Graphical presentation
Measurements should always be presented in suitable graphically coordinates, as:
1. It is easier to see from a graph if the measured quantities correlate with the theoretical
model.
2. A graph can show you if your results have some sort of systematic error
3. A graph shows you if singular datapoint is clearly flawed
Example 1
Ohm’s law states that the voltage U measured over a component is equal to the current I
running through the component times the resistance of the component
U = RI
There is a linear relation between voltage and current, voltage doubles as the current doubles.
In the case of a component the following voltages and currents were measured:
I (A)
0.1
0.2
0.3
0.4
U (V)
0.4
0.8
1.2
1.6
Graphically presented the plot looks like this:
2.0
Voltage U (V)
1.5
1.0
0.5
0.0
0.1
0.2
0.3
Current I (A)
0.4
0.5
It seems, that datapoints set on a line going through the origin. The resistance R can be
calculated for each datapoint:
I (A)
0.1
0.2
0.3
0.4
U (V)
0.4
0.8
1.2
1.6
R (Ω)
4
4
4
4
The equation U = RI tells us that in (I, U )-coordinates all plotted datapoints should set on a
line with a slope of R
2.0
Voltage U (V)
1.5
1.0
0.5
0.0
0.1
0.2
0.3
Current I (A)
0.4
0.5
What if the voltage meter was defective, so that it always showed a reading that is 2 volts too
high? Then the graph would look like this:
4
U (V)
2.4
2.8
3.2
3.6
R (Ω)
24
14
10.6
9
Voltage U (V)
3
I (A)
0.1
0.2
0.3
0.4
2
1
0.0
0.1
0.2
0.3
Current I (A)
0.4
0.5
Notice how the calculated resistances from singular (U, I)-datapoints are nonsense. Graphically
though we can see that datapoints still all set on a line with a slope of about 4 Ω
Lesson 1: If two quantities x and y have a linear dependency y = kx according to
theory, define the value of k from the slope, and not from singular datapoints.
Example 2
Graphically it is easy to see if the points set on a line. It is a lot harder to see if the points set on
a parabola or another curve. However, we can often choose coordinates where the datapoints
set on a line.
The velocity of an object that is dropped from a height h can be predicted using the energy
principle
v 2 = 2gh
Possible measurements displayed graphically look like this:
9
Max velocity v (m/s)
8
7
6
5
4
3
2
1
0.0
0.5
1.0
1.5
2.0
2.5
drop height h (m)
3.0
3.5
4.0
From this it is very hard to see if the results match the theoretical prediction. According to
theory, v 2 should be directly proportional to the height h, so the data should be plotted in h, v 2
coordinates:
Final velocity squared (m/s)^2
50
40
30
20
10
0.0
0.5
1.0
1.5
2.0
2.5
Drop height h (m)
3.0
3.5
4.0
The slope of the graph is 2g
Lesson 2: Present the correlation of two quantities in coordinates, where the relation
of the quantities are linear according to theory.
Example 3
In an experiment (I,U) - datapoints were the following:
I (A) U (V)
0.1
0.41
0.2
0.79
0.3
1.52
0.4
1.59
0.5
1.98
0.6
2.41
0.7
2.76
Presenting the datapoints graphically, it is easy to see that one point clearly differs from the
rest. An error has clearly occurred in either the measurements or logging of the results. This
point should be measured again. If measuring again is not possible, this datapoint has to be
ignored in further analysis.
2.5
Voltage U (V)
2.0
1.5
1.0
0.5
0.0
0.1
0.2
0.3
0.4
0.5
Current I (A)
0.6
0.7
0.8
References
[1] P.R. Bevington, Data reduction and error analysis for the physical sciences, McGraw-Hill,
1969