Uncertainty and Its Propagation Through Calculations

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Uncertainty and Its Propagation Through
Calculations

• Engineering Experimental Design
• ChE 408

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Uncertainty

• No measurement is perfect
• Our estimate of nearness to 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

How much more or less than xbest the true value
might reasonably be
Best estimate of the true value
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Examples
One sig fig

• 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

Two sig figs cuz the first is a 1
Scientific notation like this
xbest and ?x have the same units
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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

You cant be this certain of the uncertainty
Not rounded consistent with uncertainty
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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 estimated based on the thermometer scale.
• . . .is given by the manufacturers
  specifications for the thermometer.

These account for uncertainty due to the
measurement technique, but do not account for any
variability in the actual temperature of the
reactor during the experiment. They imply that
this variability is LESS than the measurement
uncertainty.
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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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Estimating Uncertainty from Scales
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Estimating Uncertainty from Scales
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Graphical Display of Data and Results
Error bars show uncertainty.
Axes scaled so data fills plot.
Caption for figure, NOT title.
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.
Caption interprets figure, doesnt repeat axis
labels.
Caption explains estimate of uncertainty.
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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
A and B are significantly different.
A and B are NOT significantly different.
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Comparison and Uncertainty
Consistent with the accepted value.
Significantly different from the accepted value.
May be significantly different from the accepted
value.
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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
Absolute uncertainty

• 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

Fractional uncertainty
d(ln(x))/dx 1/x
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Simple Rules

• Addition / Subtraction, q x1 x2 x3 x4
• ??q sqrt((?x1)2(?x2)2(?x3)2(?x4)2)
• Uncertainty gets bigger even when you subtract
• Multiplication / Division, q (x1x2)/(x3x4)
• ??q/q sqrt((?x1/x1)2(?x2/x2)2(?x3/x3)2(?x4/
  x4)2)
• Uncertainty gets bigger even when you divide
• 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 )

Partial derivative of q wrt x3
Absolute uncertainty in x4
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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

• Assume uncertainty on NaOH
• (NaOH VNsOH) / (VNaOHVwaterVwater))
• (0.1000 0.0014)mol / (2.100 0.002)L
• The concentration of NaOH in the reactor is
  (0.0477 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.