7. Propagation of error

7. Propagation of error
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 the area of a rectangular is estimated from replicate measurements of length and width. The area area = length x width can be computed from each replicate. The standard deviation of the reported area is estimated directly from the replicates of area.
This approach has the following advantages: proper treatment of covariances between measurements of length and width proper treatment of unsuspected sources of error that would emerge if measurements covered a range of operating conditions and a sufficiently long time period independence from propagation of error model
The formal propagation of error approach is to compute: standard deviation from the length measurements standard deviation from the width measurements and combine the two into a standard deviation for area using the approximation for products of two variables. The formula below is appropriate if there is no covariance between length and width measurements.
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: unsuspected covariances disturbances which affect the reported value and not the elementary measurements (usually a result of mis-specification of the model) mistakes in propagating the error through the defining formulas
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 Y = f(X, Z, ... ) a function of one or more variables with measurements, X, Z, ... gives the following estimate for the standard deviation of Y: where is the standard deviation of the X measurements is the standard deviation of Z measurements is the partial derivative of the function Y with respect to X, etc. is the estimated covariance between the X,Z measurements
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: If the measurements of X, Z are independent, the associated covariance term is zero. Practically speaking, covariance terms should be included in the computation only if they have been estimated from sufficient data. Generally, reported values of test items from calibration designs have non-zero covariances which must be taken into account if Y is a summation such as the mass of two weights, or the length of two gage blocks end-to-end, etc.
The partial derivatives are the sensitivity coefficients for the associated components.