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2. Measurement Process Characterization
2.3. Calibration
2.3.6. Instrument calibration over a regime

Calibration of future measurements

Purpose The purpose of creating the calibration curve is to correct future measurements made with the same instrument to the correct units of measurement. The calibration curve can be applied many, many times before it is discarded or reworked as long as the instrument remains in statistical control. Chemical measurements are an exception where frequently the calibration curve is used only for a single batch of measurements, and a new calibration curve is created for the next batch.
Notation The notation for this section is as follows:
  • \(Y'\) denotes a future measurement.
  • \(X'\) denotes the associated calibrated value.
  • \( \hat{a}, \,\, \hat{b}, \,\, \hat{c} \) are the estimates of the coefficients, \( a, \,\, b, \,\, c \).
  • \( {\large s}_a, \,\, {\large s}_b, \,\, {\large s}_c \) are standard deviations of the coefficients, \( a, \,\, b, \,\, c \).
Procedure To apply a correction to a future measurement, \(Y^*\) to obtain the calibration value \(X^*\) requires the inverse of the calibration curve.
Linear calibration line The inverse of the calibration line for the linear model $$ Y = a + bX + \epsilon $$ gives the calibrated value $$ X' = \frac{Y' - \hat{a}}{\hat{b}} $$
Tests for the intercept and slope of calibration curve -- If both conditions hold, no calibration is needed. Before correcting for the calibration line by the equation above, the intercept and slope should be tested for \( a=0 \), and \(b=1\). If both $$ \left|\frac{\hat{a}}{s_a}\right| < t_{1-\alpha/2, \nu} \,\,\, \mbox{ and } \,\,\, \left|\frac{\hat{b}-1}{s_b}\right| < t_{1-\alpha/2, \nu} $$ there is no need for calibration. If, on the other hand only the test for \( a=0 \) fails, the error is constant; if only the test for \( b=1 \) fails, the errors are related to the size of the reference standards.
Table look-up for t-factor The factor, \( t_{1-\alpha/2, \, \nu} \), is found in the t-table where \( \nu \) is the degrees of freedom for the residual standard deviation from the calibration curve, and \( \alpha \) is chosen to be small, say, 0.05.
Quadratic calibration curve The inverse of the calibration curve for the quadratic model $$ Y = a + bX + cX^2 + \epsilon $$ requires a root $$ X' = \frac{-\hat{b} \pm \sqrt{\hat{b}^2 - 4 \hat{c} \left( \hat{a} - Y' \right)}}{2 \widehat{c}} $$ The correct root (+ or -) can usually be identified from practical considerations.
Power curve The inverse of the calibration curve for the power model $$ Y = aX^b\epsilon $$ gives the calibrated value $$ X' = \mbox{exp} \left( \frac{\mbox{log}_e(Y') - \mbox{log}_e(\hat{a})}{\hat{b}} \right) $$ where \(b\) and the natural logarithm of \(a\) are estimated from the power model transformed to a linear function.
Non-linear and other calibration curves For more complicated models, the inverse for the calibration curve is obtained by interpolation from a graph of the function or from predicted values of the function.
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