Galton invented the concept of correlation to measure how variables vary together. Pearson later formalized this method. Correlation measures the strength and direction of association between two variables on a scale of -1 to 1. A value of 0 indicates no association, 1 indicates perfect positive association, and -1 indicates perfect negative association. Correlation is widely used to study relationships between variables in fields such as business, psychology, and engineering.

1. FIZZA SARFARAZ
ABBOTTABAD UNIVERSITY OF
SCIENCE & TECHNOLOGY (A.U.S.T)

3. CORRELATION

4. HISTORY
GALTON:
Obsessed with measurement
Tried to measure everything
from the weather to female
beauty
Invented correlation and
regression
KARL PEARSON
formalized Galton's method
invented method

5. CORRELATION
Measure of the degree to
which any two variables
vary together.
or
Simultaneously variation
of variables in some
direction.
e.g., iron bar

6. INTRODUCTION
Association b/w two variables
Nature & strength of
relationship b/w two variables
Both random variables
Lies b/w +1 & -1
0 = No relationship b/w
variables
-1 = Perfect Negative
correlation
+1 = Perfect positive
correlation

7. Ice-cream - Temperature

8. MATHEMATICALLY








−








−
−
=
∑ ∑∑ ∑
∑ ∑ ∑
n
y)(
y.
n
x)(
x
n
yx
xy
r
2
2
2
2

9. EXAMPLE
AnxietyAnxiety
(X)(X)
TestTest
score (Y)score (Y)
XX22
YY22
XYXY
1010 22 100100 44 2020
88 33 6464 99 2424
22 99 44 8181 1818
11 77 11 4949 77
55 66 2525 3636 3030
66 55 3636 2525 3030
∑∑X = 32X = 32 ∑∑Y = 32Y = 32 ∑∑XX22
= 230= 230 ∑∑YY22
= 204= 204 ∑∑XY=12XY=12
99

10. Calculating CorrelationCalculating Correlation
CoefficientCoefficient
( )( )
94.
)200)(356(
1024774
32)204(632)230(6
)32)(32()129)(6(
22
−=
−
=
−−
−
=r
r = - 0.94
Indirect strong correlation

11. METHODS OF STUDYING
CORRELATION
METHODS
SCATTER DIAGRAM
KARL PEARSON
COEFFICIENT
SPEARMAN'S RANK

12. SCATTER DIAGRAM
Rectangular coordinate
Two quantitative variables
1 variable: independent (X) & 2nd:
dependent (Y)
Points are not joined

13. KARL PEARSON
COEFFICIENT
• Statistic showing the degree of relationship
b/w two variables.
• represented by 'r'
• called Pearson's correlation

14. SPEARMAN'S RANK
Actual measurement of objects/individualsActual measurement of objects/individuals
not availablenot available
Accurate assesment is not possibleAccurate assesment is not possible
Arranged in orderArranged in order
Ordered arrangement: RankingOrdered arrangement: Ranking
Order Given to object: RanksOrder Given to object: Ranks
Correlation blw two sets X & Y: RankCorrelation blw two sets X & Y: Rank
correlationcorrelation

15. PROCEDURE
 Rank values of X from 1-nRank values of X from 1-n
n: # of pairs of values of X & Yn: # of pairs of values of X & Y
 Rank Y from 1-nRank Y from 1-n
 Compute value of 'di' by Xi - YiCompute value of 'di' by Xi - Yi
 Square each di & compute ∑diSquare each di & compute ∑di22
 Apply formula;Apply formula; 2
s 2
6 (di)
r 1
n(n 1)
= −
−
∑

16. EXAMPLE
In a study of the relationship between level education andIn a study of the relationship between level education and
income the following data was obtained. Find the relationshipincome the following data was obtained. Find the relationship
between them and comment.between them and comment.
sample
numbers
level education
(X)
Income
(Y)
A Preparatory.Preparatory. 25
B Primary.Primary. 10
C University.University. 8
D secondarysecondary 10
E secondarysecondary 15
F illiterateilliterate 50
G University.University. 60

17. X Y rank
X
rank
Y
di di2
A Preparato
ry
25 5 3 2 4
B Primary 10 6 5.5 0.5 0.25
C University 8 1.5 7 -5.5 30.25
D secondary 10 3.5 5.5 -2 4
E secondary 15 3.5 4 -0.5 0.25
F illiterate 50 7 2 5 25
G university 60 1.5 1 0.5 0.25
∑ di2
=64

