1. Correlation: How Strong Is the Linear Relationship?
Lecture 46
Sec. 13.7
Mon, Dec 3, 2007

2. The Correlation Coefficient
The correlation coefficient r is a number between –1 and +1.
It measures the direction and strength of the linear relationship.
If r > 0, then the relationship is positive. If r < 0, then the relationship is negative.
The closer r is to +1 or –1, the stronger the relationship.
The closer r is to 0, the weaker the relationship.

3. Strong Positive Linear Association
In this display, r is close to +1. x y

4. Strong Positive Linear Association
In this display, r is close to +1. x y

5. Strong Negative Linear Association
In this display, r is close to –1. x y

6. Strong Negative Linear Association
In this display, r is close to –1. x y

7. Almost No Linear Association
In this display, r is close to 0. x y

8. Almost No Linear Association
In this display, r is close to 0. x y

9. Interpretation of r -1
-0.8
-0.2
0.2
0.8 1

10. Interpretation of r Strong Negative Strong Positive -1 -0.8 -0.2 0.2
0.2
0.8 1

11. Interpretation of r
Weak
Negative
Weak
Positive -1
-0.8
-0.2
0.2
0.8 1

12. Interpretation of r
No Significant
Correlation -1
-0.8
-0.2
0.2
0.8 1

13. Correlation vs. Cause and Effect
If the value of r is close to +1 or -1, that indicates that x is a good predictor of y.
It does not indicate that x causes y (or that y causes x).
The correlation coefficient alone cannot be used to determine cause and effect.

14. Calculating the Correlation Coefficient
There are many formulas for r.
The most basic formula is
Another formula is

15. Example Consider again the data x y 1 8 3 12 4 9 5 14 16 20 11 17 1524

16. Example
We found earlier that
SSX = 150
SSY = 206
SSXY = 165

17. Example
Then compute r.

18. TI-83 – Calculating r To calculate r on the TI-83,
First, be sure that Diagnostic is turned on.
Press CATALOG and select DiagnosticsOn.
Then, follow the procedure that produces the regression line.
In the same window, the TI-83 reports r2 and r.

19. TI-83 – Calculating r
Use the TI-83 to calculate r in the preceding example.
Find r for the S/T Ratio vs. Graduation Rate.
Find r for SOL-Eng Passing Rate vs. Graduation Rate.

20. Another Formula for r
It turns out that
where

21. Another Formula for r
Free-lunch participation vs. graduation rate data,
SSR = ,
SST =
So we get

22. The Coefficient of Determination
r2 is called the coefficient of determination.
It is interpreted as telling us how much of the variation in y is determined by the variation in x.
So, 73% of the variation is graduation rates is determined by the variation in participation in the free-lunch program.

23. The Coefficient of Determination
What percentage of the variation in graduation rate is determined by the variation in S/T ratio?
What percentage of variation in graduation rate is determined by the variation in teachers’ average salary?

24. How Does r Work?
How does r indicate the direction of the relationship?
Consider the numerator of the formula.

25. How Does r Work? Consider the lunch vs. graduation data: District
Free Lunch
Grad. Rate
Amelia
41.2
68.9
King and Queen
59.9
64.1
Caroline
40.2
62.9
King William
27.9
67.0
Charles City
45.8
67.7
Louisa
44.9
80.1
Chesterfield
22.5
80.5
New Kent
13.9
77.0
Colonial Hgts
25.7
73.0
Petersburg
61.6
54.6
Cumberland
55.3
63.9
Powhatan
12.2
89.3
Dinwiddie
45.2
71.4
Prince George
30.9
85.0
Goochland
23.3
76.3
Richmond
74.0
46.9
Hanover
13.7
90.1
Sussex
74.8
59.0
Henrico
30.2
81.1
West Point
19.1
82.0
Hopewell
63.1
63.4

26. How Does r Work? Consider the lunch vs. graduation data: (first half)x y
x –x
y –y
(x –x)(y –y)
41.2
68.9
40.2
62.9
45.8
67.7
22.5
80.5
25.7
73.0
55.3
63.9
45.2
71.4
23.3
76.3
13.7
90.1
30.2
81.1
63.1
63.4
(first half)

27. How Does r Work? Consider the lunch vs. graduation data: (first half)x y
x –x
y –y
(x –x)(y –y)
41.2
68.9
1.9
-2.7
40.2
62.9
0.9
-8.7
45.8
67.7
6.5
-3.9
22.5
80.5
-16.8
8.9
25.7
73.0
-13.6
1.4
55.3
63.9
16.0
-7.7
45.2
71.4
5.9
-0.2
23.3
76.3
-16.0
4.7
13.7
90.1
-25.6
18.5
30.2
81.1
-9.1
9.5
63.1
63.4
23.8
-8.2
(first half)

28. How Does r Work? Consider the lunch vs. graduation data: (first half)x y
x –x
y –y
(x –x)(y –y)
41.2
68.9
1.9
-2.7
-5.13
40.2
62.9
0.9
-8.7
-7.83
45.8
67.7
6.5
-3.9
-25.35
22.5
80.5
-16.8
8.9
25.7
73.0
-13.6
1.4
-19.04
55.3
63.9
16.0
-7.7
45.2
71.4
5.9
-0.2
-1.18
23.3
76.3
-16.0
4.7
-75.2
13.7
90.1
-25.6
18.5
-473.6
30.2
81.1
-9.1
9.5
-86.45
63.1
63.4
23.8
-8.2
(first half)

29. How Does r Work? Consider the lunch vs. graduation data: (second half)x y
x –x
y –y
(x –x)(y –y)
59.9
64.1
27.9
67.0
44.9
80.1
13.9
77.0
61.6
54.6
12.2
89.3
30.9
85.0
74.0
46.9
74.8
59.0
19.1
82.0
(second half)

30. How Does r Work? Consider the lunch vs. graduation data: (second half)x y
x –x
y –y
(x –x)(y –y)
59.9
64.1
20.6
-7.5
27.9
67.0
-11.4
-4.6
44.9
80.1
5.6
8.5
13.9
77.0
-25.4
5.4
61.6
54.6
22.3
-17.0
12.2
89.3
-27.1
17.7
30.9
85.0
-8.4
13.4
74.0
46.9
34.7
-24.7
74.8
59.0
35.5
-12.6
19.1
82.0
-20.2
10.4
(second half)

31. How Does r Work? Consider the lunch vs. graduation data: (second half)x y
x –x
y –y
(x –x)(y –y)
59.9
64.1
20.6
-7.5
27.9
67.0
-11.4
-4.6
52.44
44.9
80.1
5.6
8.5
47.60
13.9
77.0
-25.4
5.4
61.6
54.6
22.3
-17.0
12.2
89.3
-27.1
17.7
30.9
85.0
-8.4
13.4
74.0
46.9
34.7
-24.7
74.8
59.0
35.5
-12.6
19.1
82.0
-20.2
10.4
(second half)

32. Scatter Plot 90 80 Graduation Rate 70 60 50 Free Lunch Rate 20 30 40

33. Scatter Plot 90 80 Graduation Rate 70 60 50 Free Lunch Rate 20 30 40

34. Scatter Plot 90 80 Graduation Rate 70 60 50 Free Lunch Rate 20 30 40

35. Scatter Plot 90 The two oddballs 80 Graduation Rate 70 60 50
Free Lunch
Rate 20 30 40 50 60 70 80