Error Propagation A Functional Approach
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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 http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm 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 http://wrap.warwick.ac.uk/50291/ length and width. The area $$ area = length \cdot width $$ can be computed from each replicate. The standard deviation of the reported area is estimated directly from error propagation the replicates of area. Advantages of top-down approach 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 Propagation of error approach error propagation a combines estimates from individual auxiliary measurements 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 (ignoring a possible covariance between length and width), $$ s_{area} = \sqrt{width^2 \cdot s_{length}^2 + length^2 \cdot s_{width}^2} $$ Exact formula 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: $$ V(\bar{x} \bar{y}) = \frac{1}{n} \left[ X^2 V(y) + Y^2 V(x) + 2XYE_{11} + 2X\frac{E_{12}}{n} + 2Y\frac{E_{21}}{n} + \frac{V(x) V(y)}{n} + \frac{Cov((\Delta x)^2, (\Delta y)^2) -E_{11}^2 }{n^2} \right] $$ with \(X = E(x)\) and \(Y = E(y)\) (corresponds to width and length, respectively, in the approximate formula) \(V(x)\) is the variance of \(x\) and \(V(y)\) is the variance \(y\) (corresponds to \(s^2\) for width and length, respectively, in the approximate fo
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Item Type: Journal Article Divisions: Faculty of Science > Physics Journal or Publication Title: Journal of Chemical Education Publisher: American Chemical Society ISSN: 0021-9584 Official Date: 8 May 2012 Dates: DateEvent8 May 2012Published Volume: Vol.89 Number: No.6 Page Range: pp. 821-822 Identification Number: 10.1021/ed2004627 Status: Peer Reviewed Publication Status: Published Access rights to Published version: Restricted or Subscription Access URI: http://wrap.warwick.ac.uk/id/eprint/50291 Request changes or add full text files to a record Actions (login required) View Item Email us: publications@warwick.ac.uk Contact Details About Us