MONT 106N -- Identifying Patterns
The Correlation Coefficient and Its Geometric Meaning
October 21, 2009
Given a scatter plot, the correlation coefficient r
is a way to quantify how close the points (x, y) are to
lying along a single straight line. The coefficient
takes values in the range 
A value of r = -1: indicates that the points all
lie along a line of negative slope
r = 0: indicates no linear relationship at all
r = +1: indicates that the points all
lie along a line of positive slope
Values between 0 and 1 or between -1 and 0 indicate
the ``strength'' of the tendency toward a linear relation
between x and y. Here are some illustrative examples: 
 |
(1) |
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_5.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_6.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_7.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_8.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_9.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_10.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_11.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_12.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_13.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_14.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_15.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_16.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_17.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_18.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_19.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_20.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_21.gif){kind=small}
![`assign`(CloudPlot, proc (a, b, p) local xs, ys, i, theta, s, C; with(Statistics); `assign`(xs, []); `assign`(ys, []); for i to 1000 do `assign`(theta, evalf(`+`(`*`(2, `*`(Pi, `*`(r1())))))); `assign...](images/Corr_22.gif){kind=small}
 |
|
 |
 |
|
 |
 |
|
 |
 |
|
 |
 |
|
 |
 |
|
 |
 |
|
 |
 |
|
 |
 |
|
 |