Correlation: How Strong Is the Linear Relationship? Lecture 46 Sec Mon, Apr 30, 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 x y In this display, r is close to +1.

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

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

Interpretation of r Strong Negative Strong Positive

Interpretation of r Weak Negative Weak Positive

Interpretation of r No Significant Correlation

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.

Mixing Populations Mixing nonhomogeneous groups can create a misleading correlation coefficient. Suppose we gather data on the number of hours spent watching TV each week and the child’s reading level, for 1 st, 2 nd, and 3 rd grade students.

Mixing Populations We may get the following results, suggesting a weak positive correlation. Number of hours of TV Reading level

Mixing Populations We may get the following results, suggesting a weak positive correlation. Number of hours of TV Reading level r = 0.26

Mixing Populations However, if we separate the points according to grade level, we may see a different picture. 1 st grade 2 nd grade 3 rd grade Number of hours of TV Reading level

Mixing Populations However, if we separate the points according to grade level, we may see a different picture. Number of hours of TV Reading level r 1 = -0.35

Mixing Populations However, if we separate the points according to grade level, we may see a different picture. Number of hours of TV Reading level r 2 = -0.73

Mixing Populations However, if we separate the points according to grade level, we may see a different picture. Number of hours of TV Reading level r 3 = -0.52

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

Example Consider again the data xy

Example Compute  x,  y,  x 2,  y 2, and  xy xyx2x2 y2y2 xy

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 r 2 and r. Use the TI-83 to calculate r in the preceding example.

Example Find the correlation coefficient for the Calorie/Cholesterol data. Calories (x) Cholesterol (y)

How Does r Work? Recall the formula We will consider the numerator.

How Does r Work? Consider the Subway data: Cal (x)Chol (y)

How Does r Work? Consider the Subway data: Cal (x)Chol (y) x –  x – – – – –76

How Does r Work? Consider the Subway data: Cal (x)Chol (y) x –  xy –  y –16– –16– –26– –16– –8 2300–76–28

How Does r Work? Consider the Subway data: Cal (x)Chol (y) x –  xy –  y(x –  )(y –  y) –16– –16– –26– –16– –8– –76–282128

How Does r Work? Consider the Subway data: Cal (x)Chol (y) x –  xy –  y(x –  )(y –  y) –16– –16– –26– –16– –8– –76–282128

How Does r Work? Consider the Subway data: Cal (x)Chol (y) x –  xy –  y(x –  )(y –  y) –16– –16– –26– –16– –8– –76–282128

How Does r Work? Consider the Subway data: Cal (x)Chol (y) x –  xy –  y(x –  )(y –  y) –16– –16– –26– –16– –8– –76–282128

How Does r Work? Calories Cholesterol

How Does r Work? Calories Cholesterol

How Does r Work? Calories Cholesterol

How Does r Work? Calories Cholesterol positive negative positive

How Does r Work? Calories Cholesterol positive negative positive