Page 173 - Vogel's TEXTBOOK OF QUANTITATIVE CHEMICAL ANALYSIS

LINEAR REGRESSION 4.17

Substituting the above values in equation (6), then

Hence, there is a very strong indication that a linear relation exists between
fluorescence intensity and concentration (over the given range of concentration).
It must be noted, however, that a value of r close to either + 1 or - 1 does
not necessarily confirm that there is a linear relationship between the variables.
It is sound practice first to plot the calibration curve on graph paper and
ascertain by visual inspection if the data points could be described by a straight
line or whether they may fit a smooth curve.
The significance of the value of r is determined from a set of tables
(Appendix 15). Consider the following example using five data (x, y,) points:
From the table the value of r at 5 per cent significance value is 0.878. If the
value of r is greater than 0.878 or less than -0.878 (if there is negative
correlation), then the chance that this value could have occurred from random
data points is less than 5 per cent. The conclusion can, therefore, be drawn that
it is likely that x, and y, are linearly related. With the value of r = 0.998,
obtained in the example given above there is confirmation of the statement that
the linear relation between fluorescence intensity and concentration is highly
likely.

4.17 LINEAR REGRESSION
Once a linear relationship has been shown to have a high probability by the
value of the correlation coefficient (r), then the best straight line through the
data points has to be estimated. This can often be done by visual inspection of
the calibration graph but in many cases it is far better practice to evaluate the
best straight line by linear regression (the method of least squares).
The equation of a straight line is

where y, the dependent variable, is plotted as a result of changing x, the
independent variable. For example, a calibration curve (Section 21.16) in atomic
absorption spectroscopy would be obtained from the measured values of
absorbance (y-axis) which are determined by using known concentrations of
metal standards (x-axis).
To obtain the regression line 'y on x', the slope of the line (a) and the intercept
on the y-axis (b) are given by the following equations.

and b = j - aX (8)
where X is the mean of all values of x, and j is the mean of al1 values of y,.
Example 10. Calculate by the least squares method the equation of the best
straight line for the calibration curve given in the previous example.
From Example 9 the following values have been determined.