Using the Calibration Function:

The main aim of deriving a calibration equation is obviously to determine the concentration of analyte in various samples (our ‘unknowns’) from the corresponding mesaured value of the instrument response. Mathematically, this requires an interpolation of the measurement (signal or response) through the calibration function. This obviously involves substituting the measured response (y0) into the regression line and solving for the concentration (x0).

The problem now is that we have uncertainty in both the measured value y0 and the parameters of our calibration function (i.e. the slope and intercept of the regression line.) As with any other experimental measurement, the uncertainty in the measured value can be reduced by performing replicate determinations; we therefore additionally need to take into account the number of such replicates in determining the uncertainty in the interpolated sample concentration.

Interpolating a Single Value:

If only one measured value is available, the uncertainty in the corresponding concentration will be higher than if replicate measurements had been performed. The standard error (or standard deviation) of the interpolated value sx0 is given by:

Remember that n here refers to the number of calibration points used in the regression calculations; clearly, the greater the number of calibration points, the lower the standard error in the interpolated value – and, therefore, the smaller its confidence interval.

Also note that the closer the measured value y0 is to the centroid of the calibration data, the lower the corresponding uncertainty in the interpolated value; it is therefore important to ensure that the range of concentrations used in calibrating an instrument fall either side of the anticipated range of sample concentrations!

Interpolating a Mean Value:

As mentioned above, performing replicate measurements on each unknown will reduce the uncertainty in the measured response y0, with a corresponding reduction in the uncertainty of the interpolated sample concentration. If a total of m replicates are performed (i.e. y0 is the mean of m values), then the previous equation for sx0 becomes:

Clearly, if only a single value was used (m = 1) then this reduces to the previous equation; as a result, the same comments apply regarding methods of reducing the uncertainty in the interpolated value.

Extrapolating a Value:

We noted above that the lowest uncertainties (and therefore smallest confidence intervals) for interpolated values are obtained when the measured value lies close to the centroid of the calibration data. If, however, the measured value lies outside the range of our calibration points we will need to extrapolate in order to obtain an estimate of the sample concentration. Under such circumstances, you would usually use this information to determine what additional calibration concentrations were required in order to obtain a more accurate value.

Another situation in which extrapolation is essential is a calibration technique referred to as the method of standard additions. This is commonly used in instrumental methods where the sample matrix may actually affect the measured response – this is known as an interference. Briefly, different known amounts of the analyte are added to portions of the same sample, and the measured response of each plotted against the concentration of added analyte. The assumptions inherent in this technique are that a) the calibration function remains linear at all times, and b) that any interference effect reduces the measured response by the same factor at all analyte concentrations.

When this is true, the sample concentration can be determined from regression line as the negative value of the intercept on the x (concentration) axis. (Any inherent dilutions performed in preparing the solutions for measurement must also be taken into account.) The extrapolated value xE is obviously obtained by substtuting y = 0 into the regresssion equation; the standard error in xE is calculated from the standard error of the regression as sxE:

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