was focused on only univariate statistics because we were interested in studying a single characteristic of a subject. In all the examples we considered, the variable of interest was either qualitative or quantitative. We now study cases involving two variables; this means examining two characteristics of a subject. The two variables of interest could be either qualitative or quantitative, but here we will consider only variables that are quantitative.
For the consideration of two variables simultaneously, the data obtained are known as bivariate data. In the examination of bivariate data, the first question is whether there is any association of interest between the two variables. One effective way to determine whether there is such an association is to prepare a graph by plotting one variable along the horizontal scale (x‐axis) and the second variable along the vertical scale (y‐axis). Each pair of observations
Example 2.9.1 (Cholesterol level and systolic blood pressure) The cholesterol level and the systolic blood pressure of 10 randomly selected US males in the age group 40–50 years are given in Table 2.1. Construct a scatter plot of this data and determine if there is any association between the cholesterol levels and systolic blood pressures.
Solution: Figure 2.9.1 shows the scatter plot of the data in Table 2.1. This scatter plot clearly indicates that there is a fairly strong upward linear trend. Also, if a straight line is drawn through the data points, then it can be seen that the data points are concentrated around the straight line within a narrow band. The upward trend indicates a positive association between the two variables, while the width of the band indicates the strength of the association, which in this case is quite strong. As the association between the two variables gets stronger and stronger, the band enclosing the plotted points becomes narrower and narrower. A downward trend indicates a negative association between the two variables.
A numerical measure of association between two numerical variables is called the Pearson correlation coefficient, named after the English statistician Karl Pearson (1857–1936). Note that a correlation coefficient does not measure causation. In other words, correlation and causation are different concepts. Causation causes correlation, but not necessarily the converse. The correlation coefficient between two numerical variables in a set of sample data is usually denoted by r, and the correlation coefficient for population data is denoted by the Greek letter
(2.9.1)
or
(2.9.2)
Table 2.9.1 Cholesterol levels and systolic BP of 10 randomly selected US males.
| Subject | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Cholesterol (x) | 195 | 180 | 220 | 160 | 200 | 220 | 200 | 183 | 139 | 155 |
| Systolic BP (y) | 130 | 128 | 138 | 122 | 140 | 148 | 142 | 127 | 116 | 123 |
Figure 2.9.1 MINITAB printout of scatter plot for the data in Table 2.9.1.
The correlation coefficient is a dimensionless measure that can attain any value in the interval
Perfect association means that if we know the value of one variable, then the value of the other variable can be determined without any error. The other special case is when
MINITAB:
1 Enter the pairs of data in columns C1 and C2. Label the columns X and Y.
2 From the Menu bar select Graph Scatterplot. This prompts a dialog box to appear on the screen. In this dialog box, select scatterplot With Regression and click OK. This prompts the following dialog box to appear:In this dialog box, under the X and Y variables, enter the columns in which you have placed the data. Use the desired options and click OK. The Scatter plot
