CORRELATION

Correlation
• Introduction:- In the previous chapter we have learnt how to construct
summary measures out of a mass of data and changes among similar
variables.
• Now we will learn how to examine the relationship between two variables.
• As a summer heat rises, hill stations, are crowed with more and more
visitors. Ice cream sales become more brisk.
• Thus , the temperature is related to number of visitors and sale of ice creams
 Definition of correlation
• Correlation analysis studies the relation between two variables.
• It deals with questions such as:
• Is there any relationship between two variables?
• It the value of one variable changes, does the value of the other also
changes?
• Do both the variables move in the same direction?
• How strong is the relationship?
 Types of relationship
• 1) Cause and effect relationship:-
• Low agricultural productivity is related to low rainfall.
• 2) Coincidence:-
• The relation between the arrival of migratory birds in a sanctuary and the
birth rate in the locality cannot be given any cause and effect interpretation.
 Types of relationship
• 3) Third variable’s impact on two variables:-
• Brisk sale of ice-creams may be related to higher number of deaths due to
drowning. The victims are not drowned due to eating of ice creams. Rising
temperature leads to brisk sale of ice cream. Moreover, large number of
people start going to swimming pools to beat the heat. This might have
raised the number of deaths by drowning.
 Types of correlation
• Positive correlation:-
• When two variable move in the same direction, that is, when one increases
the other also increases and when one decreases the other also decreases.
X Y X Y
10 100 50 250
20 150 40 200
30 200 30 150
40 250 20 100
 Types of Correlation
Negative correlation –
• When two variables changes in different directions, it is called negative
correlation. Relationship between price and demand
Price Demand
1 40
2 30
3 20
4 10
 Linear and Non-Linear Correlation
Linear Correlation:-
• When two variables change in a constant proportion, it is called linear
correlation.
• If two sets of data bearing fixed proportion to each other are shown on a
graph paper, their relationship will be indicated by a straight line.
• Thus linear correlation implies a straight line relationship.
(a) 2 4 6 8 10 12 14
(b) 5 10 15 20 25 30 35
 Linear and Non-Linear Correlation
Non – linear Correlation
• When the two variables do not change any constant proportion, the
relationship said to be non linear.
• Such a relationship does not form a straight line relationship.
(a) 2 4 6 8 10 12 14
(b) 3 7 12 18 25 35 45
 Simple and Multiple Correlation
1. Simple Correlation
• It implies the study of two variables only. Like the relationship between price and
demand.
2. Multiple Correlation
• When the relationship among three or more than three variables is studied
simultaneously, it is called multiple correlation.
• In case of such correlation the entire set of independent and dependent variables is
simultaneously studied.
 Degrees of Correlation
• Degree of Correlation refers to the Coefficient of correlation .
• Perfect correlation: When two variables are changes in the same proportion it is
called perfect correlation. It may be two kinds:
(i) Perfect Positive:- Correlation is perfectly positive when proportional change in
two variables is in the same direction.
• In this case, Coefficient of correlation is positive(+1)
(ii) Perfect Negative :- Correlation is perfectly negative when proportional change in
two variables is in the opposite direction.
• In this case, Coefficient of correlation is negative(+1)
 Degrees of Correlation
• Absence of correlation :- If there is no relation between two series or
variables, that is, change in one has no effect on the change in other, than
those series and variables lack any correlation between them.
• Limited Degree of Correlation :- Between perfect correlation and absence of
correlation there is a situation of limited degree of correlation.
• In real life, one mostly finds limited degree of correlation. Its coefficient(r) is
more than zero and less than one(r>0 but <1).
The degree of correlation between 0 and 1 may
be rated as
• High: When correlation of two series is close to one, it is called high degree
of correlation. Its coefficient lies between 0.75 and 1.
• Moderate: When correlation of two series is neither large nor small, it is
called moderate degree of correlation. Its coefficient lies between 0.25 and
0.75.
• Low: When correlation of two series is very small, it is called low degree of
correlation. Its coefficient lies between 0 and 0.25.
 Degree of Correlation

Degree Positive Negative

Perfect +1 -1
High Between +0.75 and +1 Between -0.75 and -1
Moderate Between +0.25 and +0.75 Between -0.25 and -0.75

