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
y = ax + b
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 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!):