Error Analysis (Analysis of Uncertainty)

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Error Analysis(Analysis of Uncertainty)

• Almost no scientific quantities are known exactly
• there is almost always some degree of uncertainty
  in the value
• Value Uncertainty
• Values that are measured experimentally
• Values that are calculated
• from an equation
• using other values which have their own
  uncertainty
• A value might be determined both ways calculate
  it and measure it

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Uncertainty in Measured Values

• Two components of Uncertainty
• Measured value systematic errors random
  errors
• Precision of a measurement
• variations due to random fluctuations
• power supply, angle of view of a meter, etc.
• Accuracy of a measurement
• includes uncertainty in precision
• also includes systematic errors
• incorrect experimental procedure, uncalibrated
  instrument, use a ruler with only 9 mm per cm

Add four bullseyes here next time
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Treatment of Random Errors

• Assume that systematic errors have been
  eliminated
• Simple Estimate
• Analog Gauge or Scale
• How finely divided is the readout, and how much
  more finely do you estimate that you can
  interpolate between those divisions?
• Digital Readout
• What is the smallest stable digit?

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Statistical Treatment of Random Errors

• Suppose you repeated the exact same measurement
  at the exact same conditions an infinite number
  of times
• Not every measurement will be the same due to
  random errors
• Instead there will be a distribution of measured
  values
• Could use the results to construct a frequency
  distribution or probability function

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Frequency Distribution orProbability Function

• With a finite number of measurements you get a
  frequency distribution
• Probability of a measurement falling within a
  given box is number in that box divided by total
  number
• With an infinite number you get a probability
  function
• Plot of P(x) versus x
• P(x) is the probability of a measurement being
  between x and x dx

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Characteristics of the Probability Function

• Certain kinds of experiments may naturally lead
  to a certain kind of probability function
• For example, counting radioactive decay processes
  leads to a Poisson Distribution
• Often, however, it is assumed that the errors are
  by a Normal Distribution Function
• ? is the mean (average) of the infinite number of
  measurements
• ? is the standard deviation of the infinite
  number of measurements

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Use of the Probability Function

• P(x-µ) is normalized
• That is, the total area under the P curve equals
  1.0
• If you knew ? and ? (and so you knew P) you could
  find the limits between which 95 of all
  measurements lie.
• Insert plot with shaded area at left
• Noting that P is symmetric about µ you could say
  with 95 confidence that the measured value lies
  between ? - ? and ? ?
• That is, the value is ? ? at the 95 confidence
  level

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An Infinite Number of MeasurementsIsnt Practical

• You can only make a finite number of measurements
• Therefore you do not know ? or ?
• You can calculate the average and variance for
  your set of measurements

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Average and Variance of the Data SetDo Not Equal
? and ?

• Use Students t-Table to relate the two
• Pick a confidence level, 95
• Define degrees of freedom as N-1
• Read value of t
• Be careful, t-Tables can be presented in two ways
• One is such that 95 will be less than t
• In this case if you want 95 between -t and t you
  need 97.5 less that t (the curves are symmetric)
• Another is such that 95 will be between -t and t
• Uncertainty limits are then found from the
  variance
• value average of the data set ?

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One Form of Students t-Table

• The value of t from this form of the table
  corresponds to 95 of all measurements being less
  than ? ? and therefore 5 being greater than ?
  ?
• Note that if you want 95 of all measured values
  to fall between ? - ? and ? ?
• then 97.5 of all measured values must be less
  than ? ? (or 2.5 will be greater than ? ?)
• and then due to the symmetry of P 97.5 will also
  be greater than ? - ? (or another 2.5 will be
  less than ? - ?)
• so 95 will be between ? - ? and ? ?

Add shaded bell curve here
Add abbreviated t-table here
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Another Form of Students t-Table
Add shaded bell curve here

• The value of t from this form of the table
  corresponds to 95 of all measurements being
  between ? - ? and ? ?
• Therefore 5 are either
• greater than ? ?
• or less than ? - ?

Add abbreviated t-table here
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Example

• Add Problem Statement here
• preferably use data from one of the experiments
  they are doing

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Solution

• Add solution here

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Summary Uncertainty in Measured Quantities

• Measured values are not exact
• Uncertainty must be estimated
• simple method is based upon the size of the
  gauges gradations and your estimate of how much
  more you can reliably interpolate
• statistical method uses several repeated
  measurements
• calculate the average and the variance
• choose a confidence level (95 recommended)
• use t-table to find uncertainty limits
• Next lecture
• Uncertainty in calculated values
• when you use a measured value in a calculation,
  how does the uncertainty propagate through the
  calculation
• Uncertainty in values from graphs and tables