More on Correlation
Topics . . .
• A look at the definitional
r
computation
• Four typical uses for correlations
• Problems resulting from:
–Restricted range
–Outliers
•
r
2
— coefficient of determination
• Hypothesis testing
–does the correlation exist in the population?
–hypotheses, computation,
df
• PointBiserial Correlation
• PhiCoefficient
Pearson ProductMoment Correlation Coefficient:
Pearson
r
A look at how the equation works
Why does the sign of
r
match the slope?
• Positive slope
/
–More positive products
•(+)(+) and ()()
–Positive sum
–Positive mean (
r
)
• Negative slope
\
–More negative products
•(+)() and ()(+)
–Negative sum
–Negative mean (
r
)
How is the strength of
r
explained?
•
r
is greatest (close to 1 or 1) when the
z
score products are
greatest
• The products are greatest when the points fall on along a
straight line
•the line must be tilted, at least slightly
–a horizontal or vertical line makes
r
= 0
∑
z
Y
z
X
shrinks as line becomes less perfect
1
2
3
4
5
6
7
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X
The point of correlation
• Prediction
–the stronger the correlation, the more the two variables
predict each other
–knowing one tells you about the other
–the closer
r
is to 1 or +1, the more accurate the prediction
The point of correlation
• Validity
–Do two tests measure the same thing?
–Suppose you develop a new, faster way to measure math
ability than the SAT
•If it correlates with the standard SAT math test, then a
reasonable conclusion is that it measures the same thing
(math ability)
The point of correlation
• Reliability
–Reliability of a measurement procedure refers to its ability to
produce stable and consistent measurements
•with the same individuals, under the same conditions
–With reliability, such scores should be positively, and
strongly, correlated
The point of correlation
• Theory Verification
–We can also, of course, theorize about relationships between
variables
•e.g., introverted individuals are more depressed
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 Fall '07
 Pfordesher
 Correlation, Correlation and dependence, Pearson productmoment correlation coefficient, Spearman's rank correlation coefficient

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