pa 298 measures of correlation
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MEASURES OF SIMPLE CORRELATION
Prepared by:
Dr. Maria Theresa P. Pelones, DMGraduate School Faculty
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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.
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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.
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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.
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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.
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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
80
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).
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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
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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
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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
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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
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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.
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PEARSON PRODUCT MOMENT CORRELATION COEFFICIENT (R)
Prepared by:
Dr. Maria Theresa P. Pelones, DMGraduate School Faculty
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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:
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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
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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.
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SPEARMAN RANK-ORDER CORREALATION COEFFICIENT
Prepared by:
Dr. Maria Theresa P. Pelones, DMGraduate School Faculty
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SPEA
RMAN
RAN
K-O
RDER
CO
RREA
LATI
ON
CO
EFFI
CIEN
T
SPEARMAN RANK-ORDER CORREALATION COEFFICIENT
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SPEA
RMAN
RAN
K-O
RDER
CO
RREA
LATI
ON
CO
EFFI
CIEN
T
SPEARMAN RANK-ORDER CORREALATION COEFFICIENT
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OTHER MEASURES OF RELATIONSHIP
Prepared by:
Dr. Maria Theresa P. Pelones, DMGraduate School Faculty
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SPEA
RMAN
RAN
K-O
RDER
CO
RREA
LATI
ON
CO
EFFI
CIEN
TPHI COEFFICIENT
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POIN
T BI
SERI
AL C
ORR
EALA
TIO
N
COEF
FICI
ENT
POINT BISERIAL CORREALATION COEFFICIENT
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COEF
FICI
ENT
OF
CON
TIG
ENCY
COEFFICIENT OF CONTIGENCY
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CRAM
ER’S
STA
TIST
ICS
CRAMER’S STATISTICS
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KEN
DALL
S CO
EFFI
CIEN
T O
F CO
NCO
RDAN
CEKENDALLS COEFFICIENT OF CONCORDANCE
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INTE
RPRE
TATI
ON
INTERPRETATION
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CORRELATIONAL STUDIES
Prepared by:
Dr. Maria Theresa P. Pelones, DMGraduate School Faculty
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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
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• When variable A actually causes the change in B.
Caus
ality
Causality
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• 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
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• 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)
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PROBLEM SETS
Prepared by:
Dr. Maria Theresa P. Pelones, DMGraduate School Faculty
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PRO
BLEM
SET
NO
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