Correlation in statistics

2. Correlation:-
Methods of measuring the degree of relationship existing
between two variables have been developed by Galton and
Karl Pearson. Correlation is defined as “The Relationship
which exists between two variables. In other words, when two
variables depend on each other in such a way the movements
(Increase OR Decrease) are accompanied by movements
(Increase OR Decrease) in the other , the variables are said to
be correlated For example ; Increase in height of children is
accompanied by increase in weight.
Some more examples of correlation are defined as :
1. Income and Expenditure
2. Price and Demand
Positive Correlation:
If both the variables are moving in same direction (Increase
OR Decrease) then correlation is said to be direct or Positive,
the for example , Income and Expenditure related with
positive direction.
Negative Correlation:
When the movements of the two variables are in opposite
direction then the correlation is said to be negative, for
example , Price and Demand, if price increases the demand
decreases.
Zero Correlation:
If there is no association between the two variables the
correlation is said to be zero correlation (Both the variables
are independent).
The correlation may be studied by the following two
methods.
1. Scatter Diagram
2. Coefficient of Correlation

3. Scatter Diagram:
The simplest method of investigating the relationship between
two variables is to plot a scatter diagram
Let there will be two series ‘x’ (Independent Variable) and ‘y’
(Dependent Variable) to be represented graphically.
Take the items in ‘x’ series along the axis of ‘x’ and the
corresponding items in ‘y’ series along the y-axis. The
diagram so formed will be a dotted one and scattered,
showing some relationship, such a diagram is called a
scattered.
Linear Correlation:-
If all the points on the scatter diagram tend to lie near a line,
the correlation is said to be linear.
Positive Linear Correlation:-
If all the points tend to lie near an upward sloping line , the
linear correlation is said to be positive correlation.

4. Negative Linear Correlation:-
If all the points tend to lie on a downward sloping line, the
linear correlation is said to be negative correlation.
Perfect Positive Correlation:-
If all the points tend to lie on a rising line the correlation is
perfectly positive correlation.
Perfect Negative Correlation:-
If all the points tend to lie on a falling line the correlation is
perfectly negative correlation.

5. Null Correlation:-
If the points on a scatter diagram do not show a definite
movement then there is no correlation between the variables.
Coefficient of Correlation:-
Numerical Measure of correlation is called coefficient of
correlation. It measured the degree of relationship between the
variables. The formula is called Karl Pearson’s coefficient of
correlation. There are different formulae for the calculation of
Karl Pearson coefficient of correlation.
1. r =
∑( 𝑋−𝑋̅)( 𝑌−𝑌̅)
√∑( 𝑋−𝑋̅)2∑( 𝑌−𝑌̅)2
2. r =
𝑛∑𝑥𝑦−∑𝑥∑𝑦
√( 𝑛∑𝑥2−(∑𝑥)2)( 𝑛∑𝑦2 −(∑𝑦)2)
3. r =
𝐶𝑜𝑣.(𝑥,𝑦)
√𝑉𝑎𝑟 ( 𝑥),𝑉𝑎𝑟( 𝑦)
4. r =
𝑆 𝑥𝑦
√𝑆𝑥2 𝑆𝑦2
5. r =
𝑠 𝑥𝑦
𝑆 𝑥 𝑆 𝑦

6. The Limit of correlation is to be from negative one to positive
one
−1 ≤ 𝑟 ≤ +1
Interpretation of coefficient of correlation:-
If r =1 the correlation is said to be perfect positive
correlation.
If r = -1 the correlation is said to be perfect negative
correlation.
Coefficient of
Correlation
Strength
0.90---------0.99 Very strong
0.78---------0.89 Strong
0.64---------0.77 Moderate
0.46---------0.63 Low
0.10---------0.45 Very Low
0.00---------0.09 No

7. Example-3:
An economist gives the following estimates:
Price 1 2 3 4 5
Demand 9 7 6 3 1
Calculate Karl Pearson’s coefficient of correlation and make
Comments about the type of correlation exist.
Solution:
X
Price
(Rs)
Y
Demand
(Kg)
XY 𝑿 𝟐
𝒀 𝟐
1 9 9 1 81
2 7 14 4 49
3 6 18 9 36
4 3 12 16 9
5 1 5 25 1
∑𝑿
= 𝟏𝟓
∑𝒀
= 𝟐𝟔
∑𝑿𝒚
= 𝟓𝟖
∑𝑿 𝟐
=
55
∑𝒀 𝟐
=17
6
Formula of Karl Pearson coefficient of correlation:
r =
𝑛∑𝑥𝑦−∑𝑥∑𝑦
√( 𝑛∑𝑥2−(∑𝑥)2)( 𝑛∑𝑦2 −(∑𝑦)2)
r =
5(58)−(15)(26)
√[5(55)−(15)2][5(176)−(26)2]
r =
290−390
√(275−225)(880−676)
r =
−100
√(50)(204)

8. r =
−100
√10200
r =
−100
100⋅995
r = - 0.99
Comments :
The Correlation between two variables is very high strong
negative correlation.
Example-4:
Given: ∑( 𝑋 − 𝑋̅)( 𝑌 − 𝑌̅) = 100
∑( 𝑋 − 𝑋̅)2
= 132
∑( 𝒀 − 𝒀̅) 𝟐
= 202
Calculate r = ?
Solution:
r =
∑( 𝑿−𝑿̅)( 𝒀−𝒀̅)
√∑(𝑿−𝑿̅) 𝟐∑( 𝒀−𝒀̅) 𝟐
=
𝟏𝟎𝟎
√( 𝟏𝟑𝟐)( 𝟐𝟎𝟐)
=
𝟏𝟎𝟎
√ 𝟐𝟔,𝟔𝟔𝟒
=
𝟏𝟎𝟎
𝟏𝟔𝟑⋅𝟐
r = 0.61

9. Comments:
Therefore, ‘r’ indicates that the coefficient of correlation
between two variables is low positive correlation.
Example-5:
A sample eight paired values give the following results.
Cov(x,y) = 0.5
V(x) = 15.5
V(y) = 3.8
Calculate coefficient of correlation ‘r’
Solution:
𝑆 𝑥𝑦= 0.5 , 𝑆 𝑥 = 3.94 , 𝑆 𝑦 = 1.95
r =
𝑠 𝑥𝑦
𝑆 𝑥 𝑆 𝑦
r =
0⋅5
3⋅94× 1⋅95
r =
0⋅5
7⋅683
r = 0 ⋅ 065
Comments: No Correlation exist in betweeen two variables.