Linear regression

The best line through a set of points

Lambert-Beer's law states that there is a linear relationship between the concentration of a compound and the absorbance at a certain wavelength. This property is exploited in the use of calibration curves, which enable us to estimate the concentration in an unknown sample.
A straight line is defined by the equation


Here, b denotes the intercept of the line with the y-axis and a denotes the slope. The usual method to calculate the best line is called "Least Squares (LS)" regression. This method finds values for a and b in such a way that the sum of the squared differences of the datapoints with the fitted line is minimal. This is done by setting partial derivatives for a and b to zero.
Two examples of data sets are given below. The first is a calibration line for the determination of Cd with Atomic Absorption Spectroscopy (AAS); the second a comparison of two analytical methods, a reference method and a new one. Select one of the examples and click the "Submit" button. It is also possible to fill in your own data for x and y.

LS regression: assumptions

1. Errors (in this case: deviations from the ideal straight line) are only present in the y-variable (i.e. the absorption) and not in x (concentration). Does it make a difference which variable we put on the x-axis and which on the y-axis? To check this, select one of the two prefab datasets or enter your own data, and press the "Submit" button.

2. The errors in y are independent and normally distributed with a constant variance over the whole range of the calibration line. Several violations of this assumption can be seen in practice (make a choice and be sure that you check all three options!):

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