Data Analysis
Average
When an experiment is performed, there are always errors present.
The best experiments have obviously the less amount, and results
obtained from such experiments are regarded as solid and reliable
results. From statistical reasons, random errors occur at every
measurement, and the only method to minimize them is to repeat the
measurement. Hence, one always makes the same measurement
several times in order to minimize random errors, minimize the
uncertainty of that measurement, and increase its precision. When
several measurements of the same quantity are made from a system,
one has to ensure that all the other variables affecting this system
hare held constant. In order to express the final value of all those
measurements of the same variable, one uses the average which is
defined as:
N
X 
X
i 1
i
N
Where X is the average value, N is the amount of times the
measurement has been performed, and X i are the actual
measurement. The index i is incremented from 1 up to the value N ,
and everytime i is incremented by 1 the X i values are added; the
above formula can therefore be rewritten as:
X
X 1  X 2  ...  X N
N
For instance, if one measures the length of a table 5 times, N  5 ,
length1 = 2.1m, length2 = 2.2m, length3 = 2.0m, length4 = 1.9m,
length5 = 2.2m, so the average length is:
Length 
2.1  2.2  2.0  1.9  2.2
 2.1m
5
Uncertainty
In order to express the uncertainty of this measurement, one can
perform another calculation usually noted X , whose formula is:
 X  X 
N
X 
2
i 1
i
N ( N  1)
This quantity is also known as the standard deviation of the mean
(SDM), or standard error. Hence, X is the uncertainty and SDM and
expresses how much the average value X fluctuates. Both can be
written together as:
X  X
The bigger X is, the bigger the uncertainty on the value of X , the
less precise X is, and the more random error are present.
To calculate the SDM using the above length example, X would be:
length 
2.1  2.12  2.1  2.22  2.1  2.02  2.1  1.92  2.1  2.22
5(5  1)
 0.06
Hence, the average length can be expressed as: 2.1  0.06m .
So one is confident that the average length is within 2.04m and 2.16m .
Obviously, the more trials made, the higher N becomes, and the
lower X . So to minimize the uncertainty, the amount of random
errors or to increase precision of a measurement, simply take more
trials! ;)
When computing X , make sure to keep one more significant figure
for your X than for your X i values! In other words: do not round off
your average value when using it in your X calculation!
In order to evaluate the amount of random error, a similar calculation
as percent error can be done. Let’s define it as RE.
RE 
X
*100%
X
Again, it is obvious that the smaller X , the lower RE is, and
therefore, the less random errors are present. Please note that RE is
just a number to estimate the amount of random errors, just as PE is a
number used to estimate the amount of systematic errors. RE and PE
are NOT the errors themselves! They are just indicators. One
interpretation of random errors is the irreproducibility in making
replicate measurement of the same quantity. In other words, it is
impossible to make the exact same measurement of the same quantity
over and over.
Standard deviation
The standard deviation is very similar to the standard error, but does
not evaluate how much the mean spread; instead, the standard
deviation (SD) is a measure of how spread the data points are. It is
defined in two possible ways:
 X  X 
N
X 
2
i
i 1
N 1
Or
X 
 N 
 X 
N
2
 X    i 1 

N
i 1
N 1
2
Percent Error
The percent error is number expressed in percentage used to evaluate
the amount of systematic errors in an experiment. When an
experiment is performed and a measurement is obtained, one usually
needs to evaluate how accurate this measurement is. Hence, the
percent error is also used to estimate how accurate the result is. The
bigger the percent error is, the less accurate the measurement, and
the more systematic errors are present in the experiment.
This calculation can only be done however if the accepted value of the
result is known! If not, one has to compute the percent difference
instead.
The percent error (sometimes denoted as PE) is defined as:
Percent error 
| E  A|
*100%
A
Where E is the experimental value (the value that has been measured
and/or calculated in the experiment), A is the accepted value (what
the measured value ought to be if there was no error) and || denote
the absolute value. From the functional form of the percent error, it is
clear that the closer A is to E, the lower PE is as the difference
between A and E is taken. The absolute value is only there to allow
fluctuations of E around A (A can be bigger or lower than E). Don’t
forget to express your percent error as a percentage.
The interpretation of systematic errors is the biases in measurement,
which could be due to environmental, observational, calibration errors,
etc… Those errors are called systematic as they will appear
systematically in results with a constant effect (as long as all other
factors are held constant). It is sometimes difficult to find the source
of systematic errors, but one thing must be clear: calculation or
human errors are not considered systematic errors.
Percent Difference
The percent difference is used when the accepted value of a measured
or calculated quantity is not known, and one still would like to compare
them with each other. Of course, this assume that several methods
are used to obtained this quantity. Imagine that 2 different methods
were used to obtain the variable X. The accepted value of X is not
known, but if the two values of X are closed to each other, there is
confidence that the methods used to obtained them are reliable. In
this case, the percent difference is a small value. If the percent
difference is a large value, one or more method have problems, but
one is still unable from the percent difference to figure out which one.
Percent difference 
Line equation
The equation of a line is:
| E 2  E1 |
*100%
 E1  E 2 


 2 
Y  m* X  b
Where X is the independent variable, Y is the dependent variable, m is
the slope and b is the Y-intercept. In the below example the slope is
positive and equal to 2.5 while the y-intercept is equal to 1.3
This means that the line intercepts the Y axis at the point (0,1.3) and
is going up. If the slope m was zero, it means that the variables X and
Y are totally independent from each other; that is, one can choose any
value for X, plug it into the equation and Y will always be the same
value, and dictated by the value of b. If the line goes down, the slope
m is negative.
Line Equation
Dependent Variable
(units)
y = 2.5x + 1.3
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
Independent Variable (units)
Obtaining the slope / best fit line
If you had to draw the best fit line (or regression line) by hand, the
first thing to do is to figure out, if possible, the y-intercept. If one
point is known at least, the accuracy of your line will be increased. The
y-intercept is usually obtained by looking at the functional form of the
given equation that models the experiment.
In order to draw the bet fit line by hand, try to have roughly the same
amount of points below and above your line in such way that the sum
of the perpendicular distances from every point to the line is the same
for the points below and above the line. Of course, the best fit line can
never fit ALL the experiment points (due to errors that are always
present in experiment).
Therefore, when one computes the slope, two points must be chosen
on the best fit line. Those points do not have to be the experimental
data. It is a common mistake from students to use two of their
experiment data to calculate the slope, disregarding entirely the best
fit. The best fit line has to be found first, and based on it, the slope is
computed using two points on that line (if the points on that line
correspond to experimental data, it’s OK, but it would be a rare
situation).