EML 4141L Lecture Uncertainty Analysis

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EML 4141L LectureUncertainty Analysis

• Theres no such thing as a perfect measurement!!

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Uncertainty Estimation

• When we measure some physical quantity with an
  instrument and obtain a numerical value, we want
  to know how close this value is to the true
  value. The difference between the true value and
  the measured value is the error. Unfortunately,
  the true value is in general unknown and
  unknowable. Since this is the case, the exact
  error is never known. We can only estimate
  error.

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Types of Errors

• Difference between measured result and true
  value.
• Illegitimate errors
• Blunders result from mistakes in procedure. You
  must be careful.
• Computational or calculation errors after the
  experiment.
• Bias or Systematic errors
• An error that persists and cannot be considered
  to exist entirely by chance. This type of error
  tends to stay constant from trial to trial. (e.g.
  zero offset)
• Systematic errors can be corrected through
  calibration
• Faulty equipment--Instrument always reads 3 high
  or low
• Consistent or recurring human errors-- observer
  bias
• This type of error cannot be studied
  theoretically but can be determined by comparison
  to theory or by alternate measurements.

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Types of Errors (cont.)

• Random or Precision errors
• The deviation of the measurement from the true
  value resulting from the finite precision of the
  measurement method being used.
• Instrument friction or hysteresis
• Errors from calibration drift
• Variation of procedure or interpretation of
  experimenters
• Test condition variations or environmental
  effects
• Reduce random errors by conducting more
  experiments/take more data.

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Grouping Categorizing Error Sources

• Calibration
• Laboratory certification of equipment
• Data Acquisition
• Errors in data acquisition equipment
• Data Reduction
• Errors in computers and calculators
• Errors of Method
• Personal errors/blunders

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How to combine bias and precision error?

• Rules for combining independent uncertainties for
  measurements Both uncertainties MUST be at the
  same CI
• RSS-Root-sum-square Method
• Provides 95 CI coverage
• Most commonly used/we will use this method
  throughout course
• ADD-Addition Method
• Provides 99 CI coverage
• Used in aerospace applications/more conservative

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How to Estimate Bias Error

• Manufacturers Specifications
• Assume manufacturer is giving max. error
• Accuracy - FS, reading, offset, or some
  combination (e.g., 0.1 reading0.15 counts)
• These are generally given at a 95 confidence
  interval
• Independent Calibration
• Device is calibrated to known accuracy
• Regression techniques and accuracy of standards
• Use smallest readable division
• Typically 1/2 or 1/4 smallest division
  (judgment call)
• Summing Bias Error

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General Uncertainty Analysis

• The estimate of possible error is called
  uncertainty.
• Includes both bias and precision errors.
• Need to identify all errors for the
  instrument(s).
• All measurements should be given in three parts
• Best value/average value
• Confidence limits or uncertainty interval
• Specified probability/confidence interval
  (typically 95 C.I.)
• Uncertainty can be expressed in either absolute
  terms (i.e., 5 Volts 0.5 Volts)
• or in percentage terms
• (i.e., 5 Volts 10) (relative uncertainty
  ?V/V)
• Always use a 95 confidence interval in
  throughout this course

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Propagation of Error

• Used to determine uncertainty of a quantity that
  requires measurement of several independent
  variables.
• Volume of a cylinder f(D,L)
• Volume of a block f(L,W,H)
• Density of a gas f(P,T)
• Again, all variables must have the same
  confidence interval to use this method and be in
  proper dimensions.

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RSS Method (Root Sum Squares)

• For a function y(x1,x2,...,xN), the RSS
  uncertainty is given by
• Rules
• Rule 1 - Always solve the data reduction equation
  for the experimental results before doing the
  uncertainty analysis.
• Rule 2 Always try to divide the uncertainty
  analysis expression by the experimental result to
  see if it can be simplified.
• Determine uncertainty in each independent
  variable in the form ( xN ?xN)
• Use previously established methods including
  bias and precision error.

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RSS Method (Special Function Form)

• For relationships that are pure products or
  quotients a simple shortcut can be used to
  estimate propagation of error.
• Rk X1a X2b X3c

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Example Problem Propagation of Error

• Ideal gas law
• Temperature
• T?T
• Pressure
• P?P
• RConstant

How do we estimate the error in the density?
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Apply RSS Formula to density relationship
Apply a little algebra
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Uncertainty Analysis in EES
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Uncertainty Calculation in EES
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Experimental Data Analysis References

• ASHRAE, 1996. Engineering Analysis of
  Experimental Data, ASHRAE Guideline 2-1996
• Deick, R.H., 1992. Measurement Uncertainty,
  Methods and Applications, ISA.
• Coleman, H.W. and Steele, G.W., 1989.
  Experimentation and Uncertainty Analysis for
  Engineers.

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Plotting and Data Analysis with MicroSoft Excel
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Outline

• Basic Plotting with Excel
• Regression Analysis
• Example

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Basic Plotting with Excel 97

• Plotting Experimental Data
• X-Y Plots
• RULE Data points are discreet therefore they
  should be represented by symbols. Do not connect
  symbols with lines. Functions, on the other
  hand, are continuous hence they should be
  represented by lines.

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Basic Plotting with Excel 97

• Create the basic plot.
• Format the axis and titles
• Axes should have clear labels and units
• e.g., Pressure, P (Pa)
• Adjust the scale to maximize the amount of plot
  space occupied by the data.
• Tick marks should be used
• Add Greek letters.

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Basic Plotting with Excel 97

• Format the data series
• Use open symbols before solid symbols
• Add legend if needed
• Add error bars linked to the worksheet.
• Add additional data sets.

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Plotting Common Sense

• Colors and Font
• Do not use Excel Chart Defaults
• Black points are difficult to see on a gray
  background.
• Remove unnecessary borders and headers like
  Sheet 1
• Prepare the plot in Black White only.
• Color plots look nice in presentations and
  reports, but office copiers and publishers are
  still BW only.
• To a copier red and yellow both appear gray.
• Format text for clarity
• Superscript
• Greek Symbols

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Plotting Common Sense

• Trend Line dos and donts
• Avoid using Insert Trend Line because it only
  gives, slope, intercept, and R2.
• Use Analysis Tool Pack instead.
• Use Insert Trend Line to obtain polynomial fits
  only when a curve fit for the data is required
  and one is not concerned with the underlying
  physics.
• DO NOT insert trend lines for cosmetic reasons.

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Measurements Lab Reporting Requirements

• Present the plot, clearly labeled, error bars,
  etc.
• If the plot is included directly in the body of a
  report, do not insert a title. Use figure
  captions to describe the plot.
• Present the original worksheet used to analyze
  and plot the data that we can spot mistakes and
  give partial credit. Also, neatly format and
  annotated so that we can follow your analysis.
• Sample calculations (longhand or computer
  generated) of the data and uncertainty analysis
  so that we can give partial credit.