Science & Research
Volume III - 4.4 Linear Curve Fitting
Section 4 - Basic Statistics and Presentation
|EFFECTIVE DATE: 10/01/2003||REVISED: 1-31-13|
This section deals with fitting of experimental data to a mathematical function. This situation is encountered in a variety of situations in the ORA laboratory, in particular with calibration curves. In most situations, the relationship between the variables is linear, and therefore a linear function is needed:
y = f(x) = mx + b
Where x = independent variable,
y = dependent variable,
m = calculated slope of line, and
b = calculated y-intercept of line.
The independent variable, x, is assumed to be known exactly, with no error (such as concentration, distance, time, etc.). The dependent variable, y, (instrument response for example) then depends on (is a function of) the value of x. Each value of the independent variable is assumed to follow a normal distribution and to have the same variance (i.e. square of the standard deviation). The method of linear regression (also known as linear least squares) is used to fit experimental data to a linear function (note: in certain cases, a non-linear relationship may be reduced to a linear equation by a transformation of variables; if so, the linear regression method is still applicable).
The aim of linear regression is to find the line which minimizes the sum of the squares of the deviations of individual points from that line. Once that is accomplished, the slope (m) and the intercept (b) of the ‘least squares' line is determined. It should be intuitively clear that minimizing deviations of data points from the fitted line gives the best fit of data. Given a set of data points (xi,yi), the equations used to determine the least squares parameters are:
An additional parameter, which is an indicator of the "goodness of fit" of the line to the data points, is the coefficient of determination. This coefficient denotes the strength of the linear association between x and y. The coefficient, r2, uses information on means and deviations of each data set to express variation numerically. If the two data sets correspond perfectly or exhibits no variation , a coefficient of 1 will be calculated. A coefficient of 0 indicates there is no relationship or no explanation of variation between the two data sets. Typically, for analytical work performed in the ORA laboratory, the coefficient should be very close to 1 (for example 0.999). The formula for the coefficient of determination is:
where terms have been defined previously.
The following figure illustrates several points relating to linear least squares curve fitting. Data was entered into an Excel® spreadsheet and the linear least squares regression line calculated and plotted from the data. The vertical lines indicate the distances (residuals) that are minimized in order to achieve the best fit.