# Sample Proportions

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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.
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
• 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
• 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
• 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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