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

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.

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

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

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

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

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

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

Interpretation of r -1 -0.8 -0.2 0.2 0.8 1

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

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

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

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.

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

Example Consider again the data x y 1 8 3 12 4 9 5 14 16 20 11 17 15 24

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

Example Then compute r.

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.

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.

Another Formula for r It turns out that where

Another Formula for r Free-lunch participation vs. graduation rate data, SSR = 1896.7, SST = 2598.2. So we get

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.

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?

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

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

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)

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)

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 -149.52 25.7 73.0 -13.6 1.4 -19.04 55.3 63.9 16.0 -7.7 -123.20 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 -195.16 (first half)

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)

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)

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 -154.50 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 -137.16 61.6 54.6 22.3 -17.0 -379.10 12.2 89.3 -27.1 17.7 -479.67 30.9 85.0 -8.4 13.4 -112.56 74.0 46.9 34.7 -24.7 -857.09 74.8 59.0 35.5 -12.6 -447.30 19.1 82.0 -20.2 10.4 -210.08 (second half)

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

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

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

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