2.
Measurement Process Characterization
2.3.
Calibration
2.3.6.
Instrument calibration over a regime
2.3.6.7.
Uncertainties of calibrated values
2.3.6.7.3.
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Comparison of check standard analysis and propagation of error
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Propagation of error for the linear calibration
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The analysis of uncertainty for calibrated values from a linear
calibration line can be addressed using propagation of error. On
the previous page, the uncertainty was estimated
from check standard values.
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Estimates from calibration data
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The calibration data consist of 40 measurements with an optical
imaging system on 10 linewidth artifacts. A linear fit to the data
gives a
calibration curve with the following estimates for the intercept,
a, and the
slope, b:
Parameter Estimate Std. Error t-value Pr(>|t|)
a 0.2357623 0.02430034 9.702014 7.860745e-12
b 0.9870377 0.00344058 286.881171 5.354121e-65
with the following covariance matrix.
a b
a 5.905067e-04 -7.649453e-05
b -7.649453e-05 1.183759e-05
The results shown above can be generated with
R code.
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Propagation of error
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The propagation of error is performed for the equation
so that the squared uncertainty of a calibrated value, X', is
where
The uncertainty of the calibrated value, X',
is dependent on the value of the instrument reponse Y'.
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Graph showing standard deviation of calibrated value X' plotted as
a function of instrument response Y' for a linear calibration
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Comparison of check standard analysis and propagation of error
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Comparison of the analysis of check standard
data, which gives a standard deviation of 0.119 µm, and propagation
of error, which gives a maximum standard deviation of 0.068 µm,
suggests that the propagation of error may underestimate the type A
uncertainty. The check standard measurements are undoubtedly sampling
some sources of variability that do not appear in the formal
propagation of error formula.
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