Calibration workshop

Calibration

Calibration

The quality of an analysis depends on the quality of the calibration. The quality of a calibration depends on attention to detail.

Selection of calibration samples:
- number
- concentration ranges
- families
Preparation of calibration samples
Choice of calibration function
Precision during calibration
Optimisation of calibration curve

In optical emission spectrometry a typical calibration function has the form:

where C_i is the content (or concentration) of element i; a_i to c_i and d_j are fitting parameters; I_i is the signal from element i; and I_j is the signal from an interfering element j, where there may be up to N interfering elements. -a_i is called the BEC (background equivalent concentration).

The calibration function given here is linear (in its dependence on the fitting parameters) and second order (the highest power of I_i). Sometimes higher order functions or non-linear functions are used. Several simplifications are also possible. If c_i = 0 then the calibration function is first order. If all d_j = 0 then there are no interference corrections.

To determine the best values for the fitting parameters it is usual to find the minimum of a least-squares function:

where w_i are the weights given to each calibration point i, C_i is the known content and and C^{^}_i is the content estimated using the calibration function. By minimising the least-squares function we are endeavouring to make the estimated contents as close as possible to the known contents. A higher weight means that a point has a bigger effect. Generally, it is necessary, in optical emission spectrometry, to give different weights to different points, because the precision of measurement may vary from point to point or the accuracy of the known concentration may vary from sample to sample. The software will normally set appropriate weights automatically, but occasionally it is necessary to intervene manually.

Several parameters are available to determine the goodness of fit. Unweighted equations are shown here for simplicity though, of course, weighted functions should be used.

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Correlation coefficient:

where r may vary from -1 to 1, with ±1 being a perfect correlation with y increasing (+) or decreasing (-) with x, and 0 being no correlation. Special care is required when using this parameter because it is very dependent on n, the number of calibration samples. For example, an r value of 0.96 with 5 points is comparable to an r value of 0.56 with 20 points. It is therefore better to use confidence levels or only compare r values for the same number of points.

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Standard error of estimate:

The standard error of estimate describes the "average" deviation of a data point from the fitted calibration curve. It is given by [Standard error of estimate]

the following equation. Where n is the number of calibration samples.

And the relative standard error of estimate: These important parameters indicate the accuracy possible near the centre of the calibration curve.

Once the calibration has been done, and before the first analysis, it is time to optimise the calibration curve.

The first step is to choose the order and to select any interfering elements. Generally it is better to use no interference corrections unless these are known and there is an obvious and significant improvement in the statistical parameters.

The BEC should be small and positive. If necessary adjust the weights to achieve this.
If the standard error of estimate is too high, do one or more of the following:

increase the number of calibration samples
reduce the weight of outliers
repeat the calibration with more measurements on each sample
reduce the concentration ranges or the number of families of different samples
change the calibration function or mode.

Finally the success of the calibration - the order selected, any interference corrections, etc. - should be checked with one or more validation samples, i.e. samples of known composition not included in the calibration.

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