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Correlation and Trends
Author: John M. Cimbala, Penn State University
Latest revision: 02 February 2007
Introduction
•
In engineering analysis, we often want to fit a trend line or curve to a set of
xy
data.
•
Consider a set of
n
measurements of some variable
y
as a function of another variable
x
.
•
Typically,
y
is some measured
output
as a function of some known
input
,
x
.
•
In general, in such a set of measurements, there may be:
o
Some
scatter
(precision error or random error).
o
A
trend
−
in spite of the scatter,
y
may show an
overall increase
with
x
, or perhaps an
overall decrease
with
x
.
•
The
linear correlation coefficient
is used to determine
if
there is a trend.
•
If there
is
a trend,
regression analysis
is used to find an equation for
y
as a function of
x
which provides the
best fit
to the data.
Correlation coefficient
•
The
linear correlation coefficient
r
xy
is defined as
()
1
22
11
in
ii
i
xy
xxyy
r
x
xy
y
=
=
==
−−
=
∑
∑∑
.
•
In the equation above, the
mean value of x
and the
mean value of y
are defined in the usual manner as
1
1
i
i
x
x
n
=
=
=
∑
and
1
1
i
i
yy
n
=
=
=
∑
.
•
Some observations about the linear correlation coefficient are worth noting:
o
By definition,
r
xy
must always lie between
−
1 and 1, i.e.,
xy
r
−
≤≤
.
o
The linear correlation coefficient is always
nondimensional
, regardless of the dimensions of
x
and
y
.
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