Error Analysis Module II
UCCS Physics Labs
Table of Contents
Random and Systematic error
Better results through averaging
Propagation of error
Percent Error/Difference
FYI
FYI
It is estimated that at any one time, 0.7% of the world's population are drunk.
Error Analysis II - 1
2
5
7
11
Random and Systematic Errors
Contrary to what you may think, not all error comes from inaccurate equipment.
Human beings still manage to impose errors into experimental readings. Whenever you
are asked to make a “guess” at the final digit in a measurement you introduce error. The
problem is that we do not always repeat the same “guess” for an identical measurement.
Sometimes we might guess to high and other times we might guess to low. We do not
know what the “correct” value is, so how can we say that we are guessing to high or to
low!? The point is, we cannot trust any single measurement to be exactly right. The way
to check our measurement is to perform many identical measurements. If all the
measurements give the same value, then we should be confident that it is the best value
for that measurement. Conversely, if the measurements vary in their value, then we can
use some math to determine the best value for that measurement.
The first thing I need to tell you is that there are two types of error associated with
any measurement. The first is called systematic error. It is an error that introduces a
constant offset to every measurement without your knowledge. I guess if we knew about
this offset we could fix it! Systematic errors are difficult to detect. The problem is that
all of your identical measurements can be exactly the same value. Therefore, you think
you have great data but all the data could be shifted by a huge amount. Really the only
way to reduce this type of error is to make sure all of your equipment is properly
calibrated with respect to some standard known value and zeroed-out under the
appropriate conditions. A good example of systematic error is a ruler that is worn out at
the end. This will make the ruler shorter than it should be and result in length
measurements that are too long! If you suspect that your ruler is giving some offset then
you can check it against another ruler, replace it, or do not use the first unit of the ruler.
You can begin your measurement at the second unit. Be careful, and remember to
subtract off the extra unit you added to the measurement by not starting at the zero mark.
Error Analysis II - 2
Starting
measurement
at 1 cm.
Remember to
subtract 1 cm
form the final
reading. The
ruler reads:
4.7 cm.
True length:
3.7 cm
worn
edge
There are several more examples of systematic error:
•
Voltmeter (or any meter for that matter) not zeroed, so it will always read too
high or too low.
•
Stopwatch with a bad crystal oscillator causing the time to tick by at a
different rate.
•
Can you think of any more?
The best way to detect systematic error is through the use of a graph. For example, two
lines are plotted. One represents the experimental data and the other is the theoretical
value. See the graph below:
data
The data has been shifted
up by some offset value.
theoretical line
Error Analysis II - 3
Before you jump to any conclusions and begin labeling all of your data as containing
systematic error, make sure all of the data points are shifted by the same amount.
data
theory
theory
data
No systematic error! Not all the
data is shifted by the same amount.
Possibility of some systematic error.
The lines are not perfectly parallel
but are close. Some systematic error
might be expected as well as some
other source of error, which is
causing the lines to diverge.
If you remember, I said that there are two types of error. Systematic was the first
type of error. The other type of error comes from the unpredictability in taking
measurements. Remember the human factor in predicting the last digit of a
measurement. This certainly is a source of error that cannot be predicted. This type of
error is called random error and is just what its name implies. It is a random fluctuation
in your data.
Examples of random error:
•
Starting and stopping a stopwatch
•
Human hand-eye coordination is not always exact or reproducible
•
Any measurement where the last digit has to be estimated by eye.
•
Can you think of other examples?
So if we make a graph of a number of identical measurements, ideally they should all
perfectly overlap and we should get a very boring graph with a single dot.
Error Analysis II - 4
Ideal case:
NO random error, all
the measurements
perfectly overlap
Real-world case:
Random error, all the measurements
are scattered around. The greater the
spread in data points the larger the
random error.
• •
•
• ••••• •
•
Getting better results
Since we know nothing about how error was introduced into the measurement, it
would be unwise to take a single measurement as the absolute correct result. To solve
this problem, we should repeat several of the same measurements. This will allow us to
see how much the measurement is fluctuating due to random error. For one
measurement, we could have seriously overestimated the time (example, slow hand-eye
coordination). On the second measurement we might stop timing too early. As we take
more and more data we begin to calm down. We get better at using the stopwatch.
However, there is still the possibility that we are still under- or overestimating the time.
