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Regression Help

By — McGraw-Hill Professional
Updated on Aug 26, 2011

Introduction to Regression—Paired Data

Regression is a way of defining the extent to which two variables are related. Regression can be used in an attempt to predict things, but this can be tricky. The existence of a correlation between variables does not always mean that there is a cause-and-effect link between them.

Paired Data

Imagine two cities, one named Happyton and the other named Blissville. These cities are located far apart on the continent. The prevailing winds and ocean currents produce greatly different temperature and rainfall patterns throughout the year in these two cities. Suppose we are about to move from Happyton to Blissville, and we've been told that Happyton ''has soggy summers and dry winters,'' while in Blissville we should be ready to accept that ''the summers are parched and the winters are washouts.'' We've also been told that the temperature difference between summer and winter is much smaller in Blissville than in Happyton.

We go to the Internet and begin to collect data about the two towns. We find a collection of tables showing the average monthly temperature in degrees Celsius (°C) and the average monthly rainfall in centimeters (cm) for many places throughout the world. Happyton and Blissville are among the cities shown in the tables. Table 6-1A shows the average monthly temperature and rainfall for Happyton as gathered over the past 100 years. Table 6-1B shows the average monthly temperature and rainfall for Blissville over the same period. The data we have found is called paired data, because it portrays two variable quantities, temperature and rainfall, side-by-side.

We can get an idea of the summer and winter weather in both towns by scrutinizing the tables. But we can get a more visual-friendly portrayal by making use of bar graphs.

Paired Bar Graphs

Let's graphically compare the average monthly temperature and the average monthly rainfall for Happyton. Figure 6-8A is a paired bar graph showing the average monthly temperature and rainfall there.

Regression

The graph is based on the data from Table 6-1A. The horizontal axis has 12 intervals, each one showing a month of the year. Time is the independent variable. The left-hand vertical scale portrays the average monthly temperatures, and the right-hand vertical scale portrays the average monthly rainfall amounts. Both of these are dependent variables, and are functions of the time of year. The average monthly temperatures are shown by the light gray bars, and the average monthly rainfall amounts are shown by the dark gray bars. It's easy to see from this data that the temperature and rainfall both follow annual patterns. In general, the warmer months are wetter than the cooler months in Happyton.

Now let's make a similar comparison for Blissville. Figure 6-8B is a paired bar graph showing the average monthly temperature and rainfall there, based on the data from Table 6-1B.

Regression

From this data, we can see that the temperature difference between winter and summer is less pronounced in Blissville than in Happyton. But that's not the main thing that stands out in this bar graph! Note that the rainfall, as a function of the time of year, is much different. The winters in Blissville, especially the months of January and February, are wet. The summers, particularly June, July, and August, get almost no rainfall. The contrast in general climate between Happyton and Blissville is striking. This information is, of course, contained in the tabular data, but it's easier to see by looking at the dual bar graphs.

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