Linear Regression
Handbook Chapter
Experimental Testing


Data are collected, in scientific experiments,
to test the relationship between various
measurable quantities that are predicted by a
hypothesis, either to support or invalidate the
hypothesis.
The dependent variable is measured under
varying, but preset, values of the independent
variable.
Model Fitting
Correlation =
 If the variables have a correlation, it is
convenient to express the relationship in
the form of a mathematical equation,
known as a model or natural law.
 The mathematical equation describes a
functional relationship between the
variables.

Model Fitting
It may be relatively simple to
demonstrate the functional relationship
between the dependent and
independent variables if the relationship
is direct.
 Direct relationships (linear relationships)
can be represented by a straight line.

Linear Models
A straight line is simplest statistically
analyzable function.
 One way to express the linear
relationship between an independent
variable (x) and a dependent
variable (y) is the slope –intercept
formula: y = m x + b.

Slope –Intercept Formula
y=mx+b
y = dependent variable
x = independent variable
m = slope of the line
b = y-intercept of the line
Slope
m = slope of the line
• The slope represents the amount of tilt
(slant) the straight line has relative to
the x- and y- axes.
y – Intercept
b = y-intercept of the line
• The y-intercept is the point at which a
straight line intersects the y-axis.
Slope –Intercept Formula
If the slope and y –intercept of a straight
line is given, then the y value can be
calculated, with confidence, for every
given value of x.
 The points of the straight line have the
coordinates corresponding to the (x, y)
pairs.

Linear Regression
Since measurements inherently contain
both systematic and random errors,
data points will not fit a given equation
perfectly.
 Linear regression is a statistical method
for finding the best fitting straight line to
a set of (x, y) pairs.

The expression giving the most probable
slope of the fitted straight line is:
N
N
N
N S yixi – S yi S xi
m=
i=1
i=1
N
N S
i=1
2
xi –
i=1
N
[S xi
i=1
2
]
The expression giving the most probable
y-intercept of the fitted straight line is:
N
S
b=
i=1
N
2
xi
N
N
S yi – S xi S yixi
i=1
N
N S
i=1
i=1
2
xi –
i=1
N
[S xi
i=1
2
]