Introductionto

Correlation(Dr. Monticino)

Assignment Sheet Math 1680

Read Chapters 8 and 9 Review Chapter 7 – algebra review on lines

Assignment #6 (Due Monday Feb. 28th ) Chapter 8

• Exercise Set A: 1, 5, 6• Exercise Set B: ALL• Exercise Set C: 1, 3, 4• Exercise Set D: 1

Quiz #5 – Normal Distribution (Chapter 5) Test 1 is still projected for March 2,

assuming we get through chapter 10 by then…

Correlation

The idea in examining the correlation of two variables is to see if information about the value of one variable helps in predicting the value of the other variable

To say that two variables are correlated does not necessarily imply that one causes a response in the other.

Correlation measures association. Association is not the same as causation

Scatter Diagram

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Scatter Diagram

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Correlation Coefficient

The correlation coefficient is a measure of linear association between two variables

r is always between -1 and 1. A positive r indicates that as one variable increases, so does the other. A negative r indicates that as one variable increases, the other decreases

Correlation Coefficient

The correlation coefficient is unitless

It is not affected by Interchanging the two variables Adding the same number to all the

values of one variable Multiplying all the values of one

variable by the same positive number

Correlation Coefficient

r = AVERAGE((x in standard units) (y in

standard units))

ExampleFind the correlation coefficient for

following data set

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Example

Step 1: Put x and y values into standard units Need to find respective averages and

standard deviations

Av(X) = 60.7SD of X = 30.4

Av(Y) = 43.4SD of Y = 18.1

Example

Step 1: Put x and y values into standard unitsx y

44 4518 1464 3290 5388 5696 7312 2346 3488 61

4.30

)7.6044(

1.18

)4.4361(

x: standard units y: standard units

-0.55 0.09

-1.40 -1.62

0.11 -0.63

0.96 0.53

0.90 0.70

1.16 1.64

-1.60 -1.13

-0.48 -0.52

0.90 0.97

Example

Step 2: Find (x standard units)(y standard

units)x: standard units y: standard units

-0.55 0.09

-1.40 -1.62

0.11 -0.63

0.96 0.53

0.90 0.70

1.16 1.64

-1.60 -1.13

-0.48 -0.52

0.90 0.97

)09(.)55.( x*y (standard units)

-0.049

2.282

-0.068

0.511

0.625

1.899

1.806

0.251

0.873

ExampleStep 3: Find average of (x standard

units)(y standard units) values

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)873.251.806.1899.1625.511.068.82.2049.(

x*y (standard units)

-0.049

2.282

-0.068

0.511

0.625

1.899

1.806

0.251

0.873

903.

SD Line

Standard deviation line is THE line which the correlation coefficient is measuring dispersion around

SD line passes through the point (x-average,y-average)

Slope of SD line is (SD of y)/(SD of x) if + correlation -(SD of y)/(SD of x) if - correlation

Example

Draw SD line for following data set

X 44 18 64 90 88 96 12 46 88

Y 45 14 32 53 56 73 23 34 61

Av(X) = 60.7SD of X = 30.4

Av(Y) = 43.4SD of Y = 18.1

Example

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Point on SD line(60.7 , 43.4)

Slope of SD line18.1/30.4 = .595

Equation of SD line

)()(

)()()( AvgXX

XSD

YSDAvgYY

Correlation Coefficient Definition

Visually, the definition of correlation is reasonable

Average Lines

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More on Correlation

Correlation can be confounded by outliers and non-linear associations

When possible, look at the scatter diagram to check for outliers and non-linear association

Do not be too quick to delete outliers

Do not force a linear association when there is not one

Outliers

Association Between R&D Spending and P/E Ratio

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R/S Ratio

P/E

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Outliers

Association Between R&D Spending and P/E Ratio

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R/S Ratio

P/E

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r = .72

Non-Linear Association

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r = .22

(Dr. Monticino)

Discussion Problems

Questions or Comments?Chapter 8

Review Exercises:• 1,2, 3, 5, 7, 8, 9, 11