Multivariate statistics are used to summarize large
collections of numbers into (1) a smaller

sets of numbers or (2) graphs of low dimensionality. To fully understand both of
these

applications it is helpful to understand several key ideas from trigonometry and
analytic

geometry.

In today’s lecture we will review two common metrics for
measuring angles: (1) degrees and

(2) radians. We will then review the most common trigonometric functions and
demonstrate

how correlations can be represented as cosines between directed line segments
(i.e.,

vectors).

**Measuring Angles**

The size of an angle can be described in various metrics
just like the height of a chair

can be described in various metrics (e.g., in inches or meters). Two common
metrics for

describing angles are degrees and radians. Many psychologists are familiar with
the degree

metric from various “nonmathemathical” contexts. Persons who have taken
trigonometry

or calculus are no doubt familiar with the notion of radians since radian
measures are

found often in higher mathematics.

**Degrees**

Mathematicians often teach the twin notions of degrees and
radians with the help of a

circle of radius 1 (recall that the radius of a circle is 1/2 of the diameter).
A circle of

radius 1 is often called the unit circle. Regardless of a circle’s radius, a
circle can be

divided into 360 degrees. Notice in Figure 1 that when speaking of degrees, a
right angle

defines an arc that subtends 90 degrees of the circle. Other degree markers are
denoted

on the circle. 1° is one-ninetieth of a right angle.

**Radian**

When s is the length of the arc subtended by ∠ a, then the
radian measure of . This

idea is illustrated in Figure 2.

Figure 3 shows the sizes of some common angles in degrees
and radians.

**Converting Degrees to Radians, and vice versa**

Notice in the above figure that 180° = π radians.

Thus

**Examples**

Fill in the Blanks

Degrees Radians

We could write a simple R function to convert degrees to radians.

Exercise: Write a simple R function to convert radians to degrees.

**What’s your sine?**

In a later lecture we will discuss matrix rotations (e.g.,
when we discuss Factor Analysis,

Principle Components Analysis, etc.) and two important trigonometric functions
of angles:

(1) cosines and the (2) sines. Here is a quick review of these (and other)
functions.

Note that these functions can be defined as the ratios of the sides of a right
triangle.

**Representing correlations as vectors in space**

Let’s return to the height and weight data from Chapter 2.

> options(width = 65)

> X1 <- matrix(c(57, 58, 60, 59, 61, 60, 62, 61, 62,

+ 63, 62, 64, 63, 65, 64, 66, 67, 66, 68, 69),

+ nrow = 20, ncol = 1)

> X2 <- matrix(c(93, 110, 99, 111, 115, 122, 110, 116,

+ 122, 128, 134, 117, 123, 129, 135, 128, 135,

+ 148, 142, 155), nrow = 20, ncol = 1)

> Z1 <- scale(X1)

> Z2 <- scale(X2)

> cor(Z1, Z2)

[,1]

[1,] 0.8670725

> plot(Z1, Z2, main = "Scatter plot of Height and Weight Data")

Notice that we have 20 pairs of data points for 2
variables: height and weight. The above

scatter plot shows how the data can be plotted in two dimensional space. The
paired

data fall within a relatively narrow ellipse, indicating that the data are well
described by

a linear correlation.

> library(ellipse)

> plot(ellipse(0.867), type = "l", main = "Scatter plot of Height and Weight
Data")

> points(Z1, Z2)

Alternatively, we can represent each variable (height and
weight) as vectors and depict the

correlation between the variables (vectors) as the cosine of the angle between
the vectors.

Specifically:

(4)

In the above scatter plot we were working with z scores.
What is the vector norm (vector

length) of a set of z scores?

**Representing Correlations as Angles
**To find the angle that corresponds to a correlation we need to use inverse
trig functions.

Specifically, the arc cosine function. Remember that when you are using these functions

the results will be reported in radians, not degrees.

What angle corresponds to the correlation between the height and weight data?