1. Describing Relationships
3.1 Correlation

2. Which one is stronger?
Our eyes are not always reliable when looking to see how strong a linear relationship is. This is why we rely on a number to help us.

3. Measuring Linear Association: Correlation
The correlation r measures the direction and strength of the linear relationship between two quantitative variables.
r is always a number between -1 and 1
r > 0 indicates a positive association.
r < 0 indicates a negative association.
Values of r near 0 indicate a very weak linear relationship.
strength increases as r moves away from 0 towards -1 or 1.
r = -1 and r = 1 occur only in the case of a perfect linear relationship.

6. Strength Strong Weak Moderate Moderate Weak Strong
Moderate
Weak
Strong

7. Correlation Practice 4. r ≈ 0.9
For each graph, estimate the correlation r and interpret it in context.
Answer choices:
1. r ≈
2. r ≈
3. r ≈ 0.3
4. r ≈ 0.9
5. r ≈
6. r ≈ 0.5
4. r ≈ 0.9
Pretty strong, positive relationship r ≈ 0.9

8. Correlation Practice 6. r ≈ 0.5
For each graph, estimate the correlation r and interpret it in context.
Answer choices:
1. r ≈
2. r ≈
3. r ≈ 0.3
4. r ≈ 0.9
5. r ≈
6. r ≈ 0.5
6. r ≈ 0.5
(b) Moderate, positive relationship r ≈ 0.5

9. Correlation Practice 3. r ≈ 0.3
For each graph, estimate the correlation r and interpret it in context.
Answer choices:
1. r ≈
2. r ≈
3. r ≈ 0.3
4. r ≈ 0.9
5. r ≈
6. r ≈ 0.5
3. r ≈ 0.3
(c) Weak, positive relationship r ≈ 0.3

10. Correlation Practice 2. r ≈ - 0.1
For each graph, estimate the correlation r and interpret it in context.
Answer choices:
1. r ≈
2. r ≈
3. r ≈ 0.3
4. r ≈ 0.9
5. r ≈
6. r ≈ 0.5
2. r ≈ - 0.1
(d) Weak, negative relationship r ≈ -0.1

11. Example, p. 153 r = 0.936 Interpret the value of r in context.
The correlation of confirms what we see in the scatterplot; there is a strong, positive linear relationship between points per game and wins in the SEC.

12. Example, p. 153
r = 0.936
The point highlighted in red on the scatterplot is Mississippi. What effect does Mississippi have on the correlation. Justify your answer.
Mississippi makes the correlation closer to 1 (stronger). If Mississippi were not included, the remaining points wouldn’t be as tightly clustered in a linear pattern.

13. Calculating Correlation
How to Calculate the Correlation r
Suppose that we have data on variables x and y for n individuals. The values for the first individual are x1 and y1, the values for the second individual are x2 and y2, and so on. The means and standard deviations of the two variables are x-bar and sx for the x-values and y-bar and sy for the y-values.
The correlation r between x and y is:
Notice what the formula has in it. Z-scores

14. Facts About Correlation
Correlation makes no distinction between explanatory and response variables.
r does not change when we change the units of measurement of x, y, or both.
The correlation r itself has no unit of measurement.

15. Cautions Correlation requires that both variables be quantitative.
Correlation does not describe curved relationships between variables, no matter how strong the relationship is. Because of this, r = - 1 or 1 does not guarantee a linear relationship.
Correlation is not resistant. r is strongly affected by a few outlying observations.
Correlation is not a complete summary of two-variable data.

16. This set of data has a correlation close to – 1, but we can see a slight curve in the scatterplot. Always plot your data!

17. HW Due: Block Day
p. 161 #15 – 18, 21, 30