Propagation of Error
Ch En 475
Unit Operations
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
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).
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:
Method 2:
Method 3:
Method 4:
Propagation of max error - brute force
Propagation of max error - analytical
Propagation of variance - analytical
Propagation of variance - brute force –
Monte Carlo simulation
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):
1. Estimation - we guess!
2. Discrimination - device accuracy (single data point)
3. Calibration - may not be exact (error of curve fit)
4. Technique - i.e. measure ID rather than OD
5. Constants and data - not always exact!
6. Noise - which reading do we take?
7. Model and equations - i.e. ideal gas law vs. real gas
8. Humans - transposing, …
Estimates of Error (d) for input variables
(d’s 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
Estimates of Error (d) for input variables
3. Manufacturer spec or calibration accuracy:
use given spec or accuracy data
Example: Pump spec is ± 1 ml/min, d = 1 ml/min
4. Variable from regression (i.e. calibration curve):
d ≈ 2.5*standard error (std error is stdev of residual)
Example: Velocity is slope with std error = 2 m/s
5. Judgment for a variable:
use judgment for d
Example: Read pressure to ± 1 psi, d = 1 psi
Estimate of Error for Calculated Variables
(i.e., Propagation of Error)
A. Max error:
1. propagating error with brute force
2. propagating error analytically
B. Probable error:
1. propagating variance analytically
2. propagating variance with brute force (i.e., Monte
Carlo)
Example Problem
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, it’s assumed that there is no
bias in the values used - let’s assume this for all lab projects.
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)
Example: Propagation of max error- brute force
𝑚= rAv
r = 2.0
A = 3.4 cm2
v = 2 cm/s
g/cm3
Brute force method:
min
(table)
(measured avg)
(slope of graph)
max
r
A
Additional information:
sA = 0.03 cm2
std. error (v) = 0.05 cm/s
v
All combinations
What is d for each input variable?
𝑚 min < 𝑚 < 𝑚 max
Example: Propagation of max error- brute force
𝑚= rAv
Compute d
(look at cheat sheet)
r = 2.0 g/cm3 (table)
A = 3.4 cm2 (measured avg)
v = 2 cm/s
(slope of graph)
Additional information:
sA = 0.03 cm2
std. error (v) = 0.05 cm/s
What is d for each input variable?
2.5  lowest digit = 0.25
2.5  sA = 0.075
2.5  std. error (v) = 0.125
Example: Propagation of max error- brute force
𝑚= rAv
r = 2.0
A = 3.4 cm2
v = 2 cm/s
g/cm3
Brute force method:
(table)
(measured avg)
(slope of graph)
min
max
r
1.75
2.25
A
3.325 3.475
v
1.875 2.125
Additional information:
sA = 0.03 cm2
std. error (v) = 0.05 cm/s
All combinations
What is d for each input variable?
𝑚 min < 𝑚 < 𝑚 max
10.9

16.6
13.6
2.69
3.01
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
dy  
i
y
di
xi
* Remember to take the absolute value!!
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)
Example: Propagation of max error- analytical
𝑚= r A v
y
x1 x2 x3
dy  
i
r = 2.0
A = 3.4 cm2
v = 2 cm/s
g/cm3
y
di
xi
dm 
m
m
m
dr 
dA 
dv
r
A
v
Av
(table)
(measured avg)
(slope of graph)
3.42
rv
2 2
rA
2 3.4
m = mavg ± dm
= rAv ± dm
= 13.6 ± 2.85 g/s
Remember from above:
For r, s r = 0.1 g/cm3
dr  0.25
sA = 0.03 cm2,
dA  0.075
std. error (v) = 0.05 cm/s dv  0.125
ferror,r
m
dr
r
dm
(fractional error)
= (3.4)(2)(0.25) = 0.60
(2.85)
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. >10% or more than
order of magnitude), brute force is more accurate.
• Must use brute force if errors are dependent on each
other and/or asymmetric.
• Analytical method is easier to assess if lots of inputs. Also
gives info on % contribution from each error.
Method 3: Propagation of variance- analytical
1. Maximum error can be calculated from max errors
of input variables as shown previously:
a) Brute force
b) Analytical
2. 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
Method 3: Propagation of variance- analytical
 y  2
 s xi
 xi 
s y2   
i
2
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 <10% (linear approximation)
• use propagation of max error if not much data,
use propagation of variance if lots of data
Example: Propagation of variance
Calculate r and its 95% probable error
M
r
L(D 2 / 4)
All independent variables were measured
multiple times (Rule 1); averages and s are given
M = 5.0 kg
L = 0.75 m
D = 0.14 m
s = 0.05 kg
s = 0.01 m
s = 0.005 m
Propagation of Variance
Example Problem
r
M
 D2 

L 
 4 
M av  5
Lav  0.75
Dav  0.14
s M  0.05
s L  0.01
s D  0.005
dM  2.5 s M
dL  2.5 s L
dD  2.5 s D
4
dr 
Propagation of errors:
Lav    Dav
rav 
4 M av
Lav    Dav
2
 433.075
2
 dM 
M av  4
2
  Dav  Lav
dr  102.597
2
s r 
 dL 
dr
2.5
8 M av
Lav    Dav
3
 dD
 41.039
uncertainty  1.96 s r  80.436
Method 4: Monte Carlo Simulation
(propagation of variance – brute force)
• Choose N (N is very large, e.g. 100,000) random ±δi from a
normal distribution of standard deviation σi for each variable
and add to the mean to obtain N values with errors: x '  x  d
i
i
i
– rnorm(N,μ,σ) in Mathcad generates N random numbers from a normal
distribution with mean μ and std dev σ
• Find N values of the calculated variable using the generated x’i
values.
• Determine mean and standard deviation of the N calculated
variables.
Monte Carlo
Example 1
Example Problem
r
M
2
 D

L 
 4 
M av  5
Lav  0.75
Dav  0.14
s M  0.05
s L  0.01
s D  0.005
dM  2.5 s M
dL  2.5 s L
dD  2.5 s D
Monte Carlo

M trial  rnorm N , M av , s M



Ltrial  rnorm N , Lav , s L

 

4 M trial
  434.738
rav  mean 

2
Ltrial   Dtrial




Dtrial  rnorm N , Dav , s D

 

4 M trial
  32.091
s r  stdev 

2
Ltrial   Dtrial


uncertainty  1.96 s r  62.899
Monte Carlo Simulation
Example 2
• 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?
Overall Summary
A. Measured variables: (use all)
1. Average
2. Std dev (data range)
3. Confidence interval from student t-test
(mean range and mean comparison)
B. Calculated variable: determine uncertainty (pick one)
1. Max error: propagating error with brute force
2. Max error: propagating error analytically
3. Probable error: propagating variance analytically
4. Probable error: propagating variance with brute
force (i.e., Monte Carlo)
Data and Statistical Expectations
(for UO Lab reports)
1.
2.
3.
4.
Summary of raw data (table format)
Sample calculations– including statistical calculations
Summary of all calculations- table format is helpful
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).