18. Conclusion:Conclusion:
There is an indirect weak correlationThere is an indirect weak correlation
between level of education and income.between level of education and income.
1.0
)48(7
646
1 −=
×
−=sr

19. TYPES
Types
Type 1 Type 2 Type 3

20. TYPE 1
Type 1
Negative NO Perfect
Positive

21. • POSITIVE - both
either increase or
decrease
• NEGATIVE - one
increase while other
decrease
• NO - no correlation
• PERFECT - both
variables are
independents

22. EXAMPLES
+ive Relationships
• WAter consumption
& temperature
• Study times &
grades
-ive Relationships
• Alcohol
consumption &
driving ability
• Price & Quantity
demanded

23. TYPE 2
Type 2
Linear
Non-linear

24. • LINEAR - Perfect
straight line on graph
• NON-LINEAR - Not
a perfect straight line

25. TYPE 3
Type 3
Simple Multiple Partial

26. • SIMPLE - 1 independent & 1 dependent
variable
• MULTIPLE - 1 dep & more than 1 indep
variable
• PARTIAL - 1 dep & more than 1 indep
variable bt only 1 indep variable is considered
while other const

27. COEFFICIENT OF
CORRELATION
Measure of the strength of linear relationship
b/w two variables.
Represented by 'r'
'r' lies b/w +1 & -1
-1 ≤ r ≤ +1
 +ive sign = +ive linear correlation
 -ive sign = -ive linear correlation

28. MATHEMATICALLY 'r'
2 2 2 2
( )( )
[ ( ) ][ ( ) ]
n xy x y
r
n x x n y y
−
=
− −
∑ ∑ ∑
∑ ∑ ∑ ∑

29. INTERPRETATION OF 'r'

30. INTERPRETATION
-1 ≤ r ≤ +1
-1 10-0.25-0.75 0.750.25
strong strongintermediate intermediateweak weak
no relation
perfect
correlation
perfect
correlation
Directindirect

31. CORRELATIION: LINEAR
RELATIONSHIPS
0
20
40
60
80
100
120
140
160
180
0 50 100 150 200 250
Drug A (dose in mg)
SymptomIndex
0
20
40
60
80
100
120
140
160
0 50 100 150 200 250
Drug B (dose in mg)
SymptomIndex
Srong Relationship → Good linear fit
Points clustered closely around a line show a
strong correlation. The line is a good
predictor (good fit) with the data. The more
spread out the points, the weaker the
correlation, and the less good the fit. The line
is a REGRESSSION line (Y = bX + a)

32. r : shows relationship b/w variables either
+ive or -ive
r2
: shows % of variation by best fit line

33. Example:Example:
A sample of 6 children was selected, data about their ageA sample of 6 children was selected, data about their age
in years and weight in kilograms was recorded as shownin years and weight in kilograms was recorded as shown
in the following table . It is required to find thein the following table . It is required to find the
correlation between age and weight.correlation between age and weight.
serial # Age X
I.V(years)
Weight Y
D.V(Kg)
1 7 12
2 6 8
3 8 12
4 5 10
5 6 11
6 9 13

34. Serial
n.
Age
(years)
(x)
Weight
(Kg)
(y)
xy X2
Y2
1 7 12 84 49 144
2 6 8 48 36 64
3 8 12 96 64 144
4 5 10 50 25 100
5 6 11 66 36 121
6 9 13 117 81 169
Total ∑x=
41
∑y=
66
∑xy=
461
∑x2
=
291
∑y2
=
742

35. r = 0.759r = 0.759
strong direct correlationstrong direct correlation
2 2
41 66
461
6r
(41) (66)
291 . 742
6 6
×
−
=
   
− −   
   
2 2
2 2
x y
xy
nr
( x) ( y)
x . y
n n
−
=
   
− − ÷ ÷ ÷ ÷
   
∑ ∑∑
∑ ∑∑ ∑

36. APPLICATIONS
Estimating & improving.,
Seasonal sales for departmental stores
Quantity demanded & production
Motivating tools for employees
Cost of products demanded
accuracy of estimations for demands for sails
Inflation & real wage
Oil exploration

37. Moreover, Radar system is field where
correlation is vehicle to map distance
&
in communication, for instance in digital
receivers.

38. SPSS TUTORIAL
1.Analyz
2.Correlate 3.
(Bivariate)
Points to be noted:
Confidence Level
Correlation is
highly significant
0.01**
Correlation is
significant 0.05*