Can you relate? Correlation and simple linear regression - Improving the User Experience through Practical Data Analytics - page 211

The top left plot has all the data points exactly on a straight line, and the straight
line has a positive slope. With real data, we would never see such an exact relation-
ship; we present it only for illustration purposes. The value of r is +1. This value is
the maximum value r can take on. And, the fact that the slope of the line is positive
ensures that the sign of r is positive (i.e., +). Now, if you look at the top right plot, the
relationship is also perfect—the data values are all right on the line. But, the line is
downward sloping. The value of r, correspondingly, comes out −1. The “1” indicates
that the it to a straight line is perfect, while the “−” sign indicates, indeed, that the
line is downward sloping.
This illustrates a useful property of r; while the numerical value of r tells us
about how well the data it a straight line (referred to as the [relative] strength of the
relationship), its sign tells us the direction of the relationship, if any. A + value of
r indicates a positive relationship—as one variable goes up, the other variable also
goes up; as one variable goes down, the other variable also goes down. (You may also
see this relationship described as a “positive” or “direct” correlation.)
A − [minus] value of r indicates an “inverse” relationship—as one variable
goes up, the other goes down, and vice versa. (You may also see this relationship
described as a “negative or “indirect” correlation.) A value of zero for r (or, as a
practical matter, right near it) indicates that the two variables are unrelated linearly,
and this is illustrated in the bottom left scatter plot; you can see that there is no
indication at all of a relationship between the variables. The best-itting line 1 is
horizontal—indicating zero slope, the equivalent of no linear relationship. When-
ever the best-itting line has zero slope (i.e., is horizontal), the value of r is zero,
and vice versa.
If we look at the middle left plot, the data values are not right on the line that
goes through the values, but it will, intuitively, give you a pretty accurate value
of Y from inputting the value of X into the equation of the line. The line might
be, for example, Y = 0.2 + 1.07X, and when you plug in the value of X, the Y that
comes out will be pretty close to the actual Y value for most all the data values.
Without speciic data, we cannot provide the exact value of r, but it might be in
the neighborhood of +0.8; the line clearly has a positive slope. (The “r = 0.8”
listed for this plot in Figure 9.1 is just a rough estimate by the authors when
looking at the data.)
Let's compare this plot with the middle right plot. It should be clear to the
reader that the it to a straight line is not as good as the plot on the left—the data
values are not as tightly clustered around the line as in the middle left plot—and
also, the line best-itting the data is downward sloping. Since the it is less good,
the value of r is lower, say, 0.65 (again, an estimate based on the authors' view at
the data). Also, it is negative, relecting the negative slope of the straight line best-
itting the data.
The inal plot among the six is the bottom right plot. The purpose of that plot
is to dramatically illustrate a key point that r is measuring the relative strength of a
1 We will more precisely deine “best-itting line” later in the chapter.