Transcript The Method of Maximum Likelihood as applied to a

Errors in Physical
Measurements
Error definitions
Measurement distributions
Central measures
Physics 310
Errors are uncertainties...

Every physical measurement has an
uncertainty, i.e., it has an error.
Random uncertainties (errors)
 Can be reduced but not eliminated.
 No measurement is infinitely precise.
 Determines the precision of the data.
Systematic uncertainties (errors)
 Can be reduced and eliminated
 Produces systematic shifts in the data
 Determines the accuracy of the data.
Physics 310
Errors are uncertainties...

Errors are not blunders or mistakes.
Blunders or mistakes  Must be found and corrected.
 Are not quoted in error estimates on measurements.
 “Human Error” is not a valid error.

Errors are quoted as x ± sx.
 sx needs to be estimated  From the data.
 From the measurement.
 May have asymmetric values about x
Physics 310
The parent population...
If you are measuring a physical quantity (x)
e.g., the distance a neutron penetrates into a
given material, repeating the experiment N
times produces slightly different values for
(x), i.e., (xi) :where i goes from 1 to N.
 If N goes to ∞, then this is the total of all
possible measurements of this quantity and this set is called the Parent Population.

Physics 310
The sample population...
If N is finite, this will be a sample of the
total number of all possible measurements
of this quantity (x) - and this smaller set of
measurements is called the
Sample Population.
 We will make notational distinctions
between these populations.

Physics 310
The sample population...
If N is large, the results obtained from the
sample population of measurements will
approach those of the parent population,
but- We will never know the actual values from
the parent population, even though we seek
them.
 Our goal is to find the best estimates of the
parent population values 
Physics 310
The sample population...
…and to estimate the precision and accuracy
of our estimate of (x) .
 This latter exercise is called error analysis.
 Results are often reported as
x ± sx ± ex
where x is the best representative value of
(x), sx is the estimate of the random error,
and ex is the estimate of the systematic error.

Physics 310
Deviation...
Because we cannot know (x) from the parent
population (the “true” value) we cannot
formally compare our value with a “true”
measurement of (x).
 There are few quantities whose value is
predicted exactly from theory. For example:

Physics 310
 The charge on 1 electron is 1.6 x 10-19 C.
 The speed of light is 2.998 x 108 m/s.
 The gravitational constant is 6 x 10-11 N m2/kg2
All measured!
Deviation...
Therefore, our measurement of (x) can only
be compared with the another measurement
of (x), each of which has an uncertainty
(experimental error), and neither of which is
the “true” value.
 Such a comparison results in a deviation
between the two measured quantities - or
between a measured quantity and a
theoretically expected quantity.

Physics 310
Deviation...

However, on the basis of measurement
theory, we may postulate what the expected
form of the distribution of measurements (xi)
should be expected to be.
 A plot of v vs t for a freely falling object.
 A plot of the distribution (histogram) of
measurements of neutron depth in indium.
 A plot of the angular distribution of photons
from e+ e- annihilation.
Physics 310
Deviation...
It is therefore useful to compare not only the
best estimate of (x), but also the distribution
of measured values of (x) .
 If the distributions do not appear to agree,
what does this mean?

 A problem with the experiment?
 A problem with the theory?
 Both?
Physics 310
Quantitative representations...

Given a set of N measurements, what are
quantitative ways of expressing results?
 The mean,
 The deviation,
 The variance,
 The standard deviation
m = <x>
d = (x - <x>)
s2 = <(x - <x>)2>
s = √<(x - <x>)2>
Each quantity has physical units! Don’t
forget to include them!
 Know how to compute each.

Physics 310
Computatons: the mean
_
The sample mean is defined as:
1 N
x   xi
N i
_
The parent population mean is then:
Lim 1 N 
m
 xi 

N   N i 
Physics 310
Computatons: the deviation
_
The sample deviation is defined as:
di  xi  x
_
The parent population deviation is then:
Lim 1 N 
d
 di  0

N   N i 
Physics 310
Computatons: the variance
_
The sample variance is defined as:
 1 N
2 
s  
 (xi  x ) 
N  1 i

2
_
The parent population variance is then:
Lim 1 N
2 
s 
 (xi  m ) 


N   N i
2
Physics 310
...the standard deviation
_
The sample standard deviation is defined as:
 1 N
2 
s   
 (xi  x ) 
N  1 i

_
The parent population standard deviation is
then:
Lim 1 N
2 
s
 (xi  m ) 


N   N i
Physics 310
Distributions...
Take a set of N measurements.
 Form a histogram of the measurements. (This
gives the distribution of the measurements.)

 This gives the number of measurements of
between x and x + Dx as nj, j = 1,k where k is the
total number of bins. (Dx is the fixed bin width.)
 Now, with this you can estimate m and s
because nj represents a distribution function for
the measurements xi . How do you do it?
Physics 310
Distributions...

The mean and standard deviation from a
distribution are:
k
x
 nj x j
midpoint
j
k
 nj
j
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k
s 
2
 nj (x j
midpoint
j
k
 nj
j
 x)
2
Normalized Distributions...

From a histogram of the N measurements,
you can form a normalized distribution of the
measurements.
 Take each value nj, j = 1,k and divide it by N.
This will give the fractional number fj of all
measurements in the bin j. The sum of all fj will
be 1, and hence the distribution function fj is a
normalized (discrete) distribution.
 If N is very large, his concept can be extended
to a continuous probability function P(x).
Physics 310
Normalized distributions...

The mean and standard deviation from a
normalized discrete distribution are:
k
x   fj x j
midpoint
j
j
k
 fj  1
j
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k
s   fj (x j
2
midpoint
 x)
2
Normalized distributions...

The mean and standard deviation from a
continuous distribution are:
dN(x)  N  P(x)dx

 x dN(x)
x  
 dN(x)

Physics 310
Normalized distributions...

The denominator is just:




 dN(x)   N  P(x)d(x)

 N  P(x)d(x)  N

Physics 310
Normalized distributions...

The standard deviation is then:

 x  x  dN(x)
s  
2
2

 dN(x)

Physics 310
Normalized distributions...

…or in terms of a probability function P(x):

x   x P(x)dx


s   x  x  P(x)dx
2

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Normalized distributions...

Or for any function:

g x   g(x) P(x)dx


s   g(x)  g (x) P(x)dx
2

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