Propagation of Error

Ch En 475Unit 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

Example: You take measurements of r, A, v to determine m = rAv. What is the range of m and its associated uncertainty?

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

Uncertainty of Calculated Variable

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: 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

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 OD5. Constants and data - not always exact!6. Noise - which reading do we take?7. Model and equations - i.e. ideal gas law vs. real gas8. 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 analytically2. 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.

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)

Method 1: Propagation of max error- brute force

Example: Propagation of max error- brute force

= r A v

r = 2.0 g/cm3 (table)A = 3.4 cm2 (measured avg)v = 2 cm/s (slope of graph)

sA = 0.03 cm2

std. error (v) = 0.05 cm/s

min max

r

A

v

Brute force method:

min < < max

All combinations

Additional information:

What is d for each input variable?

Example: Propagation of max error- brute force

= r A v

r = 2.0 g/cm3 (table) 2.5 lowest digit = 0.25A = 3.4 cm2 (measured avg) 2.5 sA = 0.075v = 2 cm/s (slope of graph) 2.5 std. error (v) = 0.125

sA = 0.03 cm2

std. error (v) = 0.05 cm/s

Additional information:

What is d for each input variable?

Compute d(look at cheat sheet)

Example: Propagation of max error- brute force

= r A v

r = 2.0 g/cm3 (table)A = 3.4 cm2 (measured avg)v = 2 cm/s (slope of graph)

sA = 0.03 cm2

std. error (v) = 0.05 cm/s

min max

r 1.75 2.25

A 3.325 3.475

v 1.875 2.125

Brute force method:

min < < max

All combinations

Additional information:

What is d for each input variable?10.9 16.6

3.0113.6

2.69

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!!

d d

y ii i

yx

Example: Propagation of max error- analytical

= r A vy x1 x2 x3

= (3.4)(2)(0.25) = 0.60 (2.85)

m = mavg ± dm

= rAv ± dm

= 13.6 ± 2.85 g/s

d d

y ii i

yx

rd d d dr

m A vm m m

A v

Av rv rA3.42 2 2 2 3.4r = 2.0 g/cm3 (table)

A = 3.4 cm2 (measured avg)v = 2 cm/s (slope of graph)

rdrd

m

mFor 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

Remember from above:

ferror,r

(fractional error)

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

s s

2

2 2y xi

i i

yx

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

)4/( 2DLM

r

All independent variables were measuredmultiple times (Rule 1); averages and s are given

M = 5.0 kg s = 0.05 kgL = 0.75 m s = 0.01 mD = 0.14 m s = 0.005 m

Propagation of VarianceExample Problem

rM

L D2

4

Mav 5 Lav 0.75 Dav 0.14

sM 0.05 sL 0.01 sD 0.005

dM 2.5 sM dL 2.5 sL dD 2.5 sD

dr4

Lav Dav2dM

Mav 4

Dav2 Lav

2dL

8 Mav

Lav Dav3

dD

rav4 Mav

Lav Dav2

433.075 dr 102.597 srdr2.5

41.039

uncertainty 1.96sr 80.436

Propagation of errors:

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:

– 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.

d 'i i ix x

Monte CarloExample 1

Example Problem

rM

L D2

4

Mav 5 Lav 0.75 Dav 0.14

sM 0.05 sL 0.01 sD 0.005

dM 2.5 sM dL 2.5 sL dD 2.5 sD

Monte Carlo

Mtrial rnorm N Mav, sM, Ltrial rnorm N Lav, sL, Dtrial rnorm N Dav, sD,

rav mean4 Mtrial

Ltrial Dtrial2

434.738 sr stdev4 Mtrial

Ltrial Dtrial2

32.091

uncertainty 1.96sr 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. Average2. 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 analytically3. Probable error: propagating variance analytically4. Probable error: propagating variance with brute

force (i.e., Monte Carlo)

Data and Statistical Expectations(for UO Lab reports)

1. Summary of raw data (table format)2. Sample calculations– including statistical calculations3. Summary of all calculations- table format is helpful4. If measured variable: average and standard deviation for

all, confidence of mean for at least one variable5. 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).