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36

Calibration

KEY WORDS calibration, confidence band, confidence interval, inverse prediction, least squares,
measurement error, precision, prediction, straight line, regression, Working-Hotelling.

Instrumental methods of chemical analysis (and some “wet” analyses) are based on being able to
construct a calibration curve that will translate the instrument’s highly precise physical signal (light
absorption, peak height, voltage, etc.) into an estimate of the true concentration of the specimen being
tested. The usual procedure to construct a calibration curve is for the analyst to take a series of
measurements on prepared specimens (at least three and usually more) for which the concentration is
known. It is important that these calibration standards cover the entire range of concentrations required
in subsequent analyses. Concentrations of test specimens are to be determined by interpolation and not
by extrapolation.
The calibration standards are measured under the same conditions as those that subsequently will be
used to test the unknown specimens. Also, a blank specimen should be included in the calibration curve.
The blank contains no deliberately added analyte, but it does contain the same solvents, reagents, etc.
as the other calibration standards and test specimens, and it is subjected to the same sequence of
preparatory steps. The instrumental signal given by the blank sample often will not be zero. It is, however,
subject to error and it should be treated in the same manner as any of the other points on the calibration
plot. It is wrong to subtract the blank value from the other standard values before plotting the calibration
curve.
Let us assume that the best calibration line is to be determined and that the measured concentrations
will be accompanied by a statement of their precision. This raises several statistical questions (Miller
and Miller, 1984):

1. Does the plot of the calibration curve appear to be well described by a straight line? If it
appears to be curved, what is the mathematical form of the curve?
2. Bearing in mind that each point on the calibration curve is subject to error, what is the best
straight line (or curved line) through these points?
3. If the calibration curve is actually linear, what are the estimated errors and confidence limits
for the slope and intercept of the line?
4. When the calibration curve is used for the analysis of a test specimen, what are the error and
confidence limits for the determined concentration?

The calibration curve is always plotted with the instrument response on the vertical (y) axis and the
standard concentrations on the horizontal (x) axis. This is because the standard statistical procedures
used to fit a calibration line to the data and to interpret the precision of estimated concentrations assume
that the y values contain experimental errors but the x values are error-free. Another assumption implicit
in the usual analysis of the calibration data is that the magnitude of the errors in the y values is independent
of the analyte concentrations. Common sense and experience indicate that measurement error is often
proportional to analyte concentration. This can be checked once data are available and the model has
been fitted. If the assumption is violated, fit the calibration curve using weighted least squares as explained
in Chapter 37.

© 2002 By CRC Press LLC