EXAMINING RELATIONSHIPS Section 3.2 Correlation

Recall from yesterday…  A scatterplot displays form, direction, and strength of the relationship between two quantitative variables.

Strength  Since we can’t always tell how “strong” or “weak” a relationship is by looking at a scatterplot, we can use a numerical measure... Correlation

 Measures the direction and strength of the linear relationship between two quantitative variables.  Usually written as r.

Correlation  For variables x and y for n individuals, the correlation r between x and y is x i and y i represent one individual. and represent the average of x and y. s x and s y represent the standard deviation for each variable.

Correlation, Figure 3.8 Pg. 141

Correlation  The formula for r begins by standardizing the observations

Creating Correlation in the Calculator  Insert list of numbers into L 1 and L 2 P. 142 Femur: Humerus:

Creating Correlation in the Calculator  2 nd Quit  Press STAT, choose CALC, then 2:2-VarStats.

Creating Correlation in the Calculator  Next, we’ll define L 3. List 3 will represent:  In the home screen, press ( (L 1 - )/Sx) ( (L 2 - )/Sy)  L 3 L 1 Either 2 nd 1 OR 2 nd STAT 1 L 2 Either 2 nd 2 OR 2 nd STAT 2,, Sx, and Sy can be found under VARS 5  STO>

Creating Correlation in the Calculator Look under lists. There are now numbers under List 3

Creating Correlation in the Calculator To complete, enter the following command: (1/(n – 1))*sum(L 3 ) n is under VARS 5 sum( is under 2 nd STAT MATH

Creating Correlation in the Calculator

Easier Way…  ONLY WORKS WITH LINEAR RELATIONSHIPS! Press STAT CALC 4:LinReg(ax+b) ENTER  If r does not show up: 2 nd Catalog D Scroll down to DiagnosticOn. Push ENTER twice. Try the above commands again.

What we need to know to interpret correlation  Makes no distinction between exp. & resp. variables.  Requires both variables to be quantitative.  Changing units does not change r because it does not have units. It is just a number.  Positive r indicates a positive association, negative r indicates a negative association

What we need to know to interpret correlation  r will always be between -1 and 1. Values near 0 indicate a weak linear relationship. The closer to either -1 or 1, the stronger.  Measures strength of only a linear relationship between two variables. (does not describe curved)  Not resistant. Strongly affected by a few outlying observations. Use with caution when outliers appear in the scatterplot.

Examples of correlation P. 145

Example 3.7 Page 145  Competitive divers are scored on their form by a panel of judges who use a scale from 1 to 10. The subjective nature of the scoring often results in controversy. We have the scores awarded by two judges, Ivan and George, on a large number of dives. How well do they agree? We do some calculation and find that the correlation between their scores is r = 0.9. But the mean of Ivan’s scores is 3 points lower than George’s mean. These facts do not contradict each other. They are simply different kinds of information. The mean scores show that Ivan awards much lower scores than George. But because Ivan gives every dive a score about 3 points lower than George, the correlation remains high. Adding or subtracting the same number to all values of either x or y does not change the correlation. If Ivan and George both rate several divers, the contest is high fairly scored because Ivan and George agree on which dives are better than others. The high r shows their agreement.

Practice Problems  Exercises 3.25, 3.26