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Understanding the Pearson Product Moment Correlation Coefficient (r), Slides of Advanced Data Analysis

The concept of the pearson product moment correlation coefficient (r), its calculation, and its significance. It covers topics such as standardization, regression coefficients, adjusted r-squared, and factors affecting r. The document also includes examples and explanations of concepts like covariance, correlation matrix, and whole-part correlations.

Typology: Slides

2012/2013

Uploaded on 01/01/2013

sarman
sarman 🇮🇳

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Pearson-Product Moment Correlation Coefficient (r)
COV
xy
=
A measure of the relation between x and y, but is not standardized
To standardize , we divide the covariance by the size of the standard deviations.
Given that the maximum value of the covariance is plus or minus the product
of the variance of x and the variance of y, it follows that the limits on the
correlation coefficient are + or – 1.0
rCOV
s s
xy
x y
=
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Pearson-Product Moment Correlation Coefficient (r)

COV

xy

= A measure of the relation between x and y, but is not standardized

To standardize , we divide the covariance by the size of the standard deviations.

Given that the maximum value of the covariance is plus or minus the product

of the variance of x and the variance of y, it follows that the limits on the

correlation coefficient are + or – 1.

r

COV

s s

xy x y

Example: 3 4 9 4 4 1 10 1 5 0 12 1 6 1 11 0 7 4 13 4

X (^ x^ −^ x ) Y (^ y^ − y )

x s x x = = 5

x
s

y y

COV xy = 2 25.

Adjusted r

r r N N adj =^ −^ − − − 1 1 1 2 2 ( )( ) ( )

From our example:

radj = − − − − 1 1 81 1 2 (. )(5 ) (5 )

r

= that proportion of the variance in y that is shared (accounted for) by x.

Sometimes called the “ coefficient of determination.”

Thus, r = .9 and r^

2

Or x accounts for 81% of the variance in y.

R = .2, thus r

r

= .04 or 4%

R = .4, thus = .016 or 16%

If our r is g times as large as a second r, then the proportion of the variance

associated with the first r will be g(squared) times as great as that associated

with the second.

r^ can also be misleading

Whole-Part correlations.

This is were the score for variable x contributes to the score of variable y.

Produces a + bias in r.

Again, Correlation does not imply causality. Variables may be accidentally related,

or both may be related to a third variable, or they may influence each other.

Which is more informative, the slope of the regression line or the correlation

coefficient?