Correlation Statistics for Economics Notes

CORRELATION
 Correlation is a statistical tool that helps to
measure and analyze the degree of relationship
between two variables.
 Correlation analysis deals with the association
between two or more variables.

CORRELATION
 The degree of relationship between the variables
under consideration is measure through the
correlation analysis.
 The measure of correlation called the correlation
coefficient .
 The degree of relationship is expressed by
coefficient which range from correlation
( -1 ≤ r ≥ +1)
 The direction of change is indicated by a
sign.
 The correlation analysis enable us to have an
idea about the degree & direction of the
relationship between the two variables under
study.

Correlation
Positive Correlation Negative Correlation
TYPES OF C ORRELATION - TYPE I

TYPES OF C ORRELATION TYPE I
 Positive Correlation: The correlation is said to be
positive correlation if the values of two variables
changing with same direction.
Ex. Pub. Exp. & Sales, Height & Weight.
 Negative Correlation: The correlation is said to be
negative correlation when the values of variables change
with opposite direction.
Ex. Price & Quantity demanded.

DIRECTION OF THE CORRELATION
 Positive relationship – Variables change in the
same direction.
 As X is increasing, Y is increasing
 As X is decreasing, Y is decreasing
⚫ E.g., As height increases, so does weight.
 Negative relationship – Variables change in
opposite directions.
 As X is increasing, Y is decreasing
 As X is decreasing, Y is increasing
⚫ E.g., As TV time increases, grades decrease
Indicated by
sign; (+) or (-).

EXAMPLES
Positive Correlation
 Water consumption
and temperature.
 Study time and
grades.
Negative Correlation
 Alcohol consumption
and driving ability.
 Price & quantity
demanded

Correlation
Simple
Multiple
Partial Total
TYPES OF C ORRELATION TYPE II

TYPES OF C ORRELATION TYPE II
 Simple correlation: Under simple correlation
problem there are only two variables are studied.
 Multiple Correlation: Under Multiple
Correlation three or more than three variables
are studied. Ex. Qd = f ( P,PC , PS , t, y )
 Partial correlation: analysis recognizes more
than two variables but considers only two
variables keeping the other constant.
 Total correlation: is based on all the relevant
variables, which is normally not feasible.

Types of Correlation Type III
Correlation
LINEAR NON LINEAR

TY P E S O F C O R R ELATION TY P E
III
 Linear correlation: Correlation is said to be
linear when the amount of change in one
variable tends to bear a constant ratio to the
amount of change in the other. The graph of the
variables having a linear relationship will form
a straight line.
Ex X = 1, 2, 3, 4, 5, 6, 7, 8,
Y = 5, 7, 9, 11, 13, 15,
17, 19, Y = 3 + 2x
 Non Linear correlation: The
correlation
would be non linear if the amount of
change in
one variable does not bear a constant ratio to

CORRELATION & CAUSATION
 Causation means cause & effect relation.
 Correlation denotes the interdependency among the
variables for correlating two phenomenon, it is
essential that the two phenomenon should have
cause-effect relationship,& if such relationship does
not exist then the two phenomenon can not be
correlated.
 If two variables vary in such a way that movement in
one are accompanied by movement in other, these
variables are called cause and effect relationship.
 Causation always implies correlation but correlation
does not necessarily implies causation.

DEGREE OF CORRELATION
 Perfect Correlation
 High Degree of Correlation
 Moderate Degree of Correlation
 Low Degree of Correlation
 No Correlation

METHODS OF STUDYING CORRELATION
Methods
Graphic
Methods
Scatter
Diagram
Correlation
Graph
Algebraic
Methods
Karl
Pearson’s
Coefficient
Rank
Correlation
Concurrent
Deviation

SCATTER DIAGRAM METHOD
• Scatter Diagram is a graph of
observed plotted points where each
points represents the values of X
& Y as a coordinate.
•
It portrays the relationship
between these two variables
graphically.

A P E R F E C T P O S ITIVE CORREL ATION
Height
Height
of A
Height
of B
Weight
Weight
of B
Weight
of A
A linear
relationshi
p

HIGH DE G R E E O F POSITIVE
CORREL ATION
 Positive relationship
Height
Weigh
t
r = +.80

 Moderate Positive Correlation
r = + 0.4
Shoe
Size
Weight

 Perfect Negative Correlation
r = -1.0
TV
watching
per
week
Exam score

Exam score
 Moderate Negative Correlation
r = -.80
TV
watching
per
week

 Weak negative Correlation
Weight
Shoe
Size
r = - 0.2

 No Correlation (horizontal line)
• r = 0.0
• IQ
Height

DE G R E E O F C O R RELATION (R )
r = +.80 r = +.60
r = +.40 r = +.20

DIRECTION O F THE RELATIONSHIP
 Positive relationship – Variables change in the same
direction.
 As X is increasing, Y is increasing
 As X is decreasing, Y is decreasing
⚫ E.g., As height increases, so does weight.
 Negative relationship – Variables change in opposite
directions.
 As X is increasing, Y is decreasing
 As X is decreasing, Y is increasing
⚫ E.g., As TV time increases, grades decrease
Indicated by
sign; (+) or (-).

