# Statistics for Environmental Engineers

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Calibration curve a is the most familiar (see Chapter 36) but there is no theoretical reason for assuming that a straight line will be the correct calibration model. There are factors inherent in some measurement processes that make curve b much more likely than curve a (e.g., ICP instruments). Also, there are some instruments for which a straight-line calibration is not expected (e.g., HPLC instruments). Therefore, the analyst must know how to recognize and how to treat calibration data for these four patterns.

The analyst must decide whether the data have constant variance and if not, what weights should be assigned to each у value. In addition, the analyst is also trying to decide the form of the fitted function (straight line or some curved function). These decisions will have a profound effect on confidence intervals and prediction intervals. The confidence intervals of parameter estimates will be wrong if weighted regression has not been used when there is nonconstant variance. Also, prediction intervals for the model line and specimen concentrations will be wrong unless proper weighting and model form have been used.

### Case Study: Ion Chromatograph Calibration

The nitrate calibration data in Table 37.1 were made on a Dionex-120 ion chromatograph. There are triplicate measurements at 13 concentration levels covering a range from 0 to 40 mg/L nitrate.

Suppose we assume that the calibration model is a straight line and fit it using ordinary least squares to obtain:

у = -3246 + 8795x

FIGURE 37.1 Four possible calibration curves: (a) linear with random measurement errors of equal magnitude at all standard concentrations, (b) linear with random measurement errors that are larger at high standard concentrations, (c) curvilinear with random measurement errors of equal magnitude at all standard concentrations, and (d) curvilinear with random measurement errors that are larger at high standard concentrations.

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