the secondorder polynomial y = ax2 + bx + c, where the parameters
a, b, and c represent acceleration (a = g/2), initial velocity (b = vo), and initial position (c = xo). The numbers
obtained from the result of a quadratic-curve fit allow us
to assign values to these parameters. Other functions can
be similarly analyzed.
Once we have an equation that fits the data, we can use
it to predict how an experiment might behave, either by
extrapolation or interpolation. Interpolating between
points is relatively straightforward; however, the process
of extrapolation is more risky because the regression
result is valid only within the range of data considered.
Furthermore, the predicted y-value may be completely
wrong if the true mathematical relation is unknown.
Regression analysis can only offer a mathematical
description of the data; it is up to the student to explain its
physical meaning. Caution should be exercised.
Of course any curve can be described by a sum of polynomials in the form y = ao + alx + a2x2 + a3x3 + . . . , and
even for somewhat erratic data it is possible to find a function that fits the original values very well indeed. HowExperiment
Formula
Simple pendulum
T = 2
Thermal expansion
of a rod
Refractive index of
glass
Focal length of a thin
lens
Rydberg constant
x-axis
冪莦莦
ᎏᎏ
g
= o(1+␣ ⌬t)
n=
sin␪
ᎏi
sin␪r
1
1
ᎏᎏ + ᎏᎏ = ᎏ1ᎏ
s
s⬘
f
冢
1
1
ᎏ1ᎏ = R ᎏᎏ
– ᎏᎏ
␭
2 2 n2
t
sin␪r
1
ᎏᎏ
s
冣
1
ᎏᎏ
n2
ever, a function with no physical basis will have no physical interpretation and is meaningless, regardless of the
numbers produced by computer output.
Whatever mathematical functions are used, the analysis should be kept as simple as possible and we should not
be fooled into making the
problem more complicated
y-axis
Slope
Intercept
than it needs to be. Thus, if
data appear to follow a
2
4␲ᎏ
T2
ᎏ
0
straight line, nothing more
g
ambitious than a straight
␣
o
line fit should be attempted.
(Conversely, a straight-line
fit should not be attempted
sin␪i
n
0
if the data do not follow an
1
1
obvious straight line.)
ᎏᎏ
–1
ᎏᎏ
s⬘
f
R
ᎏᎏ
4
ᎏ1ᎏ
␭
–R
n Vo
Discharge of capacitor
V =Vo
e–t/RC
t
n V
1
–ᎏᎏ
RC
Radioactive decay
N = Noe–␭t
t
n N
–␭
n No
␮oNI
B=ᎏ
ᎏ
(x2+R2) 3/2
log(x2+R2)
log B
3
–ᎏᎏ
2
log ␮oNI
magnetic-field
measurement
Table I. Common formulas from elementary physics, showing which quantities can be obtained from the slope
and intercept of a straight-line graph. Symbols have their usual meanings.
The Importance of Graphs in Undergraduate Physics
The Straight-Line
Graph
The theory behind most
experiments in the introductory laboratory is usually
simple enough to be
described by a straight-line
graph. Thus, experiments
with dynamics carts are set
up so that if we plot distance
versus time, the slope of the
Vol. 37, May 1999
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271
graph gives velocity, or a graph of velocity versus time
will give acceleration. Despite its apparent simplicity, the
applications of the straight-line graph are often underestimated. The equation of a straight line is well-known, y =
mx + b, but when trying to interpret the meaning of
“slope” many students tend to focus on the quantity y/x
because that is what was plotted, rather than m, which
leads to the desired result. By concentrating on the m term
we can take any algebraic expression and replot the x and
y values to give a straight line where the raw data alone
would not. As an example, consider the distance traveled
by a body in free fall as a function of time. Such an experiment allows for a determination of g. We know that theory predicts a t2 dependence, and so we plot distance versus t2 to obtain a straight line of slope ½g. Table I lists
some common experiments, their relevant formulas, the
quantities that should be plotted on the x- and y-axes and
the information that can be obtained from the slope and
intercept. Key points to note are:
•
•
•
The slope may be positive or negative.
Useful information can be obtained from the xintercept too. For example, in the thin-lens experiment, both the x- and y-intercepts allow the focal
length to be determined. Also, in an experiment to
determine Planck’s constant, the x-intercept will give
a value for threshold frequency.
Power laws or exponential relationships can be plotted
by calculating logarithms first. The use of logarithmic
scales simplifies the process.
