2. Measurement Process Characterization
2.5. Uncertainty analysis
2.5.5. Propagation of error considerations
|Top-down approach consists of estimating the uncertainty from direct repetitions of the measurement result||
The approach to uncertainty analysis that has been followed up to this
point in the discussion has been what is called a top-down approach.
Uncertainty components are estimated from direct repetitions of the
measurement result. To contrast this with a propagation of error
approach, consider the simple example where we estimate the area of a
rectangle from replicate measurements of length and width. The area
can be computed from each replicate. The standard deviation of the reported area is estimated directly from the replicates of area.
|Advantages of top-down approach||
This approach has the following advantages:
|Propagation of error approach combines estimates from individual auxiliary measurements||
The formal propagation of error approach is to compute:
Goodman (1960) derived
an exact formula for the variance between two products. Given
two random variables, x and y (correspond to width
and length in the above approximate formula), the exact formula for
the variance is:
To obtain the standard deviation, simply take the square root of the above formula. Also, an estimate of the statistic is obtained by substituting sample estimates for the corresponding population values on the right hand side of the equation.
|Approximate formula assumes indpendence||The approximate formula assumes that length and width are independent. The exact formula assumes that length and width are not independent.|
|Disadvantages of propagation of error approach||
In the ideal case, the propagation of error estimate above will not
differ from the estimate made directly from the area measurements.
However, in complicated scenarios, they may differ because of:
|Propagation of error formula||
Sometimes the measurement of interest cannot be replicated directly and
it is necessary to estimate its uncertainty via propagation of
error formulas (Ku). The
propagation of error formula for
a function of one or more variables with measurements, X, Z, ... gives the following estimate for the standard deviation of Y:
|Treatment of covariance terms||
Covariance terms can be difficult to estimate if measurements are not
made in pairs. Sometimes, these terms are omitted from the formula.
Guidance on when this is acceptable practice is given below:
|Sensitivity coefficients||The partial derivatives are the sensitivity coefficients for the associated components.|
|Examples of propagation of error analyses||Examples of propagation of error that are shown in this chapter are:|
|Specific formulas||Formulas for specific functions can be found in the following sections:|