2.3.6.7.1. Uncertainty for quadratic calibration using propagation of error

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

2.3.6.7.1. 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

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' can be evaluated using software capable of algebraic representation	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
Propagation of error using Mathematica	The analysis of uncertainty is demonstrated with the software package, Mathematica (Wolfram). The format for inputting the solution to the quadratic calibration curve in Mathematica is as follows: In[10]:= f = (-b + (b^2 - 4 c (a - Y))^(1/2))/(2 c)
Mathematica representation	The Mathematica representation is Out[10]= 2 -b + Sqrt[b - 4 c (a - Y)] --------------------------- 2 c
Partial derivatives	The partial derivatives are computed using the D function. For example, the partial derivative of f with respect to Y is given by: In[11]:= dfdY=D[f, {Y,1}] The Mathematica representation is: Out[11]= 1 ---------------------- 2 Sqrt[b - 4 c (a - Y)]
Partial derivatives with respect to a, b, c	The other partial derivatives are computed similarly. In[12]:= dfda=D[f, {a,1}] Out[12]= 1 -(----------------------) 2 Sqrt[b - 4 c (a - Y)] In[13]:= dfdb=D[f,{b,1}] Out[13]= b -1 + ---------------------- 2 Sqrt[b - 4 c (a - Y)] --------------------------- 2 c In[14]:=dfdc=D[f, {c,1}] Out[14]= 2 -(-b + Sqrt[b - 4 c (a - Y)]) a - Y ------------------------------ - ------------------------ 2 2 2 c c Sqrt[b - 4 c (a - Y)]
The variance of the calibrated value from propagation of error	The variance of X' is defined from propagation of error as follows: In[15]:= u2 =(dfdY)^2 (sy)^2 + (dfda)^2 (sa)^2 + (dfdb)^2 (sb)^2 + (dfdc)^2 (sc)^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. In[16]:= % /. a -> -0.183980 10^-4 % /. sa -> 0.2450 10^-4 % /. b -> 0.100102 % /. sb -> 0.4838 10^-5 % /. c -> 0.703186 10^-5 % /. sc -> 0.2013 10^-6 % /. sy -> 0.0000376353
Simplification of output	Intermediate outputs from Mathematica, which are not shown, are simplified. (Note that the % sign means an operation on the last output.) Then the standard deviation is computed as the square root of the variance. In[17]:= u2 = Simplify[%] u=u2^.5 Out[24]= 0.100102 2 Power[0.11834 (-1 + --------------------------------) + Sqrt[0.0100204 + 0.0000281274 Y] -9 2.01667 10 -------------------------- + 0.0100204 + 0.0000281274 Y -14 9 4.05217 10 Power[1.01221 10 - 10 1.01118 10 Sqrt[0.0100204 + 0.0000281274 Y] + 142210. (0.000018398 + Y) --------------------------------, 2], 0.5] Sqrt[0.0100204 + 0.0000281274 Y]
Input for displaying standard deviations of calibrated values as a function of Y'	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 displayed in Mathematica given the following input: In[31]:= Plot[u,{Y,0,2.}]
Graph showing the standard deviations of calibrated values X' for given instrument responses Y' ignoring covariance terms in the propagation of error
Problem with propagation of error	The propagation of error shown above is not correct 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 6.0049021-10 b -1.0759599-10 2.3408589-11 c 4.0191106-12 -9.5051441-13 4.0538705-14 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. In[25]:= u2 = u2 + 2 dfda dfdb sab2 + 2 dfda dfdc sac2 + 2 dfdb dfdc sbc2 % /. sab2 -> -1.0759599 10^-10 % /. sac2 -> 4.0191106 10^-12 % /. sbc2 -> -9.5051441 10^-13 u2 = Simplify[%] u = u2^.5 Plot[u,{Y,0,2.}]
	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