This document discusses error analysis in experimental measurements. It covers two types of errors - systematic errors which affect accuracy, and random errors which affect precision. Random errors follow a Gaussian distribution, and the mean and standard deviation are used to characterize these errors. Taking more measurements reduces random errors according to the central limit theorem. The document also discusses combining measurements and calculating a weighted mean to obtain the best estimate while accounting for differences in measurement precision.

1. Error Lecture pt. 2
A more complete understanding

2. Errors in the physical sciences
There are two important aspects to error analysis
1. An experiment is not complete until an analysis
of the numbers to be reported has been
conducted
2. An understanding of the dominant error is
useful when planning an experimental strategy
Chapter 1 of Measurements and their Uncertainties

3. The importance of error analysis
There are two types of error
A systematic error influences the accuracy of a
result
A random error influences the precision of a
result
A mistake is a bad measurement
‘Human error’ is not a defined term
Chapter 1 of Measurements and their Uncertainties

4. Systematic Errors
• Assumes you ‘know’ the answer – i.e. when you are
performing a comparison with accepted values or models
Pages 4 & 62 of Measurements and their Uncertainties
• Systematic errors often associated with apparatus:
• Calibration errors, incorrect assumptions, gradual heating
• Most clearly be seen graphically
• Do not vary the independent variable sequentially
No systematic error Systematic error on
Current
Systematic error on
Voltage
The intercept may be used to indicate the presence of a systematic error

5. Straight-lines
• Experiments are often designed to make use
of the gradient of a graph.
– Reduces the systematic error
– Acts to average the random errors through data
reduction
– Think about how you do the experiment

6. Finding the Gradient
All points do not count equally in heteroscedastic data sets
m=2.01±0.01
c=-0.02±0.07
m=1.99±0.01
c=0.04±0.08
m=1.97±0.03
c=0.09±0.06
m=2.04±0.04
c=-0.3±0.4
Measurements and their Uncertainties, Pg. 71

7. Rejecting Outliers
Sometimes we see data points that do not appear to follow the
trend we are expecting or trying to fit
Do not MASK the data without good reason:
Repeat the suspect point
(plot as you go)
Check experimental assumptions remain valid
Make sure you are not aliasing the data
(repeat data in the vicinity)
Check Chauvenet’s criterion for outliers:
Page 27 of Measurements and their Uncertainties
    out out1 - Erf x + x ; x, - Erf x - x ; x, N<0.5    

8. Random Errors
• Repeated measurements result in a distribution of
data indicates the presence of random error.
Chapter 2 of Measurements and their Uncertainties
Best estimate of the quantity is the mean, :
The best estimate of the precision of the
experiment is the standard deviation, :
Here:
x
Mean: =AVERAGE(A2:A6)
Standard Deviation: =STDEV(A2:A6)
Statistics on rows/columns

9. Quantifying the Width
As we take more measurements the histogram evolves
5
50
100
1000
Measurements and their Uncertainties, Pg. 10

10. Photodiode Measurement
In an experiment we assume that the shape of the curve is
a Gaussian or the Normal Distribution
Measurements and their Uncertainties, Pg. 27

11. Probability Distribution Function
• A PDF describes the probability of finding a statistical variable,
say x. It has the following key mathematical properties
Measurements and their Uncertainties, Pgs. 23-24

12. Gaussian Probability
Error Function:
Apply to find the confidence limits of a Gaussian
Chapter 2 of Measurements and their Uncertainties  NORMDIST(x,x, ,TRUE)Erf:

13. Worked Example
A box contains 100 W resistors which are
known to have a standard deviation of 2.
What is the probability of selecting a resistor with
a value of 95 W or less?
The probability of selecting a resistor with a value of 95 W
or less is simply evaluated using the error function:
What is the probability of finding a resistor in the
range 99–101 W?
The probability of selecting a resistor in a range is simply
the difference between two error functions:

14. Confidence Limits
Page 26 of Measurements and their Uncertainties       NORMDIST(x ,x, ,TRUE)-NORMDIST(x ,x, ,TRUE)

15. Range centered on Mean
Measurements within
Range
68% 95% 99.7% 99.994%
Measurements outside
Range
32%
1 in 3
5%
1 in
20
0.3%
1 in
400
0.006%
1 in
16,000
432
The error is a statement of probability. The standard deviation is used to
define a confidence level on the data.
Measurements and their Uncertainties, Pg. 26

16. NTNUJAVA Virtual Physics Laboratory
http://mw.concord.org/modeler1.3/mirror/mechanics/galton.html downloaded 7/10/2010
The Gaussian Distribution

