More About Correlation (Lesson 3.4)
Chapter 3  Day 4
Experience First
After very little debate, we decided we were not going to require our students to calculator the correlation r by hand using a formula. Students used the “2 Quantitative Variables” applet at www.statsmedic.com/applets to find r. This allowed more time within the lesson for students to discover some properties of r.
As you are monitoring the groups during the activity, ask students to predict what will happen to r before they use the applet to find out. Ask them to explain why they made their prediction.
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Formalize Later
We intentionally avoided the calculation of r by hand. Here is why:
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But…is there value in having showing students this formula? Well…maybe. The properties of correlation are nicely explained by looking at the formula:

Correlation makes no distinction between explanatory and response variables. If we decided to switch our explanatory and response variables, the z score for x and y would swap places in the formula. Because multiplication is commutative, we will get the same correlation value.

r does not change when we change units. zscores tell us the number of standard deviations above or below the mean. This value does not depend on the units. If a value measured in inches is 1.78 standard deviations above the mean, then the same value measured in centimeters will still be 1.78 standard deviations above the mean.

The correlation r has no units of measurement. Suppose the explanatory variable is measured in inches. The numerator of xxi would be measured in inches, and when divided by the standard deviation (which is also measured in inches), the units would cancel out. Thus r has no units.