Correlation Section 10.2

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. CAUTION This 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.

Finding r Formula 10-1 Known as the “shortcut” formula.

Finding r Calculator Steps – The step-up 2 nd, “0” key Scroll down to DiagnosticOn Hit enter twice

Finding r Calculator Steps – The actual process Stat Edit, Option 1: Edit … Enter values in L1 and in L2 Then … Stat Calc, LinReg(ax+b) See r = for correlation coefficient

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 a simple random sample of quantitative data. – The plotted points approximate a straight line. – There are no outliers. Requirement Check – How To – Enter data values in L1 and L2. – Y=, highlight Plot1 – Graph

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. r = 0.988

Interpreting r So r = … great, now what? We can use Table A-6 to make a meaningful conclusion… USING TABLE A-6 TO INTERPRET r If r exceeds the value in Table A-6, conclude that there is a linear correlation. Otherwise, there is not sufficient evidence to support the conclusion of a linear correlation.

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 #16, 18, 20, 23 For tomorrow’s class – find our your current GPA