r= (2)where x 1 y 1 ; x 2 y 2 ; x 3 y 3 ;....xn,ynare the co-ordinates of the plotted points, x_
and y_are the means of the xand yvalues respectively, and Âindicates sums of terms
(see standard deviation equations (1), (2) and (4), Topic B2).
The range of possible values for ris - 1 £r£+1. A value of unityindicates a
perfect linear correlationbetween xand y, all the points lying exactly on a
straight line, whilst a value of zeroindicates no linear correlation. Values may
be positive or negative depending on the slope of the calibration graph. These
alternatives are illustrated in Figure 2 (a) to(c).
Most calibration graphs have a positive slope, and correlation coefficients
frequently exceed 0.99. They are normally quoted to four decimal places. (Note
that graphs with a slight curvature may still have correlation coefficients
exceeding about 0.98 (Fig. 2(d)), hence great care must be taken before
concluding that the data shows a linear relation. Visual inspection of the plotted
points is the only way of avoiding mistakes.)When inspection of the calibration data and the value of the correlation coeffi-
cient show that there is a linear relation between the detector response and theLinear
regression
i=Ni= 1{(xi−x_
)(yi−y_
)}
i=Ni= 1(xi−x_
)^2 i=Ni= 1(yi−y_
)^2 (^1) ⁄ 2
B4 – Calibration and linear regression 43
10
8
6
4
2
0
0246810
r = +1
Detector response
Analyte mass/concentration
10
8
6
4
2
0
0246810
r = 0
Detector response
Analyte mass/concentration
10
8
6
4
2
0
0246810
r = –1
Detector response
Analyte mass/concentration
10
8
6
4
2
0
0246810
r = 0.9850
Detector response
Analyte mass/concentration
(a) (b)
(c) (d)
Fig. 2. Examples of correlation coefficients. (a) Perfect positive correlation; (b) perfect negative correlation; (c) no
correlation, and (d) curved correlation.