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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

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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?

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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.

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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.

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What does correlation measures?

Correlation studies and measures the direction and intensity of relationship among variables.
Correlation measures covariation, not causation.
Correlation should never be interpretated as implying cause and effect relation.
The presence of correlation between two variables X and Y simply means that when the value of one variable is found to change in one direction, the value of other variable is found to change either in same direction(ie positive change) or in the opposite direction(ie negative change)

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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

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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

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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

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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

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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.

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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)

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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).

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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.

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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 -0.25 and -1
Zero	0	0

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Methods of estimating correlation

Scattered Diagram Method.
Karl Pearson’s Coefficient of correlation.
Spearman Rank Correlation

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Scattered Diagram

To make scattered diagram, data are plotted on a graph paper. A dot is marked for each value. The course of these dots would indicate direction and closeness of the variable.

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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’

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Numerical by direct method

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Short-cut method

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Numerical on Short-cut method

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Step-deviation method

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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

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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. 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.

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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.

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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.

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