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Introduction to Bivariate Data

Module by: David Lane. E-mail the author

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. It helps to discuss something familiar since knowing the subject matter goes a long way in making judgments about statistical results. 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.

 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).

Table 2: Means and standard deviations of spousal ages.
Mean Standard Deviation
Husband 49 11
Wife 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 have 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.

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 (yy) increases with the second variable (vv), we say that xx and yy have a positive association. Conversely, when yy decreases as xx 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 scatterplot 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.

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 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.

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 variables that takes these factors into account is the subject of the next section.

Glossary

Quantitative Variables:
Variables that have are measured on a numeric or quantitative scale. Ordinal, interval and ratio scales are quantitative. A country's population, a person's shoe size, or a car's speed are all quantitative variables. Variables that are not quantitative are known as qualitative variables.
Histogram:
A histogram is a graphical representation of a distribution. It partitions the variable on the x-axis into various contiguous class intervals of (usually) equal widths. The heights of the bars represent the class frequencies. See also: Sturgis's Rule
Sturgis's Rule:
One method of determining the number of classes for a histogram, Sturgis's Rule is to take 1+log2 N 1 2 N classes, rounded to the nearest integer.
Bivariate:
Bivariate data is data for which there are two variables for each observation. As an example, the following bivariate data show the ages of husbands and wives of 10 married couples.
 Husband 36 72 37 36 51 50 47 50 37 41 Wife 35 67 33 35 50 46 47 42 36 41
Scatter Plot:
A scatter plot of two variables shows the values of one variable on the Y axis and the values of the other variable on the X axis. Scatter plots are well suited for revealing the relationship between two variables. The scatter plot shown in Figure 4 illustrates data from one of Galileo's classic experiments in which he observed the distance traveled balls traveled after being dropped on a incline as a function of their release height.
Positive Association:
There is a positive association between variables XX and YY if smaller values of XX are associated with smaller values of YY and larger values of XX are assoicated with larger values of YY.
Negative Association:
There is a negative association between variables XX and YY if smaller values of XX are associated with larger values of YY and larger values of XX are assoicated with smaller values of YY.
Linear Relationship:
If the relationship between two variables is a perfect linear relationship, then a scatterplot of the points will fall on a straight line as shown in Figure 6. With real data, there is almost never a perfect linear relationship between two variables. The more the points tend to fall along a straight line the stronger the linear relationship. Figure 2 shows two variables (husband's age and wife's age) that have a strong but not a perfect linear relationship.

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