Now that we have repeated the measurements several times which one do we use?
From the previous example you might think that the later data points would be better
because we were calmer, but how can we be sure? The only real option is to take an
average of all the data. If the error in the data is random, then there is no reason to
believe that it should have more error in one direction than the other, but should fluctuate
equally in both directions. Therefore, taking an average should balance out the
fluctuations in both directions and give us a better value for the measurement.
To take an average:
•
Add up all the data values
•
Divide by the number of data points
Error Analysis II - 5
Let’s do an example:
Time it takes for a ball to fall 100 feet
2.34 sec, 2.45 sec, 2.27 sec, 2.3 sec, 2.42 sec
average =
2.34 + 2.45 + 2.27 + 2.3 + 2.42
sum of data
=
= 2.36 sec
number of data points
5
This value (2.36 sec) is our best guess at the true value. We should use this value in any
calculations that may follow. In the world of error analysis, the average value of a set of
data is usually referred to as the mean of the data.
average = mean = x
Scientists, like any other group, like to define and redefine words to make
themselves feel important and to make sure no one else can understand them!
In the last example we only had to deal with five data points. There are cases when you
might have thousands of values. I do not think you would like to write out the numerator
of the average calculation with thousands of values. Mathematicians came up with a
shorthand symbol (Σ) for the summation of a series of data.
i
1
2
3
4
5
xi
2.34
2.45
2.27
2.3
2.42
5
∑x
i
= 2.34 + 2.45 + 2.27 + 2.3 + 2.42 = 11.78
i =1
i ⇒ This is the data order number the first data value is 2.34 and
has a i value of 1, the second value is 2.45 with i = 2, and so
on
xi ⇒ This is the value of the ith data point. If i equals 3, then
xi=3 = x3 = 2.27.
Error Analysis II - 6
Σ ⇒ This is the symbol for a summation of a series of data. The
bottom value is the starting point for i. The top value is the
stopping value of i. For this example we are going to sum
the entire set of data from i =1 to i =5.
We can therefore rewrite the averaging equation as:
N
∑x
x =
i =1
N
i
, where N is the total number of data points
Propagation of uncertainty
Very few experimental results will be from direct measurements. Most results are
attained from a calculation using whatever measurements you are able to gather. Density
is an example of one of these results. Density cannot be measured directly, however it is
the ratio of two measurable quantities: the mass of an object and the object’s volume.
Example:
If you were asked to find the density of a block of cheese, you
would first need to know its volume and mass. To find its volume, you need to measure
the length, width, and height of the block of cheese. All of these measurements will
contain some error. Then you need to measure the mass of the cheese, which will also
contain error. All of this information will finally go into the equation for density,
ρ =
mass
volume
. What is the uncertainty of the density? In performing this mathematical step
we appear to have lost all the information about the error in the final result. Therefore,
we need a method(s) to keep track of uncertainties as we perform calculations.
Propagation of error (Multiplying by a constant)
First, let’s examine the simplest form of error propagation through a calculation,
the uncertainty in the result of a measurement multiplied by a constant.
If y = Ax, where A is a constant and x is our measurement, then the uncertainty of
the result (y) is represented by the symbol δy. The uncertainty in the result (δy) is
calculated with the use of the following equation:
Error Analysis II - 7
δy = A δx
Example the circumference of a circle:
C = 2πr
Where the radius (r) is measured as 23.5 ± 0.2 cm and 2 and π are constants.
What is the circumference?
C = |2π| (23.5) = 147.65 cm
What is the error of the circumference?
δC = 2π(0.2) = 1.25 cm
Therefore, the final answer is written as:
C = 148 ± 1 cm (using proper significant figures)
Propagation of error (Addition and Subtraction)
The rule in a calculation involving addition and/or subtraction:
All the errors add together to give the final error.
If z = x + y, where x and y contain an uncertainty (δx and δy).
Then the uncertainty in z (δz) is given by:
δz = δx + δy
Example:
Find how much chili and cheese we will have for dinner. Will it
be enough?
