Correlation Sections 4.5, 4.6
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1 Correlation Sections 4.5, 4.6 Lecture 12 Robb T. Koether HampdenSydney College Wed, Feb 3, 2016 Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
2 Outline 1 Correlation 2 Interpreting the Correlation 3 Example 4 Practice 5 Using the TI83 6 Assignment Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
3 Outline 1 Correlation 2 Interpreting the Correlation 3 Example 4 Practice 5 Using the TI83 6 Assignment Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
4 Correlation Definition (Correlation) The correlation, denoted by r, of bivariate data measures the strength and the direction of the relationship (assuming that it is linear). The correlation is computed as ( ) ( ) x i x yi y s x s y r =. n 1 Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
5 Outline 1 Correlation 2 Interpreting the Correlation 3 Example 4 Practice 5 Using the TI83 6 Assignment Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
6 Interpreting the Correlation If r is close to 1 or 1, the relationship is strong. If r is close to 0, the relationship is weak. Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
7 Interpreting the Correlation If r is close to 1 or 1, the relationship is strong. If r is close to 0, the relationship is weak. In other words, r measures the strength of the relationship. Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
8 Interpreting the Correlation If r is close to 1 or 1, the relationship is strong. If r is close to 0, the relationship is weak. In other words, r measures the strength of the relationship. If r > 0, the relationship is positive. If r < 0, the relationship is negative. Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
9 Interpreting the Correlation If r is close to 1 or 1, the relationship is strong. If r is close to 0, the relationship is weak. In other words, r measures the strength of the relationship. If r > 0, the relationship is positive. If r < 0, the relationship is negative. In other words, the sign of r indicates the direction of the relationship. Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
10 Interpreting the Correlation If r is close to 1 or 1, the relationship is strong. If r is close to 0, the relationship is weak. In other words, r measures the strength of the relationship. If r > 0, the relationship is positive. If r < 0, the relationship is negative. In other words, the sign of r indicates the direction of the relationship. The correlation makes no distinction between explanatory and response variables. Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
11 Strong Positive Linear Association r is close to 1 Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
12 Strong Positive Linear Association r is close to 1 Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
13 Strong Positive Linear Association r is close to 0 Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
14 Strong Positive Linear Association Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
15 Strong Positive Linear Association Strong negative correlation Strong positive correlation Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
16 Strong Positive Linear Association Weak negative correlation Weak positive correlation Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
17 Strong Positive Linear Association No significant correlation Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
18 Why the Formula Works Almost all positive products Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
19 Why the Formula Works Mostly positive products Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
20 Why the Formula Works Almost all negative products Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
21 Why the Formula Works Mostly negative products Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
22 Why the Formula Works Even mix of positive and negative products Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
23 Outline 1 Correlation 2 Interpreting the Correlation 3 Example 4 Practice 5 Using the TI83 6 Assignment Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
24 Example Example (Height vs. Weight) Compute the deviations, their products, and the sums. Height (x) Weight (y) x x x x s x y y y y s y ( x x s x ) ( ) y y s y Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
25 Example Example (Height vs. Weight) Compute the deviations, their products, and the sums. Height (x) Weight (y) x x x x s x y y y y s y ( x x s x ) ( ) y y s y Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
26 Example Example (Height vs. Weight) Compute the deviations, their products, and the sums. Height (x) Weight (y) x x x x s x y y y y s y ( x x s x ) ( ) y y s y Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
27 Example Example (Height vs. Weight) Compute the deviations, their products, and the sums. Height (x) Weight (y) x x x x s x y y y y s y ( x x s x ) ( ) y y s y Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
28 Example Example (Height vs. Weight) Compute the deviations, their products, and the sums. Height (x) Weight (y) x x x x s x y y y y s y ( x x s x ) ( ) y y s y Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
29 Example Example (Height vs. Weight) Compute the deviations, their products, and the sums. Height (x) Weight (y) x x x x s x y y y y s y ( x x s x ) ( ) y y s y Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
30 Example Example (Height vs. Weight) Compute the deviations, their products, and the sums. Height (x) Weight (y) x x x x s x y y y y s y ( x x s x ) ( ) y y s y Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
31 Example Example (Height vs. Weight) Now compute r = = Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
32 We may simplify the formula: r = ( x i x s x ) ( ) yi y s y n 1 Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
33 We may simplify the formula: ( ) ( ) x i x yi y s x s y r = n 1 (xi x) (y i y) = (n 1)s x s y Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
34 We may simplify the formula: ( ) ( ) x i x yi y s x s y r = n 1 (xi x) (y i y) = (n 1)s x s y (xi x) (y i y) = (n 1) (xi x) 2 (yi y) 2 n 1 n 1 Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
35 We may simplify the formula: ) ( ) yi y s y ( x i x s x r = n 1 (xi x) (y i y) = (n 1)s x s y (xi x) (y i y) = (n 1) (xi x) 2 (yi y) 2 = n 1 (xi x) (y i y) (xi x) 2 (y i y) 2 n 1 Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
36 Outline 1 Correlation 2 Interpreting the Correlation 3 Example 4 Practice 5 Using the TI83 6 Assignment Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
37 Practice Practice Find the correlation of the data x y Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
38 Practice Practice x y x x y y (x x) 2 (y y) 2 (x x)(y y) Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
39 Practice Practice x y x x y y (x x) 2 (y y) 2 (x x)(y y) Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
40 Practice Practice x y x x y y (x x) 2 (y y) 2 (x x)(y y) Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
41 Practice Practice x y x x y y (x x) 2 (y y) 2 (x x)(y y) Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
42 Practice Practice x y x x y y (x x) 2 (y y) 2 (x x)(y y) Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
43 Practice Practice x y x x y y (x x) 2 (y y) 2 (x x)(y y) Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
44 Outline 1 Correlation 2 Interpreting the Correlation 3 Example 4 Practice 5 Using the TI83 6 Assignment Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
45 Using the TI83 The TI83 will calculate r if we put the x values and the y values into lists. Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
46 Correlation on the TI83 Correlation on the TI83 Put the x values in L 1. Put the y values in L 2. Select STAT > CALC > LinReg(abx) (item #8). Press Enter. LinReg appears in the display, followed by several numbers. The last number r is the correlation. Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
47 Correlation on the TI83 Correlation on the TI83 If you did not see r, then you must turn Diagnostic on (once and for all). To do so, press CATALOG and select DiagnosticOn, and press ENTER twice. Now the value of r should appear when you use LinReg. Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
48 Example Example District Free Lunch Grad. Rate District Free Lunch Grad. Rate Amelia King and Queen Caroline King William Charles City Louisa Chesterfield New Kent Colonial Hgts Petersburg Cumberland Powhatan Dinwiddie Prince George Goochland Richmond Hanover Sussex Henrico West Point Hopewell Find the correlation. Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
49 Outline 1 Correlation 2 Interpreting the Correlation 3 Example 4 Practice 5 Using the TI83 6 Assignment Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
50 Assignment Assignment Read Sections 4.5, 4.6. Apply Your Knowledge: 1, 3, 4, 6. Check Your Skills: 14, 15, 19, 20. Exercises: 24, 26, 27, 28, 29. Robb T. Koether (HampdenSydney College) CorrelationSections 4.5, 4.6 Wed, Feb 3, / 30
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