17. -3 -2 -1 0 +1 +2 +3
68%
x  x 
Gaussian PDF:
2
2
1 ( )
( ) exp
22
x x
P x
 
  
  
 
mean
Std. dev.
The Gaussian Distribution

18. -3 -2 -1 0 +1 +2 +3
96%
x  x 
Gaussian PDF:

p(x) 
1
 2
exp
(x  x)2
22






mean
Std. dev.
The Gaussian Distribution

19. -3 -2 -1 0 +1 +2 +3
99.7%
x  x 
Gaussian PDF:

p(x) 
1
 2
exp
(x  x)2
22






mean
Std. dev.
The Gaussian Distribution

20. Poisson PDF
• Counting rare events
• All events are independent
• Average rate does not change as a
function of time
Radioactive Decay,
Photon Counting – X-ray diffraction
   
   



N
x 0
P N;N POISSON N,N,FALSE
P x;N POISSON N,N,TRUE


Mean N
StandardDeviation N
Pages 28-30 of Measurements and their Uncertainties

21. For counting statistics with a mean count of N, the
best estimate of the error is:
N N 
N N % error
1 1 100
10 3 30
100 10 10
1000 32 3
10,000 100 1
100,000 316 0.3
“A high count rate
increases the precision
of the experiment”
Standard error for counting
errors

22. Parent and Sample Distribution
10x 
Parent Sample

23. Parent and Sample Distribution
N
10
1Parent
x




24. The Standard
Error
Parent Distribution:
Mean=10, Stdev=1
b. Average of every 5 points
c. Average of every 10 points
d. Average of every 50 points
=1.0 =0.5
=0.3 =0.14
Measurements and their Uncertainties, Pgs. 14-16

25. Error in the Error
As experimental data have statistical fluctuations there is an
error in the error.
The fractional error in the error decreases very slowly with N
Quote the error to 1 sig. fig.
If the error starts with a 1, then
consider 2 sig. fig.
Measurements and their Uncertainties, Pg. 17

26. The Central Limit Theorem
• The sum of a large number of independent random variables,
each with a finite mean and variance, will tend to be normally
distributed, irrespective of the PDF of the random variable.
• The mean is a sum of a number of independent variables, so
the CLT applies to the distribution of means in a data set
– The PDF of the variable is irrelevant.
Measurements and their Uncertainties, Pgs. 31

27. Examples
PDF with mean 0.5 1000 samples of
the parent PDF
Mean of 5 samples,
repeated 1000 times
Measurements and their Uncertainties, Pgs. 31

28. National Lottery

29. CLT in words
Smallest possible mean (3.5)
results from the draw:
1, 2, 3, 4, 5, 6
Highest (46.5):
44, 45, 46, 47, 48, 49
Mean of 25:
22,23,24,25,26,27,28
1,22,25,26,27,49
1,10,25,26,39,49
3,9,14,31,48,48
Histogram maps the number of possible ways of getting that mean value
Measurements and their Uncertainties, Pgs. 31, 34

30. CLT of Gaussian
Another example is the standard error which is defined as the
standard deviation of the mean.
In this case where 5 repeats of the mean were taken the standard
deviation of the means is (5)½ that of the parent distribution.
Measurements and their Uncertainties, Pgs. 32
1 5

31. SDOM
Parent Distribution:
Mean=10, Stdev=1
b. Average of every 5 points
c. Average of every 10 points
d. Average of every 50 points
=1.0 =0.5
=0.3 =0.14
Measurements and their Uncertainties, Pgs. 14-16

32. Gaussian Error Bars
The CLT motivates our use of:
– Gaussian Error Bars
– Gaussian Confidence
Limits
1 error bars represent the 68%
confidence level
Measurements and their Uncertainties, Pg. 55

33. Testing the Validity of a fit
Range centered on Mean
Measurements within
Range
68% 95% 99.7% 99.994%
Measurements outside
Range
32%
1 in 3
5%
1 in
20
0.3%
1 in
400
0.006%
1 in
16,000
432

34. Residuals
• The residual plot (on your origin fit page) can tell you if you
have a good fit:
Residuals should have a mean of zero and have no obvious structure.
Measurements and their Uncertainties, Pgs. 63

35. Normalised
Residuals
Is this a good fit?
Residuals not conclusive
as error scales with the
value of the data point
Normalised residuals
look random. Further
analysis shows 68%
within ±1 etc.

36. Combining Results
Not consistent at all
- don’t even try to
combine them!
Consistent
- don’t combine them
- one is clearly more
precise than the other
Consistent
- Back of the net!
- combine them
RULE OF THUMB:
If the measurements are hugely different don’t combine them.
If they are within 1  then they are consistent.
Measurements and their Uncertainties, p 49

37. The Weighted Mean
ii xx jj xx 
If the error on the ‘i’th measurement is lower than that from ‘j’th,
we would give more credence to that measurement.
We want to treat each measurement with an
importance inversely proportional to its standard error:
Measurements and their Uncertainties, p 50
withMean:
Error on Mean: