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Uncertainty Analysis
Dr. Sarah Wang
Duane Long
Department of Chemical Engineering
University of Minnesota Duluth
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Reference:
Textbook: An introduction to error
analysis: the study of uncertainties in
physical measurements by J. R.
Taylor (QA275.T38 1982)
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Presentation Outline
Objective
Why Worry About Uncertainty?
How to Calculate and Express Uncertainty
Error Presentation
• Significant Figure, Rounding
Error Origination
• Human, Fixed, Random
• Fixed Error: Propagation
• Random Error: Calibration
• Experimental Procedure
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Why Worry About Uncertainty?
No measurement is free from error
Engineers are responsible for
reporting reliability of data
Ethical consequences
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How?
1. What is a rational way of estimating
uncertainty in measurements?
2. How do you calculate propagation of
this uncertainty into the measurand?
3. How do you present the results?
Answer Three Questions
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Expressing Uncertainty
Significant Figures
Rounding
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Significant Figures
Significant Figures
6.02, 0.596, 0.000610
Multiplication & Division
Retain the significant figures of the lesser
value
Addition + Subtraction
Retain the precision of the lesser value
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Rounding
Significant figures limited by the
resolution or precision of the
measurement
When the figure ends in ‘5’ choose the
even round value (up or down)
Fixed uncertainty estimated as ½ the
precision
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Least Count vs Readability
Least Count?
10 psi
Readability?
1 to 2 psi
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Systematic (Fixed) Uncertainty
Input Precision
P(x)
x
12
xzu
10
readability For an analog readout,
unless other data is available,
uncertainty is the readability
divided by square root of 12
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Analog vs Digital Meters
Readability does not apply to
digital meters.
Unless other data is available,
precision is the least count.
Uncertainty is half of precision.
0.05 psi
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Estimating Uncertainty
1. Human Error
2. Systematic or Fixed Error
3. Random Error
Origin of Uncertainty
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Output Precision
Output = Y, Inputs are: A, B, C
e.g.: Output, Re = Duρ/µ
= (Q/t)Dρ/µ
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3 Cases of Output Precision
CBAY 222CBAY
ABCY 222
C
C
B
B
A
A
Y
Y
Y = Aα
A
A
Y
Y
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Propagation of error
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Combination Cases of Output Precision
22 ; BCACABY
22
C
C
A
A
Y
Y
B
B
C
C2
22
4
B
B
A
A
Y
Y
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Uncertainty for Complex
Equations
Previous examples are derived from
this
Work by hand or
MathCAD can help solve 16
BAfY ,
22
B
B
fA
A
fY
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Combined uncertainties
Random uncertainty with systematic
uncertainty
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2 2
i ir isu u u
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Calibration and
Experimental Uncertainty
Single Point Calibration
Multi-point Calibration
Linear Regression
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Standard Error Single-point Calibration
Sample mean
Standard Deviation
Standard error of mean
Estimate of x =
1
1 n
i
i
x xn
2/1nx
2/1
2
1
1
i i xx
n
xxy 19
i.e., measuring a single point
multiple times
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Multi-point Line Fit
cxmy ii ˆ
i i
i ii
xx
yyxxm
2
xmyc
2/122
i ii i
i ii
yyxx
yyxxR
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Linear Regression
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Multi-point Line Fit, cont.
Estimates for uncertainty
(Standard Error of regression)
for the fit of y upon x:
the slope:
and the intercept:
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Linear Regression
2/1
2ˆ
2
1
i iiyx yy
nS
2/12
i i
yx
m
xx
SS
2/1
2
2
i i
i i
yxcxxn
xSS
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Multi-point Calibration
Uncertainty in average measurement
determined from linear regression
calibration
Where n is the number of calibration
readings, k is the number of
measurements, m is the slope, 𝑦 0 is the
average of the measurements, 𝑦 is the
average of the calibration readings, and
𝑠2 is the sample variance of the x-
variable.
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𝑈𝑦 =𝑆𝑦𝑥
𝑚
1
𝑘+
1
𝑛+
𝑦 0 − 𝑦 2
𝑚2𝑠2(𝑛 − 1)
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Weighted Multi-point Line Fit
with Uncertainty
22
iiiii
iiiiiiii
xwxww
ywxwyxwwm
22
2
iiiii
iiiiiiiii
xwxww
yxwxwywxwc
222 /11 fri uuuw 23
Linear Regression when the error distribution changes with y -error distribution is weighted
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Example
Experimental results for the specific heat of air.
T/K 500 600 700
cp / (J g-1 K-1) 1.0296 1.0507 1.0743
s / (J g-1 K-1) 0.0022 0.0032 0.0046
ur / (J g-1 K-1) 0.0013 0.0018 0.0027
uf / (J g-1 K-1) 0.0001 0.0001 0.0001
u / (J g-1 K-1) 0.0013 0.0019 0.0027
w 620000 290000 140000
sn
x 2/1
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1.02
1.04
1.06
1.08
1.10
1.12
1.14
1.16
400 500 600 700 800 900 1000 1100
Cp/(J/g-K)
T/K
Specific heat of air with temperature: ∙ ♦ experimental data; — best fit with
weighting: --- best fit without weighting
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Experimental Procedure
1. Identify the data reduction equation or
sequence of calculation steps.
2. Collect (or estimate) a data point: x1, x2, x3,
... xn.
3. Estimate the uncertainties in each variable:
∆x1, ∆x2, ∆x3, … ∆xn.
4. Calculate the predicted mean value, y.
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Experimental Procedure
5. Calculate the uncertainty of the predicted
mean value according to the error
propagation equation, ∆y.
6. Identify the variables with the largest
contribution to uncertainty
7. Modify your experimental technique to
reduce the larger uncertainties.
8. Take the rest of the data and repeat these
steps if necessary.
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Present the Results
Predicted value Uncertainty (odds)
y y (95% confidence)
Significant figures
Report uncertainty with only one significant
figure .
Precision in the predicted mean value is
determined from the uncertainty
Example, compare
• 123.456 0.123 should be reported as follows
• 123.4 0.1
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Summary
How to Calculate and Express Uncertainty
Error Presentation
• Significant Figures, Rounding
Error Origination
• Human, Fixed, Random
• Fixed Error: Propagation
• Random Error: Calibration
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The End
Next week’s lecture
Technical Writing
Peer Proofreading
Each lab partner brings a hard copy of
a draft report, i.e., two for each group.
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