CORRELATIONAL RESEARCH STUDIES Describe a linear relationship between

variables Do not imply a cause-and-effect relationship Do imply that variables share something in

common

CORRELATION COEFFICIENT Expresses degree of linear relatedness

between two variables Varies between –1.00 and +1.00 Strength of relationship is

Indicated by absolute value of coefficient Stronger as shared variance increases

TWO TYPES OF CORRELATIONIf X… And Y…

The correlation is

Example

Increases in value

Increases in value

Positive or directThe taller one gets (X), the more one weighs (Y).

Decreases in value

Decreases in value

Positive or direct

The fewer mistakes one makes (X), the fewer hours of remedial work (Y) one participates in.

Increases in value

Decreases in value

Negative or inverse

The better one behaves (X), the fewer in-class suspensions (Y) one has.

Decreases in value

Increases in value

Negative or inverse

The less time one spends studying (X), the more errors one makes on the test (Y).

WHAT CORRELATION COEFFICIENTS LOOK LIKE Pearson product moment correlation

rxy Correlation between variables x and y

Scattergram representation1. Set up x and y axes

2. Represent one variable on x axis and one on y axis

3. Plot each pair of x and y coordinates

• When points are closer to a straight line, the correlation becomes stronger• As slope of line approaches 45°, correlation becomes stronger

COMPUTING THE PEARSON CORRELATION COEFFICIENT

Where

rxy = the correlation coefficient between X and Y

= the summation sign

n = the size of the sample

X = the individual’s score on the X variable

Y = the individual’s score on the Y variable

XY = the product of each X score times its corresponding Y score

X2 = the individual X score, squared

Y2 = the individual Y score, squared

AN EXAMPLE OF MORE THAN TWO VARIABLES

Grade Reading Math

Grade 1.00 .321 .039

Reading .321 1.00 .605

Math .039 .605 1.00

INTERPRETING THE PEARSON CORRELATION COEFFICIENT “Eyeball” method

Correlations between

Are said to be

.8 and 1.0 Very strong

.6 and .8 Strong

.4 and .6 Moderate

.2 and .4 Weak

0 and .2 Very weak

INTERPRETING THE PEARSON CORRELATION COEFFICIENT Coefficient of determination

Squared value of correlation coefficient Proportion of variance in one variable explained

by variance in the other Coefficient of alienation

1 – coefficient of determination Proportion of variance in one variable unexplained

by variance in the other

RELATIONSHIP BETWEEN CORRELATION COEFFICIENT AND COEFFICIENT OF DETERMINATION

The increase in the proportion of variance explained is not linear

If rxy IsAnd

rxy2 Is

Then the Change

FromIs

0.1 0.01

0.2 0.04 .1 to .2 3%

0.3 0.09 .2 to .3 5%

0.4 0.16 .3 to .4 7%

0.5 0.25 .4 to .5 9%

0.6 0.36 .5 to .6 11%

0.7 0.49 .6 to .7 13%

0.8 0.64 .7 to .8 15%

0.9 0.81 .8 to .9 17%

Doing it by hand

Make a Scatter Diagram

Student Hours Studied Test Score

A 0 52

B 10 92

C 6 75

D 8 71

E 6 64

Change all scores to Z scores Figure the cross-product Add up the cross-produces Divide by the number of people in the study

r=∑ZxZy/N

Z=X-M/SD

Prediction

Zy=(β)(Zx) So, if I know your score on x, I can predict

your score on Y.

© 2009 Pearson Prentice Hall, Salkind.

A great link

http://www.hawaii.edu/powerkills/UC.HTM#C2