2. Correlation
Definition:
In a distribution if the change in one variable
effects a change in the other variable,
the variable are said to be correlated(or there
is a correlation between the variables)
Let X and Y measure some characteristics
of a particular system .To study the overall
measure of the system it is necessary to
measure the interdependence of X and Y.
3. Correlation
If the quantities(X,Y) vary in such a way that
change in one variable corresponds to
change in the other variable then the variables
X and Y are correlated.
Types of Correlation:
The important ways of classifying the
correlation are:
1. Positive and Negative
2. Simple , Partial and Multiple
3. Linear and non-Linear.
4. Correlation
Methods of Studying Correlation:
The following are the important methods of
ascertaining between two variables.
Scatter diagram method
Karl Pearson’s Co-efficient
Spearman’s Rank Correlation Co-Efficient
• Scatter Diagram Method: The simplest device for
studying correlation between two variables is a special
type of dot