MEASURES OF SIMPLE CORRELATION

Prepared by:

Dr. Maria Theresa P. Pelones, DMGraduate School Faculty

CORR

ELAT

ION

(D

efini

tion)

Examples of variables that may be correlated:

height and shoe sizeSAT score and grade point averagenumber of cigarettes smoked per day and lung capacity

A correlation is a relationship between two

variables.

The data can be represented by the ordered

pairs (x,y) where x is the independent, or

explanatory, variable and y is the dependent, or response, variable.

CORR

ELAT

ION

(D

efini

tion)

Are two or more

variable related?

Statisticians used

numerical measure to determine

whether two or more

variables are related and

If so, what is the strength

of the relationship

? to determine

the strength of the

relationship bet and among

variables. This

measures are called

correlation coefficient.

What type of

relationship exists? There

are two types of

relationship exist. Simple

and multiple.

Correlation is a statistical method used to determine whether a relationship between variables exist. Correlations

refers to association which occurs between two or more statistical series of values.

=> In statistics correlation is usually referred to Coefficient of correlation (r). A value of (r) is the same with the mean and standard deviation which characterizes the whole set of observation and tells a story.

=>It is applicable on both descriptive and experimental researches. =>It is also a single number that tells us what extent two variables are

related. It can vary from a value of 1.00 which means perfect positive correlation, through zero, which means no correlation at all and -1.00

which means perfect negative correlation.

CORR

ELAT

ION

CO

EFFI

CIEN

TCorrelation Coefficient

A correlation coefficient indicates the extent to which two variables are related.

Independent variable: is a variable that can be controlled or manipulated while, Dependent variable: is a variable that cannot be controlled or manipulated. Its values are predicted from the independent variable.

It can range from -1.0 to +1.0

A positive correlation coefficient indicates a positive relationship, a negative coefficient indicates an inverse relationship

Correlation CANNOT be equated with causality.

USE

S O

F CO

RREL

ATIO

N

COEF

FICI

ENT

It indicates the amount of agreement between scores on any two sets of data. It is an index of predictive value of a test.

It is a form of reliability coefficient which can be obtained by correlating scores of two alternative or parallel forms of the same test.

The correlation value is always relative to the situation under which it is obtained and should be interpreted in the light of those circumstances.

Its size does not represent absolute natural fact.

SCAT

TER

PLO

TThe independent and dependent can be plotted on a graph called a scatter plot.

By convention, the independent variable is plotted on the horizontal x-axis. The dependent variable is plotted on the vertical y-axis.

• A scatter plot is a graph of the ordered pairs (x,y) of numbers consisting of the independent variables, x, and the dependent variables, y.

• Please use excel to create a scatter plot.

Scatter Plot

0

20

40

60

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100

0 1 2 3 4 5 6 7

Hours Studied

Gra

de

(%

)

A positive relationship exists when both variables increase or decrease at the same time. (Weight and height).

A negative relationship exist when one variable increases and the other variable decreases or vice versa. (Strength and age).

Correlation = +1

15

20

25

10 12 14 16 18 20

Independent variable

Dep

ende

nt v

aria

ble

Correlation = -1

15

20

25

10 12 14 16 18 20

Independent variable

Depe

nden

t var

iabl

e

RAN

GE

OF

CORR

EALT

ION

CO

EFFI

CIEN

TIn case of exact positive linear relationship the

value of r is +1. In case of a strong positive linear

relationship, the value of r will be close to + 1.

In case of exact negative linear relationship the

value of r is –1. In case of a strong negative linear

relationship, the value of r will be close to – 1.

Range of Correlation Coefficient

RAN

GE

OF

CORR

EALT

ION

CO

EFFI

CIEN

TCorrelation = 0

10

15

20

25

30

0 2 4 6 8 10 12

Independent variable

Depe

nden

t var

iabl

e

Correlation = 0

0

10

20

30

0 2 4 6 8 10 12

Independent variable

Dep

ende

nt v

aria

ble

In case of a weak relationship the value of r

will be close to 0.

In case of nonlinear relationship the value of r

will be close to 0.

