Statistics for Environmental Engineers

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Figure 37.3 shows two problems with this calibration. The enlarged view of lower concentrations (left panel) shows that the straight line does not fit the low concentrations. A plot of the residuals against the predicted peak value shows that the residuals are not random. The straight line underestimates at the low and very high values and overestimates in the mid-range. Furthermore, the vertical spread of the three residuals at each concentration increases as the peak height increases.


Either (1) the straight line must be replaced with a curve, (2) weighted least squares must be used to give the high concentrations less influence, or (3) weighted least squares must be used to fit a curved function.


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FIGURE 37.2 Plot of the nitrate calibration data and a straight line fitted by ordinary (unweighted) least squares.



(a) Fitted straight line in the region of low concentrations



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(b) Residuals of fitted straight line



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Nitrate Concentration (mg/L) Predicted Peak Height (1000s)


FIGURE 37.3 (a) Expanded view of the straight-line calibration shows lack of fit at low concentrations. (b) Residuals of the straight-line model show lack of fit and suggest that a quadratic or cubic calibration model should be tried. The greater spread of triplicates at higher values of peak height also suggests that weighting would be appropriate.

Nitrate Concentration (mg/L)


FIGURE 37.4 Residuals from the average at each of the 13 concentration levels used to construct the calibration curve show that the variance increases in proportion to nitrate concentrations.

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