Statistics Presentation

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Statistics Presentation Ch En 475 Unit
Operations
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Quantifying variables (i.e. answering a question
with a number)

1. Directly measure the variable. -
   referred to as measured variable ex.
   Temperature measured with thermocouple
2. Calculate variable from measured or
   tabulated variables - referred to as
   calculated variable ex. Flow rate m
   r A v (measured or tabulated)

Each has some error or uncertainty
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A. Error of Measured Variable

• Several measurements
• are obtained for a
• single variable (i.e. T).
• What is the true value?
• How confident are you?
• Is the value different on
• different days?

Questions
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How do you determine the error?

• Lets assume normal Gaussian distribution
• For small sampling s is known
• For large sampling s is assumed

well pursue this approach
Use t tables for this approach
Dont often have this much data
Use z tables for this approach
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Example
n Temp
1 40.1
2 39.2
3 43.2
4 47.2
5 38.6
6 40.4
7 37.7
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Standard Deviation Summary
(normal distribution)
40.9 (3.27) 1s 68.3 of data are within
this range 40.9 (3.27x2) 2s 95.4 of
data are within this range 40.9
(3.27x3) 3s 99.7 of data are within this
range
If normal distribution is questionable, use
Chebyshev's inequality At least 50 of the
data are within 1.4 s from the mean. At least
75 of the data are within 2 s from the mean. At
least 89 of the data are within 3 s from the
mean.
The above ranges dont state how accurate the
mean is - only the of data within the given
range
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Student t-test (gives confidence of where m (not
data) is located)
measured mean
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5
t
true mean
2a1- probability r n-1 6
2-tail
Prob. a t -
90 .05 1.943 2.40
95 .025 2.447 3.02
99 .005 3.707 4.58
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T-test Summary

• exact mean
• 40.9 is sample mean

41 2 90 confident m is somewhere in this
range 41 3 95 confident m is somewhere in
this range 41 5 99 confident
m is somewhere in this range
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Comparing averages of measured variables
Day 1 Day 2
What is your confidence that mx?my (i.e. they are
different)?
Larger t More likely different
1-tail
1-a confident different a confident same
nxny-2
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Example

1. Calculate average and s for both sets of data
2. Find range in which 95.4 of the data falls (for
   each set).
3. Determine range for m for each set at 95
   probability
4. At what confidence are pressures different each
   day?

Data points Pressure Day 1 Pressure Day 2
1 750 730
2 760 750
3 752 762
4 747 749
5 754 737
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B. Uncertainty of Calculated Variable
Calculate variable from multiple input
(measured, tabulated, ) variables (i.e. m
rAv) What is the uncertainty of your
calculated value? Each input variable has its
own error
Example You take measurements of r, A, v
to determine m rAv. What is the
range of m and its associated
uncertainty?
Details provided in Applied Engineering
Statistics, Chapters 8 and 14, R.M. Bethea and
R.R. Rhinehart, 1991).
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• To obtain uncertainty of calculated variable
• DO NOT just calculate variable for each set of
  data and then average and take standard
  deviation
• DO calculate uncertainty using error from input
  variables use uncertainty for calculated
  variables and error for input variables

Plan obtain max error (d) for each input
variable then obtain uncertainty of
calculated variable Method 1 Propagation of
max error - brute force Method 2 Propagation
of max error - analytical Method 3 Propagation
of variance - analytical Method 4 Propagation
of variance - brute force - Monte Carlo
simulation
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Value and Uncertainty

• Value used to make decisions - need to know
• uncertainty of value
• Potential ethical and societal impact
• How do you determine the uncertainty of the
  value?

• Sources of uncertainty (from Rhinehart, Applied
  Engineering Statistics, 1991)
• Estimation - we guess!
• Discrimination - device accuracy (single data
  point)
• Calibration - may not be exact (error of curve
  fit)
• Technique - i.e. measure ID rather than OD
• Constants and data - not always exact!
• Noise - which reading do we take?
• Model and equations - i.e. ideal gas law vs.
  real gas
• Humans - transposing,

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Estimates of Error (d) for input variables (ds
are propagated to find uncertainty)

1. Measured measure multiple times obtain s d
   2.5s Reason 99 of data is within 2.5s
   Example s 2.3 ºC for thermocouple, d 5.8
   ºC
2. Tabulated d 2.5 times last reported
   significant digit (with 1) Reason Assumes last
   digit is 2.5 ( 0 assumes perfect, 5
   assumes next left digit is fuzzy) Example r
   1.3 g/ml at 0º C, d 0.25 g/ml Example
   People 127,000 d 2500 people

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Estimates of Error (d) for input variables

1. Manufacturer spec or calibration accuracy use
   given spec or accuracy data Example Pump spec
   is 1 ml/min, d 1 ml/min
2. Variable from regression (i.e. calibration
   curve) d 2.5standard error (std error is
   stdev of residual) Example Velocity is slope
   with std error 2 m/s
3. Judgment for a variable use judgment for d
   Example Read pressure to 1 psi, d 1 psi

