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2. Measurement Process Characterization
2.3. Calibration
2.3.6. Instrument calibration over a regime Uncertainties of calibrated values

Uncertainty for quadratic calibration using propagation of error

Propagation of error for uncertainty of calibrated values of loadcells The purpose of this page is to show the propagation of error for calibrated values of a loadcell based on a quadratic calibration curve where the model for instrument response is $$ Y = a + bX + cX^2 + \epsilon $$ The calibration data are instrument responses at known loads (psi), and estimates of the quadratic coefficients, \( a, \,\, b, \,\, c \), and their associated standard deviations are shown with the analysis.

A graph of the calibration curve showing a measurement \(Y'\) corrected to \(X'\), the proper load (psi), is shown below.

quadratic calibration curve for loadcell
Uncertainty of the calibrated value X' The uncertainty to be evaluated is the uncertainty of the calibrated value, \(X'\), computed for any future measurement, \(Y'\), made with the calibrated instrument where $$ X' = \frac{-\hat{b} \pm \sqrt{\hat{b}^2 - 4 \hat{c} \left( \hat{a} - Y' \right)}}{2 \hat{c}} $$
Partial derivatives The partial derivatives are needed to compute uncertainty. $$ \frac{\partial{X'}}{\partial{Y'}} = \frac{1}{\sqrt{\hat{b}^{2} - 4\hat{c}(\hat{a}-Y')}} $$ $$ \frac{\partial{X'}}{\partial{\hat{a}}} = \frac{-1}{\sqrt{\hat{b}^{2} - 4\hat{c}(\hat{a}-Y')}} $$ $$ \frac{\partial{X'}}{\partial{\hat{b}}} = \frac{-1 + \frac{\hat{b}}{\sqrt{\hat{b}^{2} - 4\hat{c}(\hat{a}-Y')}}}{2\hat{c}} $$ $$ \frac{\partial{X'}}{\partial{\hat{c}}} = \frac{-\hat{a} + Y'}{\hat{c}\sqrt{\hat{b}^{2} - 4\hat{c}(\hat{a}-Y')}} - \frac{-\hat{b} + \sqrt{\hat{b}^{2} - 4\hat{c}(\hat{a}-Y')}}{2\hat{c}^{2}} $$
The variance of the calibrated value from propagation of error The variance of \(X'\) is defined from propagation of error as follows: $$ u^{2} = \left( \frac{\partial{X'}}{\partial{Y'}}\right) ^{2} (s_{Y'})^{2} + \left( \frac{\partial{X'}}{\partial{\hat{a}}}\right) ^{2} (s_{\hat{a}})^{2} + \left( \frac{\partial{X'}}{\partial{\hat{b}}}\right) ^{2} (s_{\hat{b}})^{2} + \left( \frac{\partial{X'}}{\partial{\hat{c}}}\right )^{2} (s_{\hat{c}})^{2} $$ The values of the coefficients and their respective standard deviations from the quadratic fit to the calibration curve are substituted in the equation. The standard deviation of the measurement, \(Y\), may not be the same as the standard deviation from the fit to the calibration data if the measurements to be corrected are taken with a different system; here we assume that the instrument to be calibrated has a standard deviation that is essentially the same as the instrument used for collecting the calibration data and the residual standard deviation from the quadratic fit is the appropriate estimate.

a  = -0.1850148e-04
sa =  0.2081e-04
b  =  0.100102
sb =  0.3681e-05
c  =  0.7030346e-05
sc =  0.1460e-06
sy =  0.0000340856
Graph showing the standard deviations of calibrated values X' for given instrument responses Y' ignoring covariance terms in the propagation of error The standard deviation expressed above is not easily interpreted but it is easily graphed. A graph showing standard deviations of calibrated values, \(X'\), as a function of instrument response, \(Y'\), is shown below.

standard deviation of calibrated value of X' for
 given instrument response Y'

Problem with propagation of error The propagation of errors shown above is not complete because it ignores the covariances among the coefficients, \( a, \,\, b, \,\, c \). Unfortunately, some statistical software packages do not display these covariance terms with the other output from the analysis.
Covariance terms for loadcell data The variance-covariance terms for the loadcell data set are shown below.

            a              b              c
a      4.330796e-10
b     -7.027501e-11  1.355352e-11
c      2.498281e-12 -5.244666e-13  2.133052e-14

The diagonal elements are the variances of the coefficients, \( a, \,\, b, \,\, c \), respectively, and the off-diagonal elements are the covariance terms.

Recomputation of the standard deviation of X' To account for the covariance terms, the variance of \(X'\) is redefined by adding the covariance terms. Appropriate substitutions are made; the standard deviations are recomputed and graphed as a function of instrument response.


sab = -7.027501e-11
sac =  2.498281e-12
sbc = -5.244666e-13
The graph below shows the correct estimates for the standard deviation of \(X'\) and gives a means for assessing the loss of accuracy that can be incurred by ignoring covariance terms. In this case, the uncertainty is reduced by including covariance terms, some of which are negative.
Graph showing the standard deviations of calibrated values, X', for given instrument responses, Y', with covariance terms included in the propagation of error standard deviation of calibrated value X' for
 given instrument response Y'
Sample code The results in this section can be generated using R code or Dataplot code.
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