Uncertainty and Its Propagation Through Calculations

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Uncertainty and Its Propagation Through
Calculations

• Engineering Experimental Design
• Winter 2003

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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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In Todays Lecture . . .

• How to report uncertainty
• Estimating uncertainty when reading scales
• Uncertainty and simple comparisons
• Propagation of error

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Reporting Uncertainty

• 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
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
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
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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Estimating Uncertainty from Scales
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Estimating Uncertainty from Scales
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Graphical Display of Data and Results
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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

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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
Absolute uncertainty

• 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

Fractional uncertainty
d(ln(x))/dx 1/x
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Use the simple rules to find the uncertainty on
log-mean-delta-T

• ?Tlm ((T2-T3)-(T1-T4)) / ln((T2-T3)/(T1-T4))
• T1 98 1 C
• T2 62 1 C
• T3 15 1 C
• T4 27 1 C

T2
T1
T4
T3
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Use the simple rules to find the uncertainty on
log-mean-delta-T
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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