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 = A v (measured or tabulated)

Each has some error or uncertainty

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

Calculate variable from multiple input (measured, tabulated, …) variables (i.e. m = Av)

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 () 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 () for input variables(’s are propagated tofind uncertainty)

1. Measured: measure multiple times; obtain s; ≈ 2.5s Reason: 99% of data is within ± 2.5s

Example: s = 2.3 ºC for thermocouple, = 5.8 ºC

2. Tabulated : ≈ 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: = 1.3 g/ml at 0º C, = 0.25 g/ml Example: People = 127,000 = 2500 people

Estimates of Error () for input variables

3. Manufacturer spec or calibration accuracy: use given spec or accuracy data Example: Pump spec is ± 1 ml/min, = 1 ml/min

4. Variable from regression (i.e. calibration curve): ≈ 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 Example: Read pressure to ± 1 psi, = 1 psi

Estimates of Error () for input variables

If none of the above rulesapply, 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 ?

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.

Estimate of Error forEstimate of Error forCalculated VariablesCalculated Variables

i.e., Propagation of Errori.e., Propagation of Error

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

m = A v

= 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

minmin maxmax

AA

vv

Brute force method:

mmin < m < mmax

All combinations

Additional information:

What is for each input variable?

Example: Propagation of max error- brute force

m = A v

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

minmin maxmax

1.751.75 2.252.25

AA 3.3253.325 3.4753.475

vv 1.8751.875 2.1252.125

Brute force method:

mmin < m < mmax

All combinations

Additional information:

What is for each input variable?10.9 16.6

3.01

13.6

2.69

sA = 0.03 cm2

std. error (v) = 0.05 cm/s

Method 2: Propagation of max error- analytical

Propagation of error: Utilizes maximum error of input variable () to estimate uncertainty range of calculated variable (y)

Uncertainty of y: y = yavg ± y

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!!

y ii i

yx

Example: Propagation of max error- analytical

m = A vy x1 x2 x3

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

m = mavg ± m

= Av ± m

= 13.6 ± 2.85 g/s

y ii i

yx

m A vm m m

A v

Av v A3.42 2 2 2 3.4= 2.0 g/cm3 (table)

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

m

mFor s = 0.1 g/cm3sA = 0.03 cm2, std. error (v) = 0.05 cm/s

Additional information:

ferror,

(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 () of each input to

propagate variance. If (stdev) is unknown, estimate = /2.5

Method 3: Propagation of variance- analytical

2

2 2y xi

i i

yx

y = yavg ± 1.96 SQRT(y) 95%

y = yavg ± 2.57 SQRT(y) 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

Method 4: Monte Carlo Simulation Method 4: Monte Carlo Simulation (propagation of variance – brute (propagation of variance – brute

force)force) Choose N (N is very large, e.g. 100,000) random ±Choose N (N is very large, e.g. 100,000) random ±δδii

from a normal distribution of standard deviation from a normal distribution of standard deviation σσii for each variable and add to the mean to obtain N for each variable and add to the mean to obtain N values with errors: values with errors: • rnorm(N,rnorm(N,μμ,,σσ) in Mathcad generates N random numbers ) in Mathcad generates N random numbers

from a normal distribution with mean from a normal distribution with mean μμ and std dev and std dev σσ

Find N values of the calculated variable using the Find N values of the calculated variable using the generated x’generated x’i i values.values.

Determine mean and standard deviation of the N Determine mean and standard deviation of the N calculated variables.calculated variables.

'i i ix x

Monte Carlo Simulation ExampleMonte Carlo Simulation Example

Estimate the uncertainty in the critical Estimate the uncertainty in the critical compressibility factor of a fluid if Tc = 514 ± 2 K, compressibility factor of a fluid if Tc = 514 ± 2 K, Pc = 61.37 ± 0.6 bar, and Vc = 0.168 ± 0.002 Pc = 61.37 ± 0.6 bar, and Vc = 0.168 ± 0.002 mm33/kmol?/kmol?

Example: Propagation of variance

Calculate and its 95% probable error

)4/( 2DLM

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 ErrorsPropagation of ErrorsExample Problem

M

L D2

4

Mav 5 Lav 0.75 Dav 0.14

sM 0.05 sL 0.01 sD 0.005

M 2.5 sM L 2.5 sL D 2.5 sD

4

Lav Dav2M

Mav 4

Dav2 Lav

2L

8 Mav

Lav Dav3D

av4 Mav

Lav Dav2

433.075 102.597 s2.5

41.039

uncertainty 1.96s 80.436

Propagation of errors:

Monte CarloMonte CarloExample Problem

M

L D2

4

Mav 5 Lav 0.75 Dav 0.14

sM 0.05 sL 0.01 sD 0.005

M 2.5 sM L 2.5 sL D 2.5 sD

Monte Carlo

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

av mean4 Mtrial

Ltrial Dtrial2

434.738 s stdev4 Mtrial

Ltrial Dtrial2

32.091

uncertainty 1.96s 62.899

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)

Data and Statistical Expectations

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 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).