Bivariate Statistics

Nominal Ordinal Interval

Nominal 2 Rank-sum t-testKruskal-Wallis H ANOVA

Ordinal Spearman rs (rho)

Interval Pearson rRegression

Y

X

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

Sir Francis Galton Karl Pearson

October 31

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

30 40 50 60 70 80

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MATH

RD

G

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

sc

sd

se

n

n

n

n n

Population

SampleA

SampleB

SampleE

SampleD

SampleC

_

XY

rXY

rXY

rXYrXY

rXY

-1 r +1

-1 +1

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

N

Population

SampleA

SampleB

SampleE

SampleD

SampleC

_

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 =

Some uses of r

• Association of two variables

• Reliability estimates

• Validity estimates

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

0 10 20 30 40Age of Arrival

0

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Eng l

i sh

Prof

icie

ncy

r=-.87 r=-.49

Spearman’s Rank Order Correlation rs

Point Biserial Correlation rpb

-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.

-1 r +1

-1 +1

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.