Error analysis

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Error analysis Measurement
Remco Zegers walk-in hour Thursday 17-18 Room
W109 Cyclotron or by appointment
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• Make No Mistake Medical Errors Can Be Deadly
  Serious
• The American Hospital Association lists these as
  some common types of medication errors
• incomplete patient information
• unavailable drug information
• miscommunication of drug orders, which can
  involve poor handwriting, confusion between drugs
  with similar names, misuse of zeroes and decimal
  points, confusion of metric and other dosing
  units, lack of or wrong precisions in dosing
  units, and inappropriate abbreviations
• The JOURNAL of the AMERICAN MEDICAL ASSOCIATION
  (JAMA) Vol 284, No 4, July 26th 2000 7000 deaths
  due to medication errors per year in the US.

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Another example
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Uncertainty Significance
1.2411 trillion digits known!!

• ?3.1415926535 8979323846 2643383279 5028841971
  6939937510 5820974944 5923078164 0628620899
  8628034825 3421170679 8214808651 3282306647
  0938446095 5058223172 5359408128 4811174502
  8410270193 8521105559 6446229489 5493038196...

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Significance Uncertainty

• Circumference2?R23.1415926... 4.2

Calculator 26.38937 m
Right answer 26 m.
R4.2 m
4.2 means that the true value lies between 4.1
and 4.3 2?4.125.76126 m 2?4.327.017627
m so Right Answer with error 261
The number of significant figures for a result of
a division or multiplication is the least
accurate of the quantities being divided or
multiplied.
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Significance

• For addition and subtraction the number of
  decimal places should be equal to the smallest
  number of decimal places of any term in the sum
• 3.0001 0.0025 3.0026
• NUMBER OF DECIMAL PLACES IS NOT THE SAME AS THE
  NUMBER OF SIGNIFICANT FIGURES!
• 0.0025 2 significant figures 4 decimal places
• Scientific notation. For example 7107 or 7E07

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Significance Error bars

• Measurement (x) and error (dx) are given by
  xdx
• Usually 70 of the time a measurement of x will
  result in a value between (x-dx) and (xdx).
• dx is given with only one significant number
• The smallest decimal place given for x
  corresponds to the decimal place of dx
• Correct 52 5 Incorrect 52.1 5 or
  52 5.3
• Correct 9.6105 0.5105 Incorrect
  96000050000
• 9.6105 5104 Incorrect
  9.63105 5.2104
• 9.6E5 5E4

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Types of errors

• Statistical errors Due to instrumental
  imprecision or statistical nature of observed
  phenomena (finite sample).
• Systematic errors Uncertainties in the bias of
  the data
• Example measurement of weight
• Statistical error scale imprecision (every
  measurement is slightly different)
• Systematic error The scale has an offset or is
  not properly calibrated
• Example Opinion poll

Systematical error poll group is biased (e.g.
only people in one town etc) Statistical error
Sample error if a Similar group was polled the
results Would be different.
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Average, Standard deviation and standard
deviation of the mean

• If we measure a quantity X N times, the best
  estimate for the value of that quantity is the
  average
• The spread in the measurements is given by the
  standard deviation
• The error in the determination of the mean is
  the standard deviation of the mean

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Example mass on a spring
We measure by how much a spring is stretched (d)
if we hang a weight from it. The measurement is
repeated 10 times.
Measurement 2.000.02 m
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Determination of a spring constant
The same measurement is repeated with different
weights (10 measurements per weight). We want to
determine the spring constant k via the
relation Fkd weightkd
d(m)
Slope1/k
w(N)
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Fitting procedure
Fit the data with the theoretical curve
x(1/k)F function type yax Use Kaleidagraph
to fit (see details on LBS272L webpage)
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Result from fitting
Results slope 1.00230.0067 so
(1/k)1.0020.007 Chisq6.49 (?2) How can we
tell how good the fit was? Use the ?2-value
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?2-value and the goodness of fit

• We have N measurements that we want to compare
  with theory (the data points with the fitted
  curve).
• The data points are xidxi i1,N
• The theoretical values are ti(1/k)fittedFi i1,N
• The the ?2-value is

If a data point is close to the theoretical
value, relative to the size of the error bar, it
does not contribute strongly to the ?2-value. If
it is far from the theoretical value, relative to
the error bar, the ?2-value rises strongly
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?2-value and degrees of freedom
Degrees of freedom (D) Number of data points (N)
minus the number of fitted parameters (Z). In
spring example N6 Z1 (1/k) So
DN-Z6-15
Degrees of Freedom
?2-value
D5
?26.49
g.o.f.26.1
Goodness of fit probability that the data
matches the theory well (0-100 use table or
calculator on LBS272L webpage)
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Goodness of fit

• From ?2-value and D we find the goodness of fit
  (0-100)
• if g.o.f. is very low (lt5) the data does not
  match the theory well
• The theory could be wrong
• The error bars are estimated too small
• We were unlucky!
• if g.o.f. is very high (gt95) the data matches
  the theory unlikely well.
• This should only happen 1 in 20 measurements!
  Perhaps the error bars were estimated too large.
• We were lucky!
• A quick check of the goodness of the fit is the
  value ?2/D.
• ?2/D?1 for random error (I.e. not too
  small/large)
• In the example ?2/D6.49/51.3 okay!

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Error propagation addition/subtraction
We have measured 2 quantities and their errors.
What is the error in the sum/subtraction?
example A51 B82 CAB5813 dC?(1222)
2.2 so C132 DA-B5-8-3 dD?(1222)2.2 so
D-32
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Error propagation product/division
We have measured 2 quantities and their errors.
What is the error in the product/division?
example A51 B-82 CAB5(-8)-40 dC-4
0?(1/5)2(2/8)212.8 so C-41011101 DA/B
5/(-8)-0.625 dD-0.625?(1/5)2(2/8)20.2 so
D-0.60.2
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Error propagation constants/polynomials
example A51 k3 CkA3515 dC313 so
C153 n-2 DAn5-20.04 dD0.04-2(1/5)0
.016 so D0.040.02
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More info on error analysis

• visit the LBS272L webpage
• http//www.nscl.msu.edu/lbs272l/lbs272l.html
• link help on error analysis
• this lecture
• extra explanations and examples
• how to fit in kaleidagraph and get the fit
  parameters
• ?2-probability calculator

Enjoy the lab!