preface to fifteenth edition

GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS 2.133
2.3.10 Curve Fitting
Very often in practice a relationship is found (or known) to exist between two or more variables. It
is frequently desirable to express this relationship in mathematical form by determining an equation
connecting the variables.
The first step is the collection of data showing corresponding values of the variables under
consideration. From a scatter diagram, a plot of Y (ordinate) versus X (abscissa), it is often possible
to visualize a smooth curve approximating the data. For purposes of reference, several types of
approximating curves and their equations are listed. All letters other than X and Y represent constants.
1. Y a 0 a 1 X
Straight line
2. Y a 0 a 1 X a 2 X 2
Parabola or quadratic curve
3. Y a 0 a 1 X a 2 X 2 a 3 X 3
Cubic curve
4. Y a 0 a 1 X a 2 · · · a n X n nth degree curve
As other possible equations (among many) used in practice, these may be mentioned:
o 5. Y (a 0 a 1 X) 1 r 1/Y a 0 a 1 X
6. Y ab X or log Y log a (log b) X
7. Y aX b or log Y log a b log X
8. Y ab X g
9. Y aX n g
Hyperbola
Exponential curve
Geometric curve
Modified exponential curve
Modified geometric curve
When we draw a scatter plot of all X versus Y data, we see that some sort of shape can be
described by the data points. From the scatter plot we can take a basic guess as to which type of
curve will best describe the X9Y relationship. To aid in the decision process, it is helpful to obtain
scatter plots of transformed variables. For example, if a scatter plot of log Y versus X shows a linear
relationship, the equation has the form of number 6 above, while if log Y versus log X shows a linear
relationship, the equation has the form of number 7. To facilitate this we frequently employ special
graph paper for which one or both scales are calibrated logarithmically. These are referred to as
semilog or log-log graph paper, respectively.
2.3.10.1 The Least Squares or Best-fit Line. The simplest type of approximating curve is a
straight line, the equation of which can be written as in form number 1 above. It is customary to
employ the above definition when X is the independent variable and Y is the dependent variable.
To avoid individual judgment in constructing any approximating curve to fit sets of data, it is
necessary to agree on a definition of a best-fit line. One could construct what would be considered
the best-fit line through the plotted pairs of data points. For a given value of X 1 , there will be a
difference D 1 between the value Y 1 and the constituent value Ŷ as determined by the calibration
model. Since we are assuming that all the errors are in Y, we are seeking the best-fit line that
minimizes the deviations in the Y direction between the experimental points and the calculated line.
This condition will be met when the sum of squares for the differences, called residuals (or the sum
of squares due to error),
N
ˆ 2 2 2 2
i i 1 2 N
i1
(Y Y ) (D D ··· D )
is the least possible value when compared to all other possible lines fitted to that data. If the sum
of squares for residuals is equal to zero, the calibration line is a perfect fit to the data. With a