Connexions module: m17092 1
Linear Regression and Correlation:
The Correlation Coefficient
∗
Susan Dean
Barbara Illowsky, Ph.D.
This work is produced by The Connexions Project and licensed under the
Creative Commons Attribution License
†
Abstract
This module provides an overview of Linear Regression and Correlation: The Correlation Coecient
as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.
Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a
good predictor? Use the correlation coecient as another indicator (besides the scatterplot) of the strength
of the relationship between
x
and
y
. The correlation coecient,
r
, is dened as:
r=n·Σx·y−(Σx)·(Σy)
q[n·Σx2−(Σx)2]·[n·Σy2−(Σy)2]
where
n
= the number of data points.
If you suspect a linear relationship between
x
and
y
, then
r
can measure how strong the linear relationship
is. One property of
r
is that
−1≤r≤1
. If
r= 1
, there is perfect positive correlation. If
r=−1
, there is
perfect negative correlation. In both these cases, the original data points lie on a straight line. Of course, in
the real world, this will not generally happen.
The formula for
r
looks formidable. However, many calculators and any regression and correlation
computer program can calculate
r
. The sign of
r
is the same as the slope,
b
, of the best t line.
Glossary
Denition 1: Coecient of Correlation
A measure developed by Karl Pearson (early 1900s) that gives the strength of association between
the independent variable and the dependent variable. The formula is:
r=nP
xy
−(Px) (Py)
rhnPx2−(Px)2ihnPy2−(Py)2i
,
(1)
where n is the number of data points. The coecient cannot be more then 1 and less then -1. The
closer the coecient is to
±1
, the stronger the evidence of a signicant linear relationship between
x
and
y
.
∗
Version 1.6: Jan 17, 2009 12:04 pm US/Central
†
http://creativecommons.org/licenses/by/2.0/
http://cnx.org/content/m17092/1.6/
Source URL: http://cnx.org/content/col10522/latest/
Saylor URL: http://www.saylor.org/courses/ma121/
Attributed to: Barbara Illowsky and Susan Dean
Saylor.org
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