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# Correlation Principles Help (page 3)

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By McGraw-Hill Professional
Updated on Sep 12, 2011

## Units (Usually) Don't Matter

Here's an interesting property of correlation. The units we choose don't matter, as long as they express the same phenomenon or characteristic. If the measurement unit of either variable is changed in size but not in essence, the appearance of a bar graph or scatter plot changes. The plot is ''stretched'' or ''squashed'' vertically or horizontally. But the correlation figure, r, between the two variables is unaffected.

Think back again to the last chapter, and the scatter plots of precipitation versus temperature for Happyton and Blissville. The precipitation amounts are indicated in centimeters per month, and the temperatures are shown in degrees Celsius. Suppose the precipitation amounts were expressed in inches per month instead. The graphs would look a little different, but upon analysis by a computer, the correlation figures would turn out the same. Suppose the temperatures were expressed in degrees Fahrenheit. Again, the graphs would look different, but r would not be affected. Even if the average monthly rainfall were plotted in miles per month and the temperatures in degrees Kelvin (where 0K represents absolute zero, the coldest possible temperature), the value of r would be the same.

We must be careful when applying this rule. The sizes of the units can be changed, but the quantities or phenomena they represent must remain the same. Therefore, if we were to plot the average rainfall in inches, centimeters, or miles per week rather than per month, we could no longer be sure the correlation would remain the same. The scatter plots would no longer show the same functions. The variable on the vertical scale – rainfall averaged over weekly periods rather than over monthly periods – would no longer represent the same thing. This is a subtle distinction, but it makes a critical difference.

## Correlation Principles Practice Problems

#### Practice 1

Suppose the distances of the outliers from the least-squares line from Fig. 7-2 are cut in half (to d/2 rather than d), as shown in Fig. 7-3. What effect will this have on the correlation?

Fig. 7-3. Illustration for Practice 1 and 2.

#### Solution 1

It will increase the correlation, because the average distances of all the points from the least-squares line will be smaller.

#### Practice 2

Suppose that one of the outliers is removed in the scenario of Fig. 7-3. Will this affect the position of the least-squares line?

Fig. 7-3. Illustration for Practice 1 and 2.

#### Solution 2

Yes. If the upper-left outlier is removed, the position of the least-squares line will be displaced slightly downward and to the right; if the lower-right outlier is removed, the least-squares line will be displaced slightly upward and to the left.

Practice problems for these concepts can be found at:

Correlation, Causation, Order, and Chaos Practice Test

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