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

2.3.6.7.2.

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.

Run software macro for computing the standard deviation

Dataplot commands for computing the standard deviation from the control data are:

read linewid2.dat day position x y
let b0 = 0.2817
let b1 = 0.9767
let w = ((y - b0)/b1) - x 
let sdcal = standard deviation w

Standard deviation of calibrated values

Dataplot returns the following standard deviation

THE COMPUTED VALUE OF THE CONSTANT SDCAL  =  0.62036246E-01 

Comparison with propagation of error

The standard deviation, 0.062 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.