Range of Correlation Coefficient

CoefficientRange

Strength ofRelationship

0.00 - 0.20 Very Low

0.20 - 0.40 Low

0.40 - 0.60 Moderate

0.60 - 0.80 High Moderate

0.80 - 1.00 Very High

CORR

EALT

ION

CO

EFFI

CIEN

TIN

TERP

RETA

TIO

NCorrelation Coefficient

interpretation

EXAM

PLE

For seven random summer days, a person recorded the temperature and their water consumption, during a three-hour period spent outside.

Temperature (F)

Water Consumption

(ounces)

75 1683 2085  2585 2792 3297 4899 48

EXAM

PLE

Temperature (F)Water Consumption

(ounces)

75 1683 2085  2585 2792 3297 4899 48

For seven random summer

days, a person recorded the

temperature and their water

consumption, during a three-hour period

spent outside.

PEARSON PRODUCT MOMENT CORRELATION COEFFICIENT (R)

Prepared by:

Dr. Maria Theresa P. Pelones, DMGraduate School Faculty

PEAR

SON

CO

RREL

ATIO

N

COEF

FICI

ENT

PEARSON CORRELATION COEFFICIENT is a linear correlation used to determine the relationship between two sets of variables, X and Y. This is the most common measure to determine the association between two sets of variables quantitatively.

This measures of relationship assumes that the two variables are both INTERVAL. The value is determined using the formula below:

PEAR

SON

CO

RREL

ATIO

N

COEF

FICI

ENT

(EXA

MPL

E)Using the data on age and blood pressure, let’s calculate the x, y, xy, x2 and y2

Student Age Blood Pressure

Age*BP age2 BP2

A 43 128 5504 1849 16384

B 48 120 5760 2304 14400

C 56 135 7560 3136 18225

D 61 143 8723 3721 20449

E 67 141 9447 4489 19881

F 70 152 10640 4900 23104

Sum 345 819 47634 20399 112443

PEAR

SON

CO

RREL

ATIO

N

COEF

FICI

ENT

(EXA

MPL

E)Substitute in the formula and solve for r:

r= {(6*47634)-(345*819)}/{[(6*20399)-3452][(6*112443)-8192]}0.5.

r= 0.897.• The correlation coefficient suggests a strong

positive relationship between age and blood pressure.

SPEARMAN RANK-ORDER CORREALATION COEFFICIENT

Prepared by:

Dr. Maria Theresa P. Pelones, DMGraduate School Faculty

SPEA

RMAN

RAN

K-O

RDER

CO

RREA

LATI

ON

CO

EFFI

CIEN

T

SPEARMAN RANK-ORDER CORREALATION COEFFICIENT

SPEA

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K-O

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EFFI

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T

SPEARMAN RANK-ORDER CORREALATION COEFFICIENT

OTHER MEASURES OF RELATIONSHIP

Prepared by:

Dr. Maria Theresa P. Pelones, DMGraduate School Faculty

SPEA

RMAN

RAN

K-O

RDER

CO

RREA

LATI

ON

CO

EFFI

CIEN

TPHI COEFFICIENT

POIN

T BI

SERI

AL C

ORR

EALA

TIO

N

COEF

FICI

ENT

POINT BISERIAL CORREALATION COEFFICIENT

COEF

FICI

ENT

OF

CON

TIG

ENCY

COEFFICIENT OF CONTIGENCY

CRAM

ER’S

STA

TIST

ICS

CRAMER’S STATISTICS

KEN

DALL

S CO

EFFI

CIEN

T O

F CO

NCO

RDAN

CEKENDALLS COEFFICIENT OF CONCORDANCE

INTE

RPRE

TATI

ON

INTERPRETATION

CORRELATIONAL STUDIES

Prepared by:

Dr. Maria Theresa P. Pelones, DMGraduate School Faculty

A correlation tells you that a relationship exists between 2 variables (aside from the 3rd variable problem), but tell you absolutely nothing about cause and effect.

Corr

ela

tional Stu

die

sCorrelational Studies

• When variable A actually causes the change in B.

Caus

ality

Causality

• Variables A and B really do NOT have anything to do with each other but happen to go up or down simultaneously.

Corr

ela

tional Stu

die

sSheer Coincidence

• Variable A is correlated with variable B but there is a third factor C (the common underlying cause) that causes the changes in both A and B.

Com

mon

Und

erly

ing

Caus

e(s)

Common Underlying Cause(s)

PROBLEM SETS

Prepared by:

Dr. Maria Theresa P. Pelones, DMGraduate School Faculty

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