Download Taking Experimental Data - Department of Chemical Engineering

Statistical Data Analysis
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Prof. Terry A. Ring, Ph. D.
Dept. Chemical & Fuels Engineering
University of Utah
http://www.che.utah.edu/~geoff/writing/index.html
Making Measurements
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Choice of Measurement Equipment
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Accuracy – systematic error associated
with measurement.
Precision – random error associated with
measurement.
Definitions
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Error – the difference between the measured
quantity and the ”true value.”
The “true value” is not known!!!
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So how do you calculate the error??
Random errors - the disagreement between
the measurements when the experiment is
repeated
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Is repeating the measurement on the same
sample a new experiment?
Systematic errors - constant errors which are the
same for all measurements.
Bogus Data – mistake reading the instrument
Random Error Sources
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Judgement errors, estimate errors, parallax
Fluctuating Conditions
Digitization
Disturbances such as mechanical vibrations or
static electricty caused by solar activity
Systematic Error Sources
Calibration of instrument
Environmental conditions different from
calibration
Technique – not at equilibrium or at steady
state.
Statistics
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Mean
Deviation
Standard Deviation
Confidence level or
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xM
xi-xM

uncertainty,
95% confidence = 1.96 
99% confidence = 2.58 
Please note that Gaussian distributions do not
rigorously apply to particles- log-normal is
better.
Mean and standard deviation have different
definitions for non-Gaussian Distributions
Comparison of means – Student’s ttest
n1n 2
v
1/2
t  (x M1 - x M2 )[
]
2
2
(n1  n 2 ) (n1  1) 1  (n2  1) 2


v= n1+n2-2
use the t-value to calculate the
probability, P, that the two means are
the same.
Estimating Uncertainties or Estimating Errors in
Calculated Quantities –with Partial Derivatives

G=f(y1,y2,y3,…)
G
2
 G
 

i
 yi
yi



2
http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm
Way around the Partial
Derivatives
This approach applies no matter how large the uncertainties (Lyons, 1991).
(i) Set all xi equal to their measured values and calculate f. Call this fo.
(ii) Find the n values of f defined by
fi = f(x1,x2,...,xi+i,...,xn)
(iii) Obtain f from
   f i  f o 
2
f
(11)
2
(12)
If the uncertainties are small this should give the same result as (10). If the uncertainties are
large, this numerical approach will provide a more realistic estimate of the uncertainty in f. The
numerical approach may also be used to estimate the upper and lower values for the uncertainty
in f because the fi in (11) can also be calculated with xi+ replaced by xi-.
Now try the same calculation using the spread sheet method. The dimensionless form of (12) is (after taking
1/ 2
2
the square root)


f

   i  1 
f0   f0
 

f
(17)
The propagated fractional uncertainties using (15) and (17) are compared in Table 4.
A further advantage of the numerical approach is that it can be used with simulations. In other words, the
function f in (12) could be a complex mathematical model of a distillation column and f might be the mole fraction
or flow rate of the light component in the distillate.
Table 4. Uncertainties in Gas Velocity Calculated from (15) and (17)
Equation Used
f/f0
(9)
0.011968
(12) with +
0.011827
(12) with -
0.012113
See web page with
sample calculation done
with Excel
Fitting Data
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Linear Equation – linear regression
Non-linear Equation
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Linearize the equation- linear regression
Non-linear least squares
Rejection of Data Points
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Maximum Acceptable Deviations
(Chauvenet’s Criterion)
Example
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xi-xM/=8.9-8.2/0.3=2.33
xi-xM/=7.9-8.2/0.3=1.0
Regression
• Linear Regression
– good for linear
equations only
• Non-linear Regression
– most accurate for nonlinear equations
• Linearize non-linear
equation first
– linearization leads to
errors
– See Mathcad example
Residence Time
Measurements
T
Time(min)
T
Time(min)
T=(To-Tin)exp(-t/tau)+Tin
105
100
95
Non-Linear Fit
Linearized Eq. Fit
Flow Calc.s
Temperature(C)
90
Ti
emp ti  p
emp ti  ul
emp ti  uc
85
80
75
70
0
1
2
3
4
5
6
ti
Time (min)
7
8
9
 u  x  exp x 
1
u 

2
F( x u ) 
 0
u 0


x

exp  

 u0 

1

Results
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Curve Fit of function
emp ( x u )  u 1  exp 
p  genfit ( t  T  ug  F)

Non-linear Fit results
 4.15 
p   36.232 


68.218



To-Tin
Tin
x  
   u2
 u0  
Linearized Fit results
Calculation from data
 4.262 
ul   35.5 


 68 
 3.803 
uc   35.5 


 68 
Standard Error of Estimate for Fit
Stderror( p  T  t)  0.488
Stderror( ul T  t)  0.675
Stderror( uc  T  t)  1.515