Experimental Design and Graphical Analysis of Data
A. Designing a controlled experiment
When scientists set up experiments they often attempt to determine how one variable affects
another variable. This requires the experiment to be designed in such a way that when the
experimenter changes one variable (the independent variable), the effects of this change on a
second variable (the dependent variable) can be measured.
If any other variable that could affect the dependent variable is changed, the experimenter
would have no way of knowing which variable was responsible for the results. For this reason,
scientists always attempt to conduct controlled experiments. This is done by choosing only one
variable to change in an experiment, observing its effect on a second variable, and holding all other
variables in the experiment constant.
In review, there are only two variables that are allowed to change in a well-designed experiment.
-
The variable manipulated by the experimenter is called the independent variable.
-
The dependent variable is the one that responds to or depends on the variable that was
manipulated.
-
Any other variable which might affect the value of the dependent value must be held constant.
We might call these variables controlled variables.
B. Recording Data
To determine if two variables are related to one another, data is collected in a controlled experiment.
The data is displayed in a table as shown in the example at right.
Below are some requirements for a DATA TABLE:
Motion of Toy Car –
Position vs Elapsed Time
1. The table should be well organized and self-explanatory.
Time
Position
(s)
(m)
There should be a column for each measured variable and for
necessary variables calculated from the measured ones.
2.0
0.34
2. Data table should have a title
4.0
0.70
3. The independent variable should be in the leftmost column.
6.0
1.12
10.0
1.80
15.0
2.63
20.0
3.69
4. Each column should be labeled with the name of the variable
and the units of measurement in parentheses.
5. At least six data points are necessary for a good graph. If
possible, the independent variable should span at least a 10 fold
range.
6. Data should be logically ordered (with indep variable in ascending
Controls:
Initial speed = 0
Same car used entire data set
or descending order)
7. All measurements in the table should have the appropriate number of sig figs which indicates
the precision of the measurement.
8. It is a good idea to construct the data table before collecting the data.
9. Controlled variables and their values
should be given
Motion of Toy Car – Position vs Elapsed Time
t (s)
X (m)
10. You may need to add columns of
Trial 1
Trial 2
Trial 3
Average
variables calculated from the raw
2.0
0.34
0.30
0.42
0.35
data for producing a graph (for eg.
4.0
0.70
0.68
0.76
0.71
average column in table below). Any
6.0
1.12
1.05
1.17
1.11
entry in your formal table that is the
10.0
1.80
1.72
1.92
1.81
result of a calculation must include an
15.0
20.0
2.63
3.69
2.83
3.55
2.77
3.74
2.74
3.66
explanation of the column and a sample
calculation (unless it is obvious like
taking an average)
Controls:
Initial speed = 0
Same car used entire data set
C. Graphing Data
Purpose of a graph: Determine the relationship between the two variables in the experiment. In
general your graphs in physics are of a type known as scatter graphs (do NOT connect the data
points). The graphs will be used to give you a conceptual understanding of the relation between the
variables, and will usually also be used to help you formulate mathematical model which describes that
relationship.
Elements of Good Graphs
1. A title which describes the experiment. It is conventional to title graphs with DEPENDENT
VARIABLE vs. INDEPENDENT VARIABLE. For example, if the experiment was designed to
show how changing the mass of a pendulum (independent variable) affects its period (dependent
variable), a good title would be PERIOD vs. MASS FOR A PENDULUM.
2. The graph must be properly scaled. The scale for each axis of the graph should always begin at
zero.
3. Each axis should be labeled with the variable being measured and the units of measurement
(not y and x).
4. Generally, the independent variable is plotted on the horizontal (or x) axis and the dependent
variable is plotted on the vertical (or y) axis.
5. The graph should contain at least 6 data points. Do not connect the data points.
6. Connect the data points with a line/curve of best fit. This line shows the overall tendency of
your data. Include the equation of best fit and the R2 value of the fit with the graph. Make
sure that the best fit equation is in terms of the variables on your graph rather than y and x.
D. Graphical Analysis and Mathematical Models
Interpreting the Graph
The purpose of doing an experiment in science is to try to find out how nature behaves given certain
constraints. In physics this often results in an attempt to try to find the relationship between two
variables in a controlled experiment. In this course, most of the graphs we make will represent one of
four basic relationships between the variables. These are
1. no relation - the independent variable has NO effect on the dependent variable
2. linear and direct relations
3. square relations
4. inverse and reciprocal relations
The relationship between measured variables becomes clear when one fits the graph to a trendline or
curve – the equation of the best fit is a mathematical model of the physical system examined in the
experiment. The information which follows will describe some of the basic types of relationships we
tend to see in physics.
Relationship between
Variables
Graph Shape
No Relation
As x increases, y remains the
same.
There is no relationship between
the variables.
Mathematical Model
y = a
y
(where a is a
constant)
x
Direct Proportion
y is directly proportional to x
y = ax
y
As x increases, y increases
proportionally.
(linear with 0 y-intercept)
x
Linear Relation
y varies linearly with x
As x increases, y increases
linearly.
(where a is a
constant)
y = ax + b
y
x
(where a and b are
constants)
Parabolic Relation
(Power function of order 2)
y = ax2 (+ bx + c)
y
y increases with the square of x
(where a, b, c are
constants)
x
Parabolic
(Power function of order ½)
y = bx½
y
y increases with the sq root of x
y2 is directly proportional to x.
y2 ax
(where a and b are
constants)
x
Inverse Proportion
(Power function of order -1)
y a/x = ax-1
y
y is inversely proportional to x
As x increases, y decreases
hyperbolically (not linearly).
(where a is a
constant)
x
n>0
Power Functions
A good number of graphs fall
in the general category of
power functions where the
vertical variable equals a
constant multiplied by the
horizontal variable raised to a
power, n.
y = axn
n>1
n=1
y
0<n<1
x
n<0
0>n>-1
n=-1
n<-1
(where a is a
constant)
E. Error Analysis
NO MEASUREMENT IS EXACT
There are 2 types of error in taking a measurement
RANDOM ERRORS
SYSTEMATIC ERRORS
Variations in measurements that have no
pattern – sometimes cause measurement to be
too low, sometimes too high (so get Avr ± ___)
Consistently cause measurement to be either too
large or too small
Some common general causes
- instrumental uncertainty
- parallax or perspective variations when
looking at device
- unavoidable variations in starting conditions
and measuring conditions
- reaction time
Can be reduced but never eliminated
Affect the PRECISION of measurements
A very precise measurement has low random
error
High precision
Repeated measurements are
close together around an
average value
High Accuracy
Average value of one or more
repeated measurements is
close to an accepted value
Total amount of Random errors in a
measurement is indicated by UNCERTAINTY
Some common general causes
- presence of unaccounted for physical effects (air
resistance, friction, …)
- improper equipment use (eg. failure to tare a
scale)
- imperfections in equipment (eg. Miscalibrated
devices)
Can be eliminated if found
Affect the ACCURACY of measurements
A very accurate measurement has low systematic
errors
Total amount of Systematic errors in a measurement
is indicated by ERROR
Uncertainty cannot be quantified, only
estimated
(average measurement)
±
UNCERTAINTY
There are several ways to estimate uncertainty
a)
instrumental uncertainty (1/2 division)
b)
(range of repeated measurements)/2
c)
standard deviation of repeated
measurements
±
±
±
%Unc 
Unc
x100
average value
Error  measured  accepted
% Error 
Error
x100
accepted