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Correlation and Simple Linear Regression
There are many different types of “correlation coefﬁcients,” but the one we will use
in this topic is the Pearson product-moment correlation which we will call: r .
6.1 What is a “Correlation?”
Basically, a correlation is a number between
1 that summarizes the
relationship between two variables, which we will call X and Y.
A correlation can be either positive or negative. A positive correlation means
that as X increases, Y increases. A negative correlation means that as X increases, Y
decreases . In statistics books, this part of the relationship is called the direction of
the relationship (i.e., it is either positive or negative).
The correlation also tells us the magnitude of the relationship between X and Y.
As the correlation approaches closer to
1, we say that the relationship is strong
and positive .
As the correlation approaches closer to ─ 1, we say that the relationship is strong
and negative .
A zero correlation means that there is no relationship between X and Y. This
means that neither X nor Y can be used as a predictor of the other.
A good way to understand what a correlation means is to see a “picture” of the
scatterplot of points produced in a chart by the data points. Let’s suppose that you
want to know if variable X can be used to predict variable Y. We will place the
predictor variable X on the x-axis (the horizontal axis of a chart) and the criterion
variable Y on the y-axis (the vertical axis of a chart). Suppose, further, that you have
collected data given in the scatterplots below (see Fig. 6.1 through Fig. 6.6 ).
Fig. 6.1 shows the scatterplot for a perfect positive correlation of r
means that you can perfectly predict each y-value from each x-value because the
data points move “upward-and-to-the-right” along a perfectly-ﬁtting straight line
(see Fig. 6.1 )
T.J. Quirk et al., Excel 2007 for Biological and Life Sciences Statistics ,
DOI 10.1007/978-1-4614-6003-9_6, # Springer Science+Business Media New York 2013
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