xiv
______________________________________________________________________________
Physics 1020
Plotting and Interpreting Graphs
Plotting and Interpreting Graphs
Introduction
In many labs you will be plotting data in order to obtain the value of a physical quantity. It is
imperative to get a good understanding of this technique in order to obtain the best results in the
lab.
Plotting your results
To plot data points we use a software package called “Graphical Analysis”.
Let’s start by plotting some data, consisting of a set of points (x, y). The important thing here is to
notice that if you plot a graph of 𝑦 vs. 𝑥 then your quantity 𝑥 is called an independent variable and
is represented on the horizontal axis. The quantity 𝑦 is called a dependent variable and is plotted
on the vertical axis.
Consider the following data points of x and y.
x
1
2
3
4
5
Plotting 𝑦 vs. 𝑥 produces the graph below.
y
4.25
5.48
6.74
7.89
9.10
xv
______________________________________________________________________________
Physics 1020
Plotting and Interpreting Graphs
Interpreting Graphs
In physics your dependent and independent variables have a physical meaning i.e. they are
physical quantities like mass (𝑀), volume (𝑉), position (𝑥) or time (𝑡) to name a few. These
quantities are related to each other via physical laws generally expressed as equations. When
interpreting your graph you have to know that equation. Here is an example.
Consider the definition of density. It is the mass per unit volume and is expressed by the following
equation:
𝑀
𝑉
You can find the density of any material by taking a piece of it and measuring its mass and volume.
However, a more precise method of doing it involves making the same measurements but for a set
of a few objects, made out of the same material, each of different mass and volume. Rearranging
the above equation gives:
𝜌=
𝑀 = 𝜌𝑉
This shows that the mass of an object is linearly dependent on its volume or in other words if the
volume doubles so does the mass. That in turn means that the relation between 𝑀 and 𝑉 can be
described by an equation of a straight line i.e. 𝑦 = 𝑚𝑥 + 𝑏, and a plot of M (on 𝑦-axis) versus V (on
𝑥-axis) should be a straight line. Our task is to find the best line going through our data points. We
do that by performing a straight line fit (or linear regression). The results of that analysis will result
in numerical values of the parameters 𝑚 and 𝑏. They will also have a physical meaning, which you
can obtain by comparing the fit equation with the physics equation, which relates the two variables
in question (in this case 𝑀 and 𝑉)
Let’s look at an example. Consider the following measurements.
Volume (m3)
0.250
0.350
0.440
0.490
0.600
Mass (kg)
0.191
0.268
0.341
0.375
0.449
Plot of that data together with the fit parameters obtained by the software’s fitting routine are shown
below.
Comparing:
𝑀 = 𝜌𝑉
with
𝑦 = 𝑚𝑥 + 𝑏
xvi
______________________________________________________________________________
Physics 1020
Plotting and Interpreting Graphs
tells us that if we choose 𝑀 as the 𝑦 variable and 𝑉 as the 𝑥 variable then the slope of our straight
line 𝑚 will be equal to density 𝜌, and we would expect the intercept 𝑏 to be equal zero. Therefore in
this example 𝜌 = (0.74 ± 0.02)𝑘𝑔/𝑚3, where we take the standard deviation of the slope as its
absolute uncertainty.
IMPORTANT:
Whenever you are asked to find the physical meaning of the slope of your graph you will
have to write the physics equation and the fit equation side by side and identify which
physics variables correspond to which symbols in the fit equation.
Summary
For plotting and interpreting graphs follow the steps below and refer to the above example as
needed:
1. Identify which physical variable is the x (x-axis) and which is the y (y-axis).
[e.g. V, M]
2. Identify a physics equation that relates the variables. [e.g. M=V]
3. Note the relationship of your data i.e. linear, quadratic etc. [e.g. linear]
4. Write down the general form of the equation i.e. y=mx+b, y=ax2+bx+c, etc.
[e.g. y=mx+b]
5. Compare with physical equation to the equation of the fit. [e.g. Comparing
M=V to y=mx+b gives m=, b=0]