ADVANTAGES O F SCATTER DIAGRAM
Simple & Non Mathematical method
Not influenced by the size of extreme
item
First step in investing the
relationship between two variables

DISADVANTAGE O F SCATTER DIAGRAM
Can not adopt the an exact
degree of correlation

CORRELATION GRAPH
100
50
0
2012 2013 2014 2015 2016
2017
150
300
250
200
Consumption
Production

KARL PEARSON’S COEFFICIENT OF
CORRELATION
 It is quantitative method of measuring
correlation
 This method has been given by Karl
Pearson
 It’s the best method

CALCULATION OF COEFFICIENT OF
CORRELATION – ACTUAL MEAN METHOD
 Formula used is:
⚫ r =
Σ 𝑥𝑦
Σ𝑥2 .
Σ𝑦2
where x = X – 𝑋 ; y = Y
– 𝑌
Q1: Find Karl Pearson’s coefficient of correlation:
Ans: 0.96
Summation of product of deviations of X & Y series from their respective
arithmetic means = 122 Ans: 0.89
X 2 3 4 5 6 7 8
Y 4 7 8 9 10 14 18
X- Series Y-series
No. of items 15 15
AM 25 18
Squares of deviations from mean 136 138

PRACTICE PROBLEMS - CORRELATION
Arithmetic Means of X & Y are 6 & 8 respectively. Ans: – 0.92
Q4: Find the number of items as per the given data: r
= 0.5, xy
Ʃ = 120, σy = 8, x
Ʃ 2 = 90
where x & y are deviations from arithmetic
means
Ans: 10
Q5: Find r:
X
Ʃ = 250, Y
Ʃ = 300, (X
Ʃ – 25)2 = 480, (Y
Ʃ –
30)2 = 600
X 6 2 10 4 8
Y 9 11 ? 8 7

CORRELATION – ASSUMED MEAN METHOD
⚫ r =
𝑁 .Σ𝑑𝑥𝑑𝑦 − Σ𝑑𝑥.Σ𝑑𝑦
𝑁.Σ𝑑𝑥2 −(Σ𝑑𝑥)2 𝑁.Σ𝑑𝑦2
−(Σ𝑑𝑦)2
Q6:Find r:
Ans: 0.98
Q7: Find r, when deviations of two series from assumed mean
are as follows: Ans: 0.895
X 10 12 18 16 15 19 18 17
Y 30 35 45 44 42 48 47 46
Dx +5 -4 -2 +20 -10 0 +3 0 -15 -5
Dy +5 -12 -7 +25 -10 -3 0 +2 -9 -15

CORRELATION – ACTUAL DATA METHOD
⚫ r =
𝑁.Σ𝑋𝑌 − Σ𝑋.Σ𝑌
𝑁.Σ𝑋2 −(Σ𝑋)2 𝑁.Σ𝑌2 −
(Σ𝑌)2
Q8:Find r:
Ans: 0.98
Q9: Calculate product moment correlation coefficient from the
following data: Ans: 0.996
X 10 12 18 16 15 19 18 17
Y 30 35 45 44 42 48 47 46
X -5 -10 -15 -20 -25 -30
Y 50 40 30 20 10 5

IMPORTANT TYPICAL PROBLEMS
Q10: Calculate the coefficient of correlation from the following
data and interpret the result: Ans: 0.76
N = 10, XY
Ʃ = 8425, 𝑋 = 28.5, 𝑌 = 28.0,
𝜎𝑥 = 10.5, 𝜎𝑦 = 5.6
Q11: Following results were obtained from an analysis:
N = 12, XY
Ʃ = 334, X
Ʃ = 30, Y
Ʃ = 5, X
Ʃ 2 = 670, Y
Ʃ 2 = 285
Later on it was discovered that one pair of values (X = 11,
Y = 4) were wrongly copied. The correct value of the pair
was (X = 10, Y = 14).
Find the correct value of correlation
coefficient.
Ans: 0.774

VARIANCE – COVARIANCE METHOD
 This method of determining correlation coefficient is based on
covariance.
Q12: For two series X & Y, Cov(X,Y) = 15, Var(X)=36, Var (Y)=25.
Find r.
Q13: Find r when N = 30, 𝑋 = 40, 𝑌 = 50, 𝜎𝑥 =
6,
Ans: 0.5
𝜎𝑦 = 7, Σ𝑥𝑦 =
360
Ans: 0.286
Q14: For two series X & Y, Cov(X,Y) = 25, Var(X)=36, r = 0.6.
Find 𝜎𝑦. Ans: 6.94

CALCULATION OF CORRELATION
COEFFICIENT – GROUPED DATA
Q15: Calculate Karl Pearson’s coefficient of correlation:
Ans: 0.33
X / Y 10-25 25-40 40-55
0-20 10 4 6
20-40 5 40 9
40-60 3 8 15