A more complicated example from our laboratory concerns the determination of the ratio of the principle specific heats for air by Rüchardt’s method. In this experiment a metal ball is released at the top of a glass tube,
which is connected to a large glass bottle, and undergoes
damped harmonic oscillations. The period, T, of the oscillations is approximated by the expression
T2 = 4␲ 2mV/PoA2 ␥
(1)
where Po is atmospheric pressure, m is the mass of the ball,
A is the cross-sectional area of the tube, and V is the volume of air in the container.1 The volume of air is varied by
adding water to the container. If we write the volume of air
as
V = Vo – Vw
(2)
where Vo is the volume of the container and V␻ is the volume of water added, then Eq. (1) may be rewritten as
4␲2m Vo Vw
T 2 = ᎏᎏ
ᎏᎏ – ᎏᎏ
Po A 2
␥
␥
冢
冣
(3)
4␲2m
PoA ␥
4␲2m V
PoA ␥
o
equal to – ᎏᎏ
2 ᎏᎏ. Some sample data
2 and intercept ᎏᎏ
are plotted in Fig. 1. The slope gives an acceptable value
for ␥ (␥ = 1.4 for a diatomic gas) and the y-intercept
allows Vo to be determined, though we note that the volume of the container can be read directly from the xintercept without further calculation. Just because we have
a numerical value for the y-intercept doesn’t mean we
have to use it.
General Presentation and Organization
In addition to using the graph as a tool to extract information about an experiment, it is equally important to
emphasize its presentation. In the student’s laboratory
notebook, the graph is essentially an aid to calculation; in
a more formal report the graph represents the conclusion
of a project and needs to be presented in a neat, professional manner and should not be cluttered with calculations of slope and the like.
Plotting the Data
Raw data should be plotted using whatever numbers
are easiest to visualize: it is far easier to picture a 20-g
mass hanging from a spring than a 0.020-kg mass, even
though the kilogram is the correct SI unit. Thus, in a
Hooke’s law experiment, I can plot mass in grams versus
extension in centimeters and obtain the spring constant
from the slope, but need to perform one extra calculation
to put the answer into the correct units. The process is
even quicker if I plot the data without drawing an intermediate table first. Of course we should use the correct SI
units throughout in a more formal presentation, but in a
regular lab report we should try to keep everything as simple as possible, reducing the likelihood of making
mistakes.
Independent vs Dependent Variable (or is it the
other way around)?
A popular rule of thumb is to plot the independent variable along the x-axis, while the dependent variable is plotted along the y-axis. While this works well in many applications, there is potential for confusion. I prefer to say that
the x-axis contains the quantity that I control and that the
y-axis contains the quantity that varies as a result of x
varying. Thus in an Ohm’s law experiment I vary current
in a circuit and measure the voltage across a resistor. This
does not mean that all current-voltage graphs have current
plotted along the x-axis. In a Franck-Hertz experiment we
vary voltage and measure the current.2 Why is there a difference? In the first case we use the slope of the graph to
determine resistance. In the second case we are determining the excitation energy of the mercury atom—it’s a completely different experiment and there is no reason to
expect the same shape of graph.
Plotting T 2 versus Vw will yield a straight line of slope
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THE PHYSICS TEACHER
Vol. 37, May 1999
The Importance of Graphs in Undergraduate Physics
How do I decide what to plot on
each axis? The answer is it depends on
what information I want to extract
from the graph. Thus, I will normally
plot the independent variable along
the x-axis, except where the analysis
would be simplified by switching the
axes. For example, to determine
refractive index by Snell’s law, I vary
the angle of incidence, i, and measure
the angle of the refracted beam, r. I
could plot sin i along the x-axis and
sin r along the y-axis because the
angle of incidence is the quantity that
I control. This would lead to a straight
line of slope 1/n which, while not
impossible to deal with, is not as simple as plotting sin i versus sin r and
obtaining a straight line of slope n.
To Title or Not to Title?
Graphs published in professional
journals (including The Physics
Teacher) do not normally have titles.
Instead, a descriptive figure caption is
used. When graph-plotting software
provides space for a title, what should
we write in it? We don’t need to use it
at all, but if we choose to give the
graph a title, it should be clear and descriptive. “Graph of
X versus Y” is not appropriate if the axes are already
labeled X and Y. It gives the reader no more information
about the graph. A graph that investigates Stefan’s law by
looking at filament temperature as a function of electrical
power supplied could have as its title, “Verification of
Stefan’s Law.” Simply writing “Graph of power versus
temperature” does not say what the experiment is about.
Error Bars
Fig. 1. Graph plotted to obtain Cp/Cv for air by Rüchardt’s method. A least-squares
fit to the data gives = 1.42 0.01 and Vo = (10.10 0.16) liter.
The Importance of Graphs in Undergraduate Physics
No measurement has any physical meaning unless
an estimate of the uncertainty is assigned to it. We
use error bars on a graph to denote uncertainties, so
that the true value of a measured quantity lies
between some upper and lower limit as the data
points in Fig. 1 show. Error bars may be omitted only
if they are too small to be shown on the scale.