Mass of pot
= 72 ± 1 grams
Mass of cheese and pot
= 540 ± 10 grams
Mass of chilly added to pot
= 940 ± 20 grams
Mass of chili and cheese = (Mass of chili) + (Mass of cheese and pot) - (Mass of pot)
Mass of chili and cheese = 940 g + 540 g - 72 g = 1408 g
Value of error = 1 g + 10 g + 20 g = 31 g
Final result = 1408 ± 31 g
Error Analysis II - 8
You have plenty of food; you should invite me over for dinner!
Propagation of error (Multiplication and division)
Both multiplication and division are treated the same and follow this rule:
If z = xy or
z = xy
then the uncertainty in z is given by:
δz
z
=
δx
x
+
δy
y
Solving for the error in z:
⎛ δx
δy ⎞⎟
δz = z ⎜
+
y⎠
⎝x
What if you combine multiplication and division in the same calculation? No problem!
A=
BC
D
then δA is:
⎡ δB
δC
δD ⎤
and so on, if more variables were present.
δA = A ⎢
+
+
C
D ⎦⎥
⎣B
Lets try some examples of what I’m talking about:
Length
Density of cheese
Length = 10 ± 2 cm
Width
Height = 5 ± 1 cm
Mass = 700 ± 1 g
Height
Width = 20 ± 5 cm
The value for the density is:
ρ =
mass
700
M
=
=
= 0.7 g 3
cm
volume
10 × 20 × 5
L ×W× H
Error Analysis II - 9
The value of the error
⎡ δM
δρ = ρ ⎢
δL
δW
δH ⎤
⎥
H⎦
5 1⎤
2
⎡ 1
= 0.7 ⎢
+ +
+ = 0.455
⎣ 700 10 20 5 ⎥⎦
⎣M
+
L
+
W
+
g
cm
3
Therefore, the final result for the density of the cheese: 0.7 ± 0.5
g
3
cm
You may notice in our example the error was almost as
large as the result. This can happen. To reduce the error
we need to improve the accuracy of our measurements!
This was a brief introduction to the propagation of error. If you do not understand
where these equations came from, don’t worry about it. In the upper division labs a more
in-depth explanation will be given concerning error and its propagation through
equations.
You will be required to use these equations and methods of
error analysis for the first few labs.
Even though the method just described is the correct method of
error analysis it will become very time consuming and counter
productive in subsequent labs. Other alternatives will be
presented as needed throughout the semester. You have only
seen the tip of the iceberg and hopefully this will help when the
time comes to explore the topic fully.
It may seem like a huge waste to spend so much time and energy on
determining the uncertainty. In fact experimental scientists spend as
much time if not more on calculating their uncertainty as they do on
actually getting the measurement. This is because the uncertainty is
representation of the quality of the measurement. It tells the researcher
if it is necessary to further improve the experimental procedure and
gives him/her clues on where the problems are arising.
Error Analysis II - 10
Percent Error/ Percent Difference
There will be instances when proper methods of determining your uncertainty
(propagation of error, standard deviation, etc.) is too time consuming (I will let you when
that is case). We will still need a method of determining how much our experimental
value differs from an accepted value.
In the case of comparing our results to a standard, known value (like g = 9.81
m/s2) we will use the percent error. Multiplying by 100 at the end converts the error into
a percentage.
percent error (%) =
x exp - x th
x th
× 100
where, xexp is the experimental result and xth is the theoretical value.
Therefore, using the example of g:
Through a brilliantly designed and carefully executed experiment you arrived at a
value of 9.95 m/s2 for your value of g. Therefore: xexp = 9.95 and xth = 9.81
9.95 - 9.81
× 100 = 1.4 % error
9.81
percent error (%) =
What happens when we do not have a fundamental constant for comparison? What if we
want to compare two values both arrived at the same results by different means.
Example, Moment of inertial of a disk measured in two different ways.
x1 – moment of inertial measured using the linear acceleration of a falling mass.
x2 – moment of inertial calculated using the geometric dimensions of the disk.
Both methods yielded a value for the moment of inertia, but there is no reason to suspect
that either one is more accurate than the other. This is a perfect for a percent difference
calculation.
percent difference (%) =
x1 - x 2
1
(x 1 +
2
x2 )
× 100
lets say that our two measurements were: x1 = 0.006719 kg⋅m2 x2 = 0.006788 kg⋅m2
percent difference (%) =
0.006719 - 0.006788
1
(0.006719 +
2
0.006788 )
Error Analysis II - 11
× 100 = 1%