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					The Correlation Constant

       Section 3.2.1
             Starter 3.2.1
                            5
                                i
Evaluate the expression:   2
                           i 1
         Today’s Objectives
• Students will evaluate the strength of a
  linear association by calculating the
  CORRELATION CONSTANT (r) from its
  formula.
      Comparing Scatterplots
• Consider the two
  scatterplots to the
  right.
• They both show
  positive linear
  association.
• Are they about the
  same strength?
• Clearly not, so how
  can we quantify the
  difference?
       The Correlation Constant
• We use mean or median to quantify the center of
  a distribution.
• We use standard deviation or IQR to quantify the
  spread of a distribution.
• In the same spirit, we quantify the strength of a
  linear association with a number.
  – It is called the Correlation Constant
  – It is abbreviated by the letter r
  – It is calculated by the formula

           1    X  X Y Y                 
       r      S  S
          n 1 
                                            
                                            
                   x     y                 
    What does the formula mean?
                  1    X  X Y Y                    
              r      S  S
                 n 1 
                                                      
                                                      
                          x     y                    
• Essentially we are averaging the product of the x and y
  z-scores
• r is the correlation constant
   – It is a measure of how closely the data fit a straight line
   – A perfect fit would give an r value of 1 or -1
   – A completely random pattern would give r = 0
       • Certain non-random patterns also have r = 0; you will see one on
         HW
   – Most r values are somewhere in between
       • The closer |r| is to 1, the stronger the linear association
       • A positive r value means a positive slope: an increase in x leads to
         an increase in y
       • A negative r value means a negative slope
                     Example
• Consider the data from problem 3.13 (p.125)
  about the femur and humerus of archaeopteryx.
  – You already did the scatterplot for homework
  – Now let’s find the correlation constant
• Let the femur measurements be x and the
  humerus measurements be y.
  – Put x values in L1    38     56    59     64     74
  – Put y values in L2    41     63    70     72     84
• Run STAT:CALC:2-Var Stats L1, L2
  – The calculator will show you both means and both
    standard deviations
  – It also stores them under VARS:Statistics for later use
• Now use list operations to evaluate the formula
            Example Continued
• After running 2-Var Stats, do these list ops:
  – Define L3 = (L1 - )/Sx
     • Note: Find the symbols , , Sx, and Sy in
       VARS:Statistics menu
     • If you did not run 2-Var Stats first, these values will
       be wrong
  – Define L4 = (L2 - )/Sy
  – Define L5 = (L3 x L4)
  – In the compute screen, divide sum(L5) by n-1
    (which is 4 in this case)
     • You should get r = .994
            Save your lists
• We will use these data again
• Save L1 in a list called FEMUR
  – L1→FEMUR
• Save L2 in a list called HUMER
  – L2→HUMER
            Fathom Demo
• How does the correlation constant become
  1 for perfect straight lines and 0 for no
  association?
  – Run Fifty Fathoms Demo 8: Correlation.ftm
         Today’s Objectives
• Students will evaluate the strength of a
  linear association by calculating the
  CORRELATION CONSTANT (r) from its
  formula.
             Homework
• Read pages 128 – 133
• Do problems 20 – 24

				
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posted:5/8/2013
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