# EML 4141L Lecture Uncertainty Analysis

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```					      EML 4141L Lecture
Uncertainty Analysis

There’s no such thing as a perfect
measurement!!
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.
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.
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.
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
How to combine bias and precision error?
Rules for combining independent uncertainties for
measurements: Both uncertainties MUST be at the
same CI
Provides 95% CI coverage
Most commonly used/we will use this method throughout
course

U x  Bx  Px2 or U x  Bx  Px2
2                 2

Provides 99 % CI coverage
Used in aerospace applications/more conservative

U x, ADD  Bx  Px or U x , ADD  Bx  Px
How to Estimate Bias Error
Manufacturers Specifications
Assume manufacturer is giving max. error
Accuracy - %FS, %reading, offset, or some combination (e.g.,
These are generally given at a 95% confidence interval
Independent Calibration
Device is calibrated to known accuracy
Regression techniques and accuracy of standards
Typically ± 1/2 or ± 1/4 smallest division (judgment call)

Summing Bias Error

Btotal  ( Bi )
2 12
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
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.
For a function y(x1,x2,...,xN), the RSS uncertainty is given by:

 R       2
  R        
2
 R       
2

 x U x1    x U x2   ...  x U xN  
U R                                        
 1
                2                N       
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.
For relationships that are pure products
or quotients a simple shortcut can be
used to estimate propagation of error.
R=k X1a X2b X3c…

2            2             2
2                      2        
UR       U x1  2
U x2         U x3
 x  b  x
 a                  c 
     x       ......

R       1       2           3      
Example Problem: Propagation of Error
Ideal gas law:            P

RT
Temperature
T±T           How do we
estimate the error
Pressure
P±P
in the density?

R=Constant
Apply RSS Formula to density relationship:
2
2
   
2
 1          P2

 RSS     p      T       P      T
p  T   RT  RT 2    

Apply a little algebra:              P

RT

2
  p  T 2
         
    p      T
Uncertainty Analysis in EES
Uncertainty Calculation in EES
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.
Plotting and Data Analysis with
MicroSoft Excel
Outline
Basic Plotting with Excel
Regression Analysis
Example
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.
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
Basic Plotting with Excel 97
Format the data series
Use open symbols before solid symbols

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 B&W only.
To a copier red and yellow both appear gray.
Format text for clarity
Superscript
Greek Symbols
Plotting Common Sense
Trend Line do’s and don’ts
Avoid using “Insert Trend Line” because it
only gives, slope, intercept, and R2.
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.
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