Correlation

Chapter 7

The Basics

• A correlation exists between two variables when the values of one variable are somehow

associated with the values of the other variable.

• What correlations can you think of in your life?

Variety is the Spice of Life

• There are many different types of correlations.

POSITIVE CORRELATION

NEGATIVE CORRELATION

Variety is the Spice of Life

• There are many different types of correlations.

NO CORRELATION

NON-LINEAR CORRELATION

How Strong Is That?

• The linear correlation coefficient r measures the strength of the linear correlation between

the paired quantitative x and y values in a sample.

Properties of r

1. The value of r is always between -1 and 1 inclusive. • -1 ≤ r ≤ 1

2. r measures the strength of a linear relationship

3. r is very sensitive to outliers.

CAUTIONThis section ONLY applies to linear correlations. If you conclude

there does not appear to be a correlation, know that it is possible that there might be some other association that is not linear.

Practice Round

• Eric Bram, a NY teenager, noticed that the cost of a slice of cheese pizza was typically the same as the cost of a subway ride. Over the years, he noticed that as one increased, so did the other. When the cost of a slice of pizza increased, he told the New York Times that the cost of subway fares would rise as well.

Before we start …

• Requirement Check– The data are quantitative. – The plotted points approximate a straight line. – There are no outliers.

Back to the Problem

• Eric Bram, a NY teenager, noticed that the cost of a slice of cheese pizza was typically the same as the cost of a subway ride. Over the years, he noticed that as one increased, so did the other. When the cost of a slice of pizza increased, he told the New York Times that the cost of subway fares would rise as well.

Practice Round• Here is some of his data:

• Check the requirements, if met then … • Find the correlation coefficient.

Cost of Pizza

$0.15 $0.35 $1.00 $1.25 $1.75 $2.00

Subway Fare

$0.15 $0.35 $1.00 $1.35 $1.50 $2.00

r = 0.988

Interpreting r

• So r = 0.988 … great, now what? • Describe the correlation.

Ask YourselfIs r positive or negative?

Where does it fit on the chart you wrote down (other power point)?

Common Errors

1. A common error is to conclude that correlation implies causality.

There is a correlation between the costs of pizza and subway fares, but we cannot conclude that increases in pizza cost (massive cheese price spike) causes subways to increase their rates. Both costs might be affected by some other variable lurking (creepily) in the background.

Common Errors

2. Another error arises with data based on averages.

Averages suppress individual variations and may inflate the correlation coefficient. One study produced a 0.4 linear correlation coefficient for paired data relating income and education among individuals, but the linear correlation coefficient became 0.7 when regional averages were used.

Common Errors

3. A third error involves the property of linearity. Remember, this section only deals with linear

correlations. Just because we find that there is no correlation between two data sets supplied in this section, that does not mean there is no correlation PERIOD. There might be a non-linear relationship.

Class Activity

• Collect data from each student consisting of the number of credit cards and the number of

keys that the student has in his or her possession. Is there a correlation?

• Try to identify at least one reasonable explanation for the presence or absence of a

correlation.

Homework

• Pg. 532-533 #16, 18, 20, 23

• For tomorrow’s class – find our your current GPA