Page 14 - Industrial Process Plant Construction Estimating and Man Hour Analysis

Introduction xxix

Method of least squares
Least squares line

The least-squares line approximating the set of points (x1, y1), (x2, y2) …
(xn, yn) has the equation
Y ¼ bx + a

where b¼the slope of the line and a¼y-intercept.
The best fit line for the (x 1 ,y 1 ), (x 2 ,y 2 ) … (x n ,y n ) is given by

ð
y ybar ¼ bx xbarÞ
where the slope is
X X 2
ð
b ¼ ð xi xbarÞ yi ybarÞ= ð xi xbarÞ
and the y-intercept is
a ¼ ybar bxbar

Correlation
Correlation coefficient
(1) Positive correlation: If x and y have a strong positive linear correlation, r is
close to +1. An r value exactly +1 indicates a perfect positive fit. Positive
values indicate a relationship between x and y variables such that as values
for x increase, values for y also increase.
(2) A perfect correlation of + or 1 occurs only when all the data points all lie
exactly on a straight line. If r¼+1, the slope line is positive. If r¼ 1, the
slope line is negative.
(3) A correlation greater than 0.8 is generally described as strong, whereas a
correlation less than 0.5 is generally described as weak.

2
Coefficient of determination, r or R 2
(1) Itisusefulbecauseitgivestheproportionofthevarianceofonevariablethatis
predictable from the other variable. It is a measure that allows us to determine
how certain one can be of making predications from a certain model or graph.
(2) The coefficient of determination is the ratio of the explained variation to
the total variation.
2
(3) The coefficient of determination is such that (0<¼r <¼1) and denotes
the strength of the linear association between x and y.
(4) The coefficient of determination is a measure of how well the regression
line represents the data. If the regression line passes exactly through every
point on the scatter plot, it explains all of the variation. The further the line
is away from the points, the less it is able to explain the variation.