2. Measurement Process Characterization
2.5. Uncertainty analysis
2.5.5. Propagation of error considerations

2.5.5.3.

Propagation of error for many variables

Simplification for dealing with many variables

Propagation of error for several variables can be simplified considerably if:

• The function, Y, is a simple multiplicative function of secondary variables
• Uncertainty is evaluated as a percentage

Example of three variables

For three variables, X, Z, W, the function

has a standard deviation in absolute units of

In % units, the standard deviation can be written as

if all covariances are negligible. These formulas are easily extended to more than three variables.

Software can simplify propagation of error

Propagation of error for more complicated functions can be done reliably with software capable of algebraic representations such as Mathematica (Wolfram).

Example from fluid flow of non-linear function

For example, discharge coefficients for fluid flow are computed from the following equation (Whetstone et al.)

where

Representation of the defining equation

The defining equation is input as

Cd=m(1 - (d/D)^4)^(1/2)/(K d^2 F p^(1/2) delp^(1/2))

Mathematica representation

and is represented in Mathematica as follows:

Out[1]=
               4
              d
    Sqrt[1 - ---] m
               4
              D
-----------------------
 2
d  F K Sqrt[delp] Sqrt[p]

Partial derivatives - first partial derivative with respect to orifice diameter

Partial derivatives are derived via the function D where, for example,

D[Cd, {d,1}]

indicates the first partial derivative of the discharge coefficient with respect to orifice diameter, and the result returned by Mathematica is

Out[2]=

               4
              d
 -2 Sqrt[1 - ---] m
               4
              D
--------------------------    -  
 3
d  F K Sqrt[delp] Sqrt[p]



 
                 2 d m
  ------------------------------------
             4
            d    4
  Sqrt[1 - ---] D  F K Sqrt[delp] Sqrt[p]
             4
            D

First partial derivative with respect to pressure

Similarly, the first partial derivative of the discharge coefficient with respect to pressure is represented by

D[Cd, {p,1}]

with the result

Out[3]=


                 4
                d
  -  (Sqrt[1 - ---] m)
                 4
                D
----------------------
   2                 3/2
2 d  F K Sqrt[delp] p

Comparison of check standard analysis and propagation of error

The software can also be used to combine the partial derivatives with the appropriate standard deviations, and then the standard deviation for the discharge coefficient can be evaluated and plotted for specific values of the secondary variables.