- Introduction
- → Introduction to Linear Regression and Calibration Curves
- The Correlation Coefficient
- → Definition and use of the R, the correlation coefficient
- Linear portions of the curves
- → Using linear portion of curves for equation estimation
- The Regression Equation
- → Calculation of a calibration curve, its error, and its use to determine an uknown concentration
- Limits of detection
- → Determing the limit of detection of an instrumental method
- Outliers
- → Evaluation of outliers with the Q-test
Every instrument used in chemical analysis can be characterised by a specific reponse function, that is an equation relating the instrument output signal (S) to the analyte concentration (C). This response function may be linear, logarithmic, exponential, or any other appropriate mathematical form, depending on the nature of the behaviour of the system being measured, and the measurement process itself. While this may be known theoretically, various factors (such as the specific analyte being measured, interference effects caused by other components of the sample matrix, or random experimental errors) require that we calibrate each instrument for the specific analyte and measurement conditions to be used in a particular experiment.
A calibration curve is an empirical equation that relates the response of a specific instrument to the concentration of a specific analyte in a specific sample matrix (the chemical background of the sample). As with the instrument response function, the calibration curve can have a number of mathematical forms, depending on the type of measurement being performed. Some common examples are listed below:
| Type | Equation |
|---|---|
| Linear (zero intercept) | S = bC |
| Linear (non-zero intercept) | S = bC + a |
| Logarithmic | S = a + blnC OR S = a + 2.303b logC |
The calibration curve is obtained by fitting an appropriate equation to a set of experimental data (calibration data) consisting of the measured responses to known concentrations of analyte. For example, in molecular absorption spectroscopy, we expect the instrument response to follow the Beer-Lambert equation, A = εbC, and so we would fit a linear equation with zero intercept to the data. On the other hand, if we were measuring electrochemical cell potentials (i.e. potentiometry) we would expect the response to be given by the Nernst equation, which is logarithmic in form. We would therefore either fit a logarithmic equation to the calibration data, or linearise the data by calculating the signal response S as 10E (where E is the cell potential).
The most common response function encountered in instrumental analytical chemistry is linear, so we require some means of determining and qualifying the best-fit straight line through our calibration data. Before discussing this in detail, however, a word of caution: even when we expect a linear instrument response function, we should not assume that the calibration data must always be linear. In fact, a moment of reflection reveals that we already know that this cannot be true. For example, stray light and polychromatic radiation cause non-linear deviations from Beer's law at higher concentrations; quenching and self-absorption can cause fluorescence intensities to start decreasing with increasing concentration; and column- or detector-overload can cause non-linearities in chromatography.
In this section, we further develop the concept of the correlation coefficient R2, followed by an example of using linear portions of non-linear curves to estimate a calibration equation. We provide an in-depth introduction to the regression equation and how it should be calculated in this course. We will learn how to determine the limit-of-detection of an instrument, then close with an introduction to the concept of outliers and how they should be treated in analytical chemistry.