Introduction to Correlation (Dr. Monticino)

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Introductionto Correlation(Dr. Monticino)
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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

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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

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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

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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

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Correlation Coefficient

• r AVERAGE((x in standard units) ? (y in
  standard units))

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Example

• Find 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

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Example

• Step 1 Put x and y values into standard units

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Example

• Step 2 Find
• (x standard units)?(y standard units)

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Example

• Step 3 Find average of (x standard units)?(y
  standard units) values

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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

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Example

• Draw SD line for following data set

Av(X) 60.7 SD of X 30.4
Av(Y) 43.4 SD of Y 18.1
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Example
Point on SD line (60.7 , 43.4) Slope of SD
line 18.1/30.4 .595 Equation of SD line
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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

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Outliers

• r .31

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Outliers

• r .72

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Non-Linear Association

• r .22

(Dr. Monticino)
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Discussion Problems

• Questions or Comments?
• Chapter 8
• Review Exercises
• 1,2, 3, 5, 7, 8, 9, 11