http://www.york.ac.uk/depts/maths/histstat/people/

Sir Francis Galton Karl Pearson

October 28 and 29

Source: Raymond Fancher, Pioneers of Psychology. Norton, 1979.

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A correlation coefficient is a numerical expression of the degree of relationship between two continuous variables.

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What might be some practical uses of such a statistic?

A correlation coefficient is a numerical expression of the degree of relationship between two continuous variables.

-1 r +1

-1 +1

Pearson’s r

Population

SampleA XA

µ

_

SampleB XB

SampleE XE

SampleD XD

SampleC XC

_

_

_

_

sa

sb

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sd

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n n

Population

SampleA

SampleB

SampleE

SampleD

SampleC

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XY

rXY

rXY

rXYrXY

rXY

-1 r +1

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Pearson’s r

Pearson’s r is a function of the sum of the cross-product of z-scores for x and y.

Pearson’s r

r = zxzy

NWhere z is based on an uncorrected standard deviation, SS N

Pearson’s r

r = zxzy

N-1if z is based on a corrected standard deviation, SS N-1

Pearson’s r

N XY - X Y

[N X2 - ( X)2] [N Y2 - ( Y)2] r =

… or, for your convenience,

Population

SampleA

SampleB

SampleE

SampleD

SampleC

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XY

rXY

rXY

rXYrXY

rXY

The familiar t distribution, at N-2 degrees of freedom, can be used to test the probability that the statistic r was drawn from a population with = 0

H0 : XY = 0

H1 : XY 0

where

r N - 2

1 - r2

t =

-1 r +1

-1 +1

Pearson’s r

Pearson’s r can also be interpreted as how far the scores of Y individuals tend to deviate from the mean of X when they are expressed in standard deviation units.

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Pearson’s r

Pearson’s r can also be interpreted as the expected value of zY given a value of zX.

tend to deviate from the mean of X when they are expressed in standard deviation units.

The expected value of zY is zX*r

If you are predicting zY from zX where there is a perfect correlation (r=1.0), thenzY=zX.. If the correlation is r=.5, then zY=.5zX.

Factors that affect r

Non-linearity

Restriction of range / variability

Outliers

Reliability of measure / measurement error

Johnson & Newport, scaled properly,with new ranges age <20 and >20.

All Subjects

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r=-.87 r=-.49

Spearman’s Rank Order Correlation rs

Point Biserial Correlation rpb