Presenting Numerical Values from an Investigation

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Presenting Numerical Values from an Investigation

• Version 080104

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Outline

• Uncertainty
• Propagation of Uncertainty
• Least Squares Fit to Lines

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Online References

• www.av8n.com/physics/uncertainty.htm
• mathworld.wolfram.com/LeastSquaresFitting.html
• www.physics.csulb.edu/151lab/exp4/lsf.html

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Which is the best measurement?Which measurement
is consistent with the previously published
result?

• 6.63 E-34 Js
• 6.7 E-34 Js
• 6.623 E-34 Js

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Which is the best measurement?Which measurement
is consistent with the previously published
result?

• 6.613 0.003 E-34 Js
• 6.7 0.2 E-34 Js
• 6.6220.001 E-34 Js

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UNCERTAINTY
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Uncertainty
CJs bad karma
Best Power Supply money can buy
30 yr old piece of precision equipment
Best measurements that can be performed
Voltage stabilization circuit built by Floyd
Vibrations from Levi Center construction
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Cumulative Set of Measurementsby Lab Unit 1
of Occurances
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Cumulative Set of Measurementsby Lab Unit 2
of Occurances
should report center and width of distribution
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may look like numbers, but theyre not.Its a
way to specify a probability distribution
451
452
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Relative Uncertainty
An experiment measures the value of the
fundamental charge to be (1.84 0.09) x 10-19
Coulombs or 1.14 0.06 e
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SignificantFigures
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Sig Figs are for Losers
Non-experts tend to talk about significant
digits, but this approach is heavily flawed.
Professionals general prefer to speak in terms of
quantifying the uncertainty.
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Why Sig Fig approach is bad
An experiment measures the value of the
fundamental charge to be 1.1 e
1.149
1.050
10 rel unc. is implied
Sig Fig approach 1) misrepresents the true
uncertainty (actually the uncertainty is not
specified) 2) throws
away information 3)
can produce erroneous results (see linear fits)
Physicists present a reasonable number of
digits in published work,
but many more digits in
calculations.
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Example
Calculational tools for an investigation return
the result
1.64578359 0.05385672 meters
Q In a professional paper this should be
presented as (?)
1.6458 0.0539 meters
1.646 0.054 meters
1.65 0.05 meters
1.6 0.1 meters
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Notation

• 1.646 0.054 meters
• 1.646 (54) meters
• 1.646 3 meters

preferred for written work
preferred for conversation
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• If you have something worth saying, dont say it
  in terms of sig figs.

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Types of Uncertainty

• Statistical
• Systematic

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What is the Correct Answer?

• In classroom settings, people often get the idea
  that the goal is to report an uncertainty that
  reflects the difference between the measured
  value and the correct value.
• That idea certainly doesnt work in real life
  if you knew the correct value you wouldnt need
  to make measurements.

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PROPAGATION OF UNCERTAINTY
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?
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LEAST SQUARES FITTING
Figures and equations from mathworld.wolfram.com/
LeastSquaresFitting.html
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y a0 a1 x a2 x2
The Excel trendline function isnt good
enough. One must always present uncertainties to
be a member of the physics community.
Use linest to get the parameters and their
corresponding uncertainities.
Use trendline to draw the graph.
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Two possible ways to select the best line
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Minimize square-deviation wrt choice of
coefficients
assume linear fn
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intercept
a
slope
b
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Using the Excel function LINEST
Slope INDEX(LINEST(y-values,x-values,TRUE,TRU
E),1,1) Uncertainty in Slope
INDEX(LINEST(y-values,x-values,TRUE,TRUE),2,1) In
tercept INDEX(LINEST(y-values,x-values,TRUE,TR
UE),1,2) Uncertainty in Intercept
INDEX(LINEST(y-values,x-values,TRUE,TRUE),2,2)