Introduction to Correlation (Dr. Monticino). Assignment Sheet Math 1680 Read Chapters 8 and 9 ...

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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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Midterm Score
Fin
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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 (xaverage,yaverage)
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 nonlinear associations
When possible, look at the scatter diagram to check for outliers and nonlinear 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
Rat
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Outliers
Association Between R&D Spending and P/E Ratio
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R/S Ratio
P/E
Rat
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r = .72
NonLinear 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