Low Between 0 and +0.25 Between 0 and -0.25

Zero 0 0
 Methods of estimating correlation
• Scattered Diagram Method.
• Karl Pearson’s Coefficient of correlation.
• Spearman Rank Correlation
 Karl Pearson’s Coefficient of Correlation
• Karl Pearson's has given a quantitative method of calculating correlation. It is
an important and widely used method of studying correlation .
• Karl Pearson’s Coefficient of Correlation is generally written as ‘r’
• There are four methods
1. Actual Mean Method
2. Direct Method
3. Assumed mean/Short cut method
4. Step Deviation method
ACTUAL MEAN METHOD

X Y
ഥ=
X ഥ
Y=
𝑁 𝑁
CALCULATE USING ACTUAL MEAN
METHOD

YEAR 2011 2012 2012 2014 2015 2016

BIRTH 24 26 32 33 35 30
RATE
DEATH 15 20 22 24 27 24
RATE
YEAR BR(X) DR(Y) ഥ
x= X-X ഥ
y= Y-Y 𝑥2 𝑦2 XY
2011 24 15 -6 -7 36 49 42
2012 26 20 -4 -2 16 4 8
2013 32 22 2 0 4 0 0
2014 33 24 3 2 9 4 6
2015 35 27 5 5 25 25 25
2016 30 24 0 2 0 4 0
TOTAL 180 132 0 0 90 86 81
81
• 𝑟=− = 81/87.97= 0.92
90∗86
• Hence it is a high positive correlation
 DIRECT METHOD

X= GIVEN OBSERVATION
Y= GIVEN OBSERVATION
N= NO. OF OBSERVATIONS
Marks in Economics Marks in English
4 6
6 8
8 10
10 12
12 14

CALCULATE THE CORRELATION USING DIRECT METHOD

Marks in Economics Marks in English 𝐱𝟐 𝐘𝟐 XY
X Y
4 6 16 36 24
6 8 36 64 48
8 10 64 100 80
10 12 100 144 120
12 14 144 196 168
40 50 360 540 440
Short-cut method
Numerical on Short-cut method
Step-deviation method
 Numerical on Step-deviation Method
Calculate the coefficient of Correlation between the price and quanitiy
demanded-

Price (Rs.) 5 10 15 20 25
Demand(kg) 40 35 30 25 20
 Properties of Correlation Coefficient
• R has no units. It is a pure number.
• A negative value of r indicates an inverse relation, and if r is positive, the two
variables moves in a same direction.
• If r = 0, the two variables are uncorrelated( NIL correlation). There is no
linear relationship between them.
• If r = 1 or r = -1, the correlation is perfect or proportionate.
• The value of correlation coefficient lies between -1 and +1 ie -1< r < +1.
 Merits and demerits of Karl Pearson’s method

Merits
• Most popular method and gives exact measurement of correlation
• Expresses the direction and degree of change in two variables
• Interpretation of the result becomes easy as this method gives us precise
magnitude of the correlation.
Demerits :
• Cannot be used to quantify the qualities of two variables
• The values of correlation are largely affected by the values of extreme items
as this method is based on arithmetic mean and standard deviation
 Spearman’s Rank Correlation Coefficient
• In 1904, Charles Edward Spearman developed a formula to calculate,
coefficient of correlation of qualitative variables.
• It is popularly known as Spearman’s Rank Difference Formula or Method.
• There are some variables whose qualitative measurement is not possible.
• These variables are known as qualitative variables such as beauty, bravery,
wisdom, ability, virtue, etc.
• For eg- consider a situation in a rural area where neither measuring/weighing
scales are not available. The students can be ranked in terms of height and
weight without using measuring rods and scales
 Rank correlation in three different situations

i) When Ranks are given

ii) When Ranks are not given.
iii) When the values of the series are the same.
FORMULA – 1st TWO SITUATIONS
When ranks are given
Ten entries are submitted for a competition. Three judges study each entry and then list the
ten in rank order.
Their ranking is as given in the table :

a) Which pair of Judges agree the most ( or have the same taste in the attribute) ?
b) Which pair of Judges disagree the most ?
Entry 1 2 3 4 5 6 7 8 9 10
no
Judge A 9 3 7 5 1 6 2 4 10 8

Judge B 9 1 10 4 3 8 5 2 7 6

Judge C 6 3 8 7 2 4 1 5 9 10
When Ranks are not given
When the values of the series are the same
Numerical
X 25 45 35 40 15 19 35 42
Y 55 60 30 35 40 42 36 48
1-
1- 0 .
 Importance of correlation
• Formation of laws and concepts.
• Cause and Effect relationship
• Business Decisions
• Policy formulation