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Suppose we are interested in predicting concentration from a measured value of y = 30. For m = 1,
= 0.24. would be reduced to 0.17. This is
s x 0 If we had m = 4 measurements that average y o = 30, s x 0
a considerable gain in precision.
The effect of increasing the number of calibration points (n) is more complex because changing n
2
also changes the value of the t and the χ statistics that are used to compute the confidence intervals.
The motivation to reduce n is to limit the amount of work involved with preparing standards at many
concentrations. It is clear, however, that large values of n will decrease the confidence interval by making
the term 1/n small, while simultaneously decreasing the corresponding t statistic. In most calibrations,
six or so calibration points will be adequate and if extra precision is needed, it can be gained most
efficiently by making repeated measurements at some or all calibration levels.
The calibration model may not be a straight line, and the variance of the signal may increase as the
values of x and y increase. This is the case study in Chapter 37 which deals with weighted regression.
There are other interesting problems associated with calibrations. In some analytical applications (e.g.,
photometric titrations), it is necessary to locate the intersection of two regression lines. In the standard
addition method, the concentration is estimated by extrapolating a straight line down to the abscissa.
References at the end of the chapter provide guidance on these problems.
2 < 2
A special case is where there are errors in both x and y. Errors in x can be ignored if σ x < σ y . Carroll
and Spiegelman (1986) examine this criteria in some detail. The effect of errors in x is to pull the regression
line down so the estimated slope is less than would be estimated if errors in x were taken into account.
In going from y to x, this could badly overestimate x. This emphasizes the importance of using accurate
standards in preparing calibration curves.
Values of the correlation coefficient r() and the coefficient of determination R ) are often cited as
(
2
evidence that the calibration relation is strong and useful, or that the calibration is in fact a straight line.
2
An R value near 1.00 does not prove these points and in the context of calibration curves it has little
2
meaning of any kind. Values of R = 0.99+ are to be expected in calibration curves. If the relation between
standard and instrumental response is not clean and strong, there is simply no useful measurement
2
method. Second, the value of R value can be increased without increasing the precision of the mea-
surements or of the predictions. This is done simply by expanding the range of concentrations covered
2
by the standards. Third, R can be large (>0.98) although the curve deviates slightly from the linear.
2
The coefficient of determination (R ) is discussed in Chapter 39.

References
Bailey, C. J., E. A. Cox, and J. A. Springer (1978). “High Pressure Liquid Chromatographic Determination
of the Immediate/Side Reaction Products in FD&C Red No. 2 and FD&C Yellow No. 5: Statistical
Analysis of Instrument Response,” J. Assoc. Off. Anal. Chem., 61, 1404–1414.
Carroll, R. J. and C. H. Spiegelman (1986). “The Effect of Ignoring Small Measurement Errors in Precision
Instrument Calibration,” J. Quality Tech., 18(3), 170–173.
Danzer, K. and L. A. Currie (1998). “Guidelines for Calibration in Analytical Chemistry,” Pure Appl. Chem.,
70, 993–1014.
Hunt, D. T. E. and A. L. Wilson (1986). The Chemical Analysis of Water, 2nd ed., London, The Royal Society
of Chemistry.
Hunter, J. S. (1981). “Calibration and the Straight Line: Current Statistical Practice,” J. Assoc. Off. Anal.
Chem., 64(3), 574–583.
Leiberman, G. J., R. G. Miller, Jr., and M. A. Hamilton (1967). “Unlimited Simultaneous Discrimination
Intervals in Regression,” Biometrika, 54, 133–145.
Mandel, J. (1964). The Statistical Analysis of Experimental Data, New York, Interscience Publishers.
Miller, J. C. and J. N. Miller (1984). Statistics for Analytical Chemistry, Chichester, England, Ellis Horwood Ltd.
Sharaf, M. A., D. L. Illman, and B. Kowalski (1986). Chemometrics, New York, John Wiley.
Wynn, H. P. and P. Bloomfield (1971). “Simultaneous Confidence Bands in Regression Analysis,” J. Royal
Stat. Soc. B, 33, 202–217.

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