Distinguish between a linear and a nonlinear relationship

Identify positive and negative associations from a scatter plot

Measures of central tendency, variability, and
spread summarize a single variable by providing important information
about its distribution. Often, more than one variable is collected
on each individual. For example, in large health studies of
populations it is common to obtain variables such as age, sex,
height, weight, blood pressure, and total cholesterol on each
individual. Economic studies may be interested in, among other
things, personal income and years of education. As a third example,
most university admissions committees ask for an applicant's
high school grade point average and standardized admission test
scores (e.g., SAT). In this chapter we consider bivariate data,
which for now consists of two quantitative
variables for each individual. Our first interest is in
summarizing such data in a way that is analogous to summarizing
univariate (single variable) data.

By way of illustration, let's consider something
with which we are all familiar: age. Let’s begin by asking
if people tend to marry other people of about the same age.
Our experience tells us "yes," but how good is the correspondence?
One way to address the question is to look at pairs of ages
for a sample of married couples. Table 1 below shows the ages
of 10 married couples. Going across the columns we see that,
yes, husbands and wives tend to be of about the same age, with
men having a tendency to be slightly older than their wives.
This is no big surprise, but at least the data bear out our
experiences, which is not always the case.

Table 1. Sample of spousal ages of 10 White American Couples.

Husband

36

72

37

36

51

50

47

50

37

41

Wife

35

67

33

35

50

46

47

42

36

41

The pairs of ages in Table 1 are from a dataset
consisting of 282 pairs of spousal ages, too many to make sense
of from a table. What we need is a way to summarize the 282 pairs
of ages. We know that each variable can be summarized by a histogram
(see Figure 1) and by a mean and standard deviation (See Table
2).

Figure 1. Histograms of spousal ages.

Table 2. Means and standard deviations of spousal ages.

Mean

Standard Deviation

Husbands

49

11

Wives

47

11

Each distribution is fairly skewed with a long right
tail. From Table 1 we see that not all husbands are older than
their wives and it is important to see that this fact is lost
when we separate the variables. That is, even though we provide
summary statistics on each variable, the pairing within couple
is lost by separating the variables. We cannot say, for example,
based on the means alone what percentage of couples has younger
husbands than wives. We have to count across pairs to find this
out. Only by maintaining the pairing can meaningful answers
be found about couples per se. Another example of information
not available from the separate descriptions of husbands and
wives' ages is the mean age of husbands with wives of a certain
age. For instance, what is the average age of husbands with
45-year-old wives? Finally, we do not know the relationship
between the husband's age and the wife's age.

We can learn much more by displaying the bivariate
data in a graphical form that maintains the pairing. Figure 2
shows a scatter
plot of the paired ages. The x-axis represents the age of
the husband and the y-axis the age of the wife.

Figure 2. Scatter plot showing wife's age as a function of husband's age.

There are two important characteristics of the data
revealed by Figure 2. First, it is clear that there is a strong
relationship between the husband's age and the wife's age: the
older the husband, the older the wife. When one variable (Y) increases
with the second variable (X), we say that X and Y have a positive
association. Conversely, when Y decreases as X increases,
we say that they have a negative
association.

Second, the points cluster along a straight line.
When this occurs, the relationship is called a linear
relationship.

Figure 3 shows a scatter plot of Arm Strength and
Grip Strength from 149 individuals working in physically demanding
jobs including electricians, construction and maintenance workers,
and auto mechanics. Not surprisingly, the stronger someone's
grip, the stronger their arm tends to be. There is therefore
a positive association between these variables. Although the
points cluster along a line, they are not clustered quite as
closely as they are for the scatter plot of spousal age.

Figure 3. Scatter plot of Grip Strength and Arm Strength.

Not all scatter plots show linear relationships.
Figure 4 shows the results of an experiment conducted by Galileo
on projectile motion. In the experiment, Galileo rolled balls
down an incline and measured how far they traveled as a function
of the release height. It is clear from Figure 4 that the relationship
between "Release Height" and "Distance Traveled"
is not described well by a straight line: If you drew a line
connecting the lowest point and the highest point, all of the
remaining points would be above the line. The data are better
fit by a parabola.

Figure 4. Galileo's data showing a non-linear relationship.

Scatter plots that show linear relationships between
variables can differ in several ways including the slope of the
line about which they cluster and how tightly the points cluster
about the line. A statistical measure of the strength of the relationship
between two quantitative variables that takes these factors into account is the
subject of the section "Values of Pearson's Correlation."