Uncertainty and Its Propagation Through Calculations

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Uncertainty and Its Propagation Through
Calculations

• Engineering Experimental Design
• Valerie L. Young

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Uncertainty

• No measurement is perfect
• Our estimate of a range likely to include the
  true value is called the uncertainty (or error)
• Uncertainty in data leads to uncertainty in
  calculated results
• Uncertainty never decreases with calculations,
  only with better measurements
• Reporting uncertainty is essential
• The uncertainty is critical to decision-making
• Estimating uncertainty is your responsibility

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Todays Topics . . .

• How to report uncertainty
• The numbers
• The text
• Identifying sources of uncertainty
• Estimating uncertainty when collecting data
• Uncertainty and simple comparisons
• Propagation of error in calculations

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Reporting Uncertainty The Numbers

• Experimental data and results always shown as
• xbest ?x
• Uncertainty gets 1 significant figure
• Or 2 if its a 1, if you like
• Best estimate gets rounded consistent with
  uncertainty
• Keep extra digits temporarily when calculating

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Examples

• Right
• (6050 30) m/s
• (10.6 1.3) gal/min
• (-16 2) C
• (1.61 0.05) ? 1019 coulombs
• Wrong
• (6051.78 32.21) m/s
• (-16.597 2) C

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Fractional Uncertainty

• ?x / xbest
• Also called relative uncertainty
• ?x is absolute uncertainty
• ?x / xbest is dimensionless (no units)
• Example
• (-20 2) C ? 2 / -20 0.10
• -20 C 10

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Reporting Uncertainty The Text

• You must explain how you estimated each
  uncertainty. For example
• The reactor temperature was (35 2) C. The
  uncertainty. . .
• . . .is estimated based on the thermometer scale.
• . . .is given by the manufacturers
  specifications for the thermometer.
• . . .is the standard deviation of 10 measurements
  made over the 30 minutes of the experiment.
• . . .represents the 95 confidence limits for 10
  measurements made over the 30 minutes of the
  experiment.

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Reporting Uncertainty The Text

• You must explain how you estimated each
  uncertainty. For example
• The reactor temperature was (35 2) C. The
  uncertainty. . .
• . . .is the standard deviation of 10 measurements
  made over the 30 minutes of the experiment.
• . . .represents the 95 confidence limits for 10
  measurements made over the 30 minutes of the
  experiment.

These estimates of uncertainty include both the
precision of temperature control on the reactor
and the precision of the measurement technique.
They do not account for the accuracy of the
measurement technique.
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Precision vs. Accuracy

• Precision
• Accuracy

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Estimating Uncertainty from Scales
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Estimating Uncertainty from Scales
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Graphical Display of Data and Results
Figure 1. Cell reproduction declines
exponentially as the mass of growth inhibitor
present increases. Vertical error bars represent
standard deviation of 5 replicate measurements
for one growth plate.
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Experimental Results and Conclusions

• A single measured number is uninteresting
• An interesting conclusion compares numbers
• Measurement vs. expected value
• Measurement vs. theoretical prediction
• Measurement vs. measurement
• Do we expect exact agreement?
• No, just within experimental uncertainty

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Comparison and Uncertainty
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Comparison and Uncertainty
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Comparison and Uncertainty

• xbest ?x means . . .
• xtrue is probably between xbest - ?x and xbest
  ?x
• (later well make probably quantitative)
• Two values whose uncertainty ranges overlap are
  not significantly different
• They are consistent with one another
• A value just outside the uncertainty range may
  not be significantly different
• More on this later (hypothesis testing)

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Propagation of Uncertainties

• We often do math with measurements
• Density (m ?m) / (V ?V)
• What is the uncertainty on the density?
• Propagation of Error estimates the uncertainty
  when we combine uncertain values mathematically
• NOTE dont use error propagation if you can
  measure the uncertainty directly (as variation
  among replicate experiments)

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Simple Rules

• Addition / Subtraction, q x1 x2 x3 x4
• ??q sqrt((?x1)2(?x2)2(?x3)2(?x4)2)
• Multiplication / Division, q (x1x2)/(x3x4)
• ??q/q sqrt((?x1/x1)2(?x2/x2)2(?x3/x3)2(?x4/
  x4)2)
• 1-Variable Functions, q ln(x)
• ??q dq/dx ?x ? 1/x ?x

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General Formula for Error Propagation

• q f(x1,x2,x3,x4)
• ?q sqrt(((?q/ ?x1) ?x1)2 ((?q/ ?x2) ?x2)2
  ((?q/ ?x3) ?x3)2 ((?q/ ?x4) ?x4)2 )

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User Beware!

• Error propagation assumes that the relative
  uncertainty in each quantity is small
• Weird things can happen if it isnt, particularly
  for functions like ln
• e.g., ln(0.5 0.4) -0.7 0.8
• In this case, I suggest assuming that the
  relative error in x is equal to the relative
  error in f(x)
• Dont use error propagation if you can measure
  the uncertainty directly (as variation among
  replicate experiments)

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Sample Calculation

• You pour the following into a batch reactor
• (100 1) ml of 1.00 M NaOH in water
• (1000 1) ml of water
• (1000 1) ml of water
• What is the concentration of NaOH in the batch
  reactor?

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Sample Calculation

• (NaOH VNsOH) / (VNaOHVwaterVwater))
• The concentration of NaOH in the reactor is
  (0.0476 0.0007)M. The uncertainty was
  estimated by propagation of error, using the
  measurement uncertainties in the volumes added,
  and assuming an uncertainty of 0.01 M in the
  concentration of the 1.00 M NaOH solution. Note
  that writing 1.00 M implies an uncertainty of
  0.01 M.