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

## Uncertainty for linear calibration using check standards

Check standards provide a mechanism for calculating uncertainties The easiest method for calculating type A uncertainties for calibrated values from a calibration curve requires periodic measurements on check standards. The check standards, in this case, are artifacts at the lower, mid-point and upper ends of the calibration curve. The measurements on the check standard are made in a way that randomly samples the output of the calibration procedure.
Calculation of check standard values The check standard values are the raw measurements on the artifacts corrected by the calibration curve. The standard deviation of these values should estimate the uncertainty associated with calibrated values. The success of this method of estimating the uncertainties depends on adequate sampling of the measurement process.
Measurements corrected by a linear calibration curve As an example, consider measurements of linewidths on photomask standards, made with an optical imaging system and corrected by a linear calibration curve. The three control measurements were made on reference standards with values at the lower, mid-point, and upper end of the calibration interval.
Compute the calibration standard deviation For the linewidth data, the regression equation from the calibration experiment is $$Y = a + bX + \epsilon$$ and the estimated regression coefficients are the following. $$\hat{a} = 0.2357$$ $$\hat{b} = 0.9870$$ Next, we calculate the difference between the "predicted" $$X$$ from the regression fit and the observed $$X$$. $$W_i = \frac{(Y_i - \hat{a})}{\hat{b}} - X_i$$ Finally, we find the calibration standard deviation by calculating the standard deviation of the computed differences. $$S = \sqrt{\frac{\sum \left( W_i - \overline{W} \right)^2}{n-1}}$$ The calibration standard deviation for the linewidth data is 0.119 μm.

The calculations in this section can be completed using Dataplot code and R code. The reader can download the data as a text file.

Comparison with propagation of error The standard deviation, 0.119 μm, can be compared with a propagation of error analysis.
Other sources of uncertainty In addition to the type A uncertainty, there may be other contributors to the uncertainty such as the uncertainties of the values of the reference materials from which the calibration curve was derived. 