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Estimates of Error (d) for input variables
If none of the above rules apply, give your best
guess
Example Data from a computer show that the flow
rate is 562 ml/min 3 ml/min (stdev of computer
noise). Your calibration shows 510 ml/min 8
ml/min (stdev). What flow rate do you use and
what is d?
In the following propagation methods, its
assumed that there is no bias in the values used
- lets assume this for all lab projects.
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Method 1 Propagation of max error- brute force

• Brute force method obtain upper and lower
  limits of all input variables (from maximum
  errors) plug into equation to get uncertainty
  of calculated variable (y).
• Uncertainty of y is between ymin and ymax.
• This method works for both symmetry and
• asymmetry in errors (i.e. 10 psi 3 psi or -
  2 psi)

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Example Propagation of max error- brute force
m r A v
Brute force method
max min
r
A
v
r 2.0 g/cm3 (table) A 3.4 cm2 (measured
avg) v 2 cm/s (slope of graph)
Additional information
sA 0.1 cm2 std. error (v) 0.1 cm/s
All combinations
What is d for each input variable?
mmin lt m lt mmax
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Method 2 Propagation of max error- analytical

• Propagation of error Utilizes maximum error
  of input variable (d) to estimate uncertainty
  range of calculated variable (y)
• Uncertainty of y y yavg dy
• Assumptions
• input errors are symmetric
• input errors are independent of each other
• equation is linear (works o.k. for non-linear
  equations if input errors are relatively
  small)

Remember to take the absolute value!!
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Example Propagation of max error- analytical
m r A v
y x1 x2 x3
Av rv rA
r 2.0 g/cm3 (table) A 3.4 cm2 (measured
avg) v 2 cm/s (slope of graph)
m mavg dm rAv dm 13.6 4.4 g/s
Additional information
sA 0.1 cm2 std. error (v) 0.1 cm/s
(3.4)(2)(0.25) 0.39 (4.4)
ferror,r
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Propagation of max error

• If linear equation, symmetric errors, and input
  errors are independent ? brute force and
  analytical are same
• If non-linear equation, symmetric errors, and
  input errors are independent ? brute force and
  analytical are close if errors are small. If
  large errors (i.e. gt10 or more than order of
  magnitude), brute force is more accurate.
• Must use brute force if errors are dependant on
  each other and/or asymmetric.
• Analytical method is easier to assess if lots of
  inputs. Also gives info on contribution from
  each error.

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Method 3 Propagation of variance- analytical

• Maximum error can be calculated from max errors
  of input variables as shown previously
• Brute force
• Analytical
• Probable error is more realistic
• Errors are independent (some may be and some
  -). Not all will be in same direction.
• Errors are not always at their largest value.
• Thus, propagate variance rather than max error
• You need variance (s2) of each input to propagate
  variance. If s (stdev) is unknown, estimate s
  d/2.5

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Method 3 Propagation of variance- analytical
y yavg 1.96 SQRT(s2y) 95
y yavg 2.57 SQRT(s2y) 99

• gives propagated variance of y or (stdev)2
• gives probable error of y and associated
  confidence
• error should be lt10 (linear approximation)
• use propagation of max error if not much data,
  use propagation of variance if lots of data

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Method 4 Monte Carlo Simulation (propagation of
variance brute force)

• Choose N (N is very large, e.g. 100,000) random
  di from a normal distribution of standard
  deviation si for each variable and add to the
  mean to obtain N values with errors
• rnorm(N,µ,s) in Mathcad generates N random
  numbers from a normal distribution with mean µ
  and std dev s
• Find N values of the calculated variable using
  the generated xi values.
• Determine mean and standard deviation of the N
  calculated variables.

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Monte Carlo Simulation Example

• Estimate the uncertainty in the critical
  compressibility factor of a fluid if Tc 514 2
  K, Pc 61.37 0.6 bar, and Vc 0.168 0.002
  m3/kmol?

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Example Propagation of variance
Calculate r and its 95 probable error
All independent variables were measured multiple
times (Rule 1) averages and s are given
M 5.0 kg s 0.05 kg L 0.75 m s
0.01 m D 0.14 m s 0.005 m
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Propagation of Errors
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Monte Carlo
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Overall Summary

• measured variables use average, std dev (data
  range), and student t-test (mean range and
  mean comparison)
• calculated variable determine uncertainty
  -- Max error propagating error with brute force
  -- Max error propagating error
  analytically -- Probable error propagating
  variance analytically
• -- Probable error propagating variance with
  brute force
• (Monte Carlo)

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Data and Statistical Expectations

1. Summary of raw data (table format)
2. Sample calculations including statistical
   calculations
3. Summary of all calculations- table format is
   helpful
4. If measured variable average and standard
   deviation for all, confidence of mean for at
   least one variable
5. if calculated variable 1 of the 4 methods.
   Please state in report. If messy equation, you
   may show 1 of 4 methods for small part and then
   just average (with std dev.) the value
   (although not the best method).