Technically, a weighted fit should be used since we
expect points with small error bars to be more reliable than data with large error bars. The regression
line is expected to pass closest to the points with the
smallest errors. The problem is that regression formulas are based on the statistics of random variables.
Each point is assigned a weight according to its standard deviation. With the exception of experiments
that involve the counting of radioactive decays, the
standard deviation is unknown. Even multiple measurements of the same point can only provide an estimate of the variance and thus, in the majority of
experiments that students encounter, the standard
formulas for weighted regression cannot be applied.
According to Myers,3 ignoring the weights may be
Vol. 37, May 1999
THE PHYSICS TEACHER
273
the most effective course of action, though Lyons4 suggests that the observed experimental error (i.e., ␦yi ) associated with each data point can be used on the grounds that
it makes the algebra of determining the best line very
much simpler. In practice, the weighted and unweighted
regression lines obtained from a set of experimental data
have similar slopes and intercepts even if the points are
widely scattered about the line.5 Miller further suggests
that this may account for the widespread neglect of
weighted regression in practice.
In the introductory physics laboratory we should plot
the points with all the error bars and draw the connecting
curve by eye. If the relationship is a straight line, the slope
and its error can be easily measured.6
Putting It All Together
A good, well-drawn graph needs good data and good
thought behind it. Irrespective of the quality of a leastsquares fit, insight is needed to translate the equation of a
straight line or curve into a meaningful description of the
behavior of a physical phenomenon. I suggest that the art
of graph plotting requires five fundamental steps.
Following this sequence will enable a student to get the
most out of any experiment. These are listed below.
1. Before plotting, ask “What am I plotting and what do
I want to find out?” The aim of the experiment should
be set out clearly enough so that this can be answered.
2. Plot the data using units that are easy to visualize.
Don’t perform unnecessary calculations before plotting. Choose axes that are long enough to accommodate the entire range of the data.
3. Find the line of best fit. If it’s a straight line, use the
slope and intercept to obtain the quantities of interest;
if it’s a polynomial, remember that each of the coefficients must have physical meaning. If the line does
not behave as predicted, ask if some unexpected physical phenomenon is manifesting itself.
4. Determine if there is a better way of plotting the data.
Does the underlying theory suggest an alternative?
Would a logarithmic or semilogarithmic scale be more
appropriate?
5. Show how the graph relates to the experiment performed. Can further predictions be made from the
graph?
When all is said and done, the graph is a tool for data
analysis. Like many tools, its effectiveness is only as good
as the proficiency of the user. A competent student should
be able to draw and interpret a graph using the most basic
computer software (or no computer at all if circumstances
require). On the other hand, top-quality software and highspeed computers are no guarantee that a useful result will
be obtained if the reasons for drawing the graph are not
understood. In summary, the science of physics concerns
the study of naturally occurring phenomena. The science
of mathematics has allowed us to describe these phenomena in terms of equations that allow us to predict the
behavior of physical systems as conditions change. A
well-drawn graph provides a bridge between the two disciplines, since it provides a convenient way of visualizing
the mathematics and the physics together. The challenge
for the instructor is to make the graph-plotting and interpretation process as meaningful as possible so that students will develop these skills well, in preparation for
more advanced courses later on.
References
1. C. G. Deacon and J. P. Whitehead, “Determination of
the ratio of the principal specific heats for air,” Am. J.
Phys. 60 (9), 859 (1992).
2. The Franck-Hertz experiment is described in most
modern physics texts; for example, A. Beiser,
Concepts of Modern Physics, 4th ed. (McGraw-Hill,
1987), pp. 153-154.
3. R. H. Myers, Classical and Modern Regression with
Applications, 2nd ed. (PWS-Kent Publishing, 1990),
p. 280.
4. L. Lyons, A Practical Guide to Data Analysis for
Physical Science Students (Cambridge University
Press, 1991), pp. 46-47.
5. J. C. Miller and J. N. Miller, Statistics for Analytical
Chemistry (Ellis Horwood Limited, 1984), pp. 109110.
6. C. E. Swartz, private communication.
et cetera...
Matter/Antimatter
“One of the bearing walls of modern physics is that particles of antimatter and those of matter are
perfect counterparts, down to their mass. That wall is standing strong, according to new results. The
international team has caged a proton and an antiproton in a trap and deduced that they have
the same mass to within one part in 10 billion.”1
1. A. Hellemans, Sci. 280, 1526 (June 5, 1998).
et cetera...
Column Editor: Albert A. Bartlett,
Department of Physics, University of Colorado,
Boulder, CO 80309-0390
274
THE PHYSICS TEACHER
Vol. 37, May 1999
The Importance of Graphs in Undergraduate Physics