PROPERTIES OF COEFFICIENT OF
CORRELATION
 Karl Pearson’s coefficient of correlation lies between - 1 & 1, i.e. –
1 ≤ r ≤ +1
 If the scale of a series is changed or the origin is shifted, there is
no effect on the value of ‘r’.
 ‘r’ is the geometric mean of the regression coefficients byx & bxy, i.e.
r = 𝑏𝑥𝑦 . 𝑏𝑦𝑥
 If X & Y are independent variables, then coefficient of correlation
is zero but the converse is not necessarily true.
 ‘r’ is a pure number and is independent of the units of
measurement.
 The coefficient of correlation between the two variables x & y is
symmetric. i.e. ryx = rxy

PROBABLE ERROR &
STANDARD ERROR

PRACTICE PROBLEM – PROBABLE
ERROR

SPEARMAN’S RANK
CORRELATION METHOD

THREE CASES
Spearman’s
Rank
Correlation
Method
When ranks are
given
When ranks are
not given
When equal or
tied ranks exist

PRACTICE PROBLEMS – RANK
CORRELATION (WHEN RANKS ARE GIVEN)
Q18: In a fancy dress competition, two judges accorded the
following ranks to eight participants:
Calculate the coefficient of rank correlation. Ans: .62
Q19: Ten competitors in a beauty contest are ranked by three judges X,
Y, Z:
Use the rank correlation coefficient to determine which pair of
judges has the nearest approach to common tastes in beauty.
Ans: X & Z
Judge X 8 7 6 3 2 1 5 4
Judge Y 7 5 4 1 3 2 6 8
X 1 6 5 10 3 2 4 9 7 8
Y 3 5 8 4 7 10 2 1 6 9
Z 6 4 9 8 1 2 3 10 5 7

CORRELATION
(WHEN RANKS ARE NOT GIVEN)
Q20: Find out the coefficient of Rank Correlation
between X & Y:
Ans: 0.48
X 15 17 14 13 11 12 16 18 10 9
Y 18 12 4 6 7 9 3 10 2 5

CORRELATION (WHEN RANKS ARE
EQUAL OR TIED)

PRACTICE PROBLEMS – RANK CORRELATION
(WHEN RANKS ARE EQUAL OR TIED)
Q21: Calculate R:
Ans: – 0.37
Q22: Calculate Rank Correlation:
Ans: 0.43
X 15 10 20 28 12 10 16 18
Y 16 14 10 12 11 15 18 10
X 40 50 60 60 80 50 70 60
Y 80 120 160 170 130 200 210 130

IMPORTANT TYPICAL PROBLEMS –
RANK CORRELATION
Q23: Calculate Rank Correlation from the following data:
Ans: 0.64
Q24: The coefficient of rank correlation of marks obtained by 10
students in English & Math was found to be 0.5. It was later
discovered that the difference in the ranks in two subjects was
wrongly taken as 3 instead of 7. Find the correct rank correlation.
Ans: 0.26
Q25: The rank correlation coefficient between marks obtained by
some students in English & Math is found to be 0.8. If the total of
squares of rank differences is 33, find the number of students.
Ans: 10
Serial No. 1 2 3 4 5 6 7 8 9 10
Rank
Difference
-2 ? -1 +3 +2 0 -4 +3 +3 -2

PRACTICE PROBLEMS – COEFFICIENT OF
CONCURRENT DEVIATIONS
Q26: Find the Coefficient of Concurrent Deviation from
the following data:
Ans: – 1
Q27: Find the Coefficient of Concurrent Deviation from
the following data:
Ans: – 0.75
Year 2001 2002 2003 2004 2005 2006 2007
Demand 150 154 160 172 160 165 180
Price 200 180 170 160 190 180 172
X 112 125 126 118 118 121 125 125 131 135
Y 106 102 102 104 98 96 97 97 95 90

COEFFICIENT OF DETERMINATION
(COD)

PRACTICE PROBLEMS – COD
Q28: The coefficient of correlation between consumption
expenditure (C) and disposable income (Y) in a study was
found to be +0.8. What percentage of variation in C are
explained by variation in Y? Ans: 64%

CLASS TEST
Q1: In a fancy dress competition, two judges accorded the
following ranks to eight participants:
Calculate the coefficient of rank correlation.
Q2: Following results were obtained from an analysis:
N = 12, XY
Ʃ = 334, X
Ʃ = 30, Y
Ʃ = 5, X
Ʃ 2 = 670, Y
Ʃ 2 = 285
Later on it was discovered that one pair of values (X = 11,
Y = 4) were wrongly copied. The correct value of the pair
was (X = 10, Y = 14).
Find the correct value of correlation coefficient.
Judge X 8 7 6 3 2 1 5 4
Judge Y 7 5 4 1 3 2 6 8

Correlation Statistics for Economics Notes

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