3.2 Correlation3.2 Correlation

CorrelationCorrelation• Measures direction and strength of the Measures direction and strength of the

linear relationship in the scatterplot.linear relationship in the scatterplot.• Correlation coefficient is r.Correlation coefficient is r.

y

i

x

i

syy

sxx

nr

11

In words, r is the sum of the product of the standardized values of the observations divided by the degrees of freedom.

Calculating the Correlation Calculating the Correlation CoefficientCoefficient

Exercise 3.24Exercise 3.24

Find the correlation coefficient, r, step – by – Find the correlation coefficient, r, step – by – step. step.

Femur:Femur: 3838 5656 5959 6464 7474

Humerus:Humerus: 4141 6363 7070 7272 8484

y

i

x

i

syy

sxx

nr

11

Can’t we do this on the calculator?!

Facts about correlation Facts about correlation coefficientcoefficient• No distinction between explanatory and No distinction between explanatory and

response variables.response variables.• Both variables must be quantitative.Both variables must be quantitative.• r has no unit of measure. Changing units r has no unit of measure. Changing units

will not change the value for r.will not change the value for r.• +r = positive linear association; -r = +r = positive linear association; -r =

negative linear association.negative linear association.• -1 < r < 1. The closer the values are to -1 -1 < r < 1. The closer the values are to -1

or 1 indicate how close the points lie to a or 1 indicate how close the points lie to a straight line.straight line.

Correlation facts continued Correlation facts continued ……• r = -1 or r = 1 shows a perfect linear r = -1 or r = 1 shows a perfect linear

relationship.relationship.• Correlation measures the strength of Correlation measures the strength of linear linear

relationshipsrelationships between two variables. For curved between two variables. For curved relationships we will use another determiner.relationships we will use another determiner.

• The correlation is NOT RESISTANT.The correlation is NOT RESISTANT.• Correlation is not an end all solve all. We use it Correlation is not an end all solve all. We use it

in part to help describe the data along with the in part to help describe the data along with the means and standard deviations of two means and standard deviations of two variables.variables.

r: r: Paper/Pencil vs. Calculator Paper/Pencil vs. Calculator almighty!almighty!• At this point, we should have an At this point, we should have an

appreciation of what our handy-dandy appreciation of what our handy-dandy calculator can do …calculator can do …

• The most efficient way to obtain the The most efficient way to obtain the correlation coefficient, r, is to ensure the correlation coefficient, r, is to ensure the calculator’s diagnostics is turned on and calculator’s diagnostics is turned on and run a linear regression on the scatterplot.run a linear regression on the scatterplot.

• Complete exercises 3.26 – 3.28, 3.32, 3.35 Complete exercises 3.26 – 3.28, 3.32, 3.35 – 3.37 (Quiz 3.2: Monday, 10/4).– 3.37 (Quiz 3.2: Monday, 10/4).

Correlation ActivityCorrelation Activity• UNIVERSITY OF ILLINOIS AT URBANA-UNIVERSITY OF ILLINOIS AT URBANA-

CHAMPAIGN (UIUC)CHAMPAIGN (UIUC)• http://www.stat.uiuc.edu/courses/stat100/java/GCApplet/GCAppletFrame.ht

ml