preface to fifteenth edition

2.134 SECTION 2
mathematical treatment known as linear regression, one can find the “best” straight line through
these real world points by minimizing the residuals.
This calibration model for the best-fit fit line requires that the line pass through the “centroid”
of the points (X, Y). It can be shown that:
i
i
i
(X X) (Yi
Y)
b
i
(X X) 2
(2.17)
a Y bX (2.18)
The line thus calculated is known as the line of regression of Y on X, that is, the line indicating how
Y varies when X is set tochosen values.
If X is the dependent variable, the definition is modified by considering horizontal instead of
vertical deviations. In general these two definitions lead to different least square curves.
Example 13 The following data were recorded for the potential E of an electrode, measured
against the saturated calomel electrode, as a function of concentration C (moles liter 1 ).
log C E,mV log C E,mV
1.00 106 2.10 174
1.10 115 2.20 182
1.20 121 2.40 187
1.50 139 2.70 211
1.70 153 2.90 220
1.90 158 3.00 226
Fit the best straight line tothese data; X i represents log C, and Y i represents E. We will perform
the calculation manually, using the following tabular lay-out.
X i Y i (X X) i
(Xi X) 2 (Yi Y) (Xi
X)(Yi
Y)
1.00 106 0.975 0.951 60 58.5
1.10 115 0.875 0.766 51 44.6
1.20 121 0.775 0.600 45 34.9
1.50 139 0.475 0.226 27 12.8
1.70 153 0.275 0.076 13 3.6
1.90 158 0.075 0.006 8 0.6
2.10 174 0.125 0.016 8 1.0
2.20 182 0.225 0.051 16 3.6
2.40 187 0.425 0.181 21 8.9
2.70 211 0.725 0.526 45 32.6
2.90 220 0.925 0.856 54 50.0
3.00 226 1.025 1.051 60 61.5
X i 23.7 Y i 1992 0 5.306 0 312.6
X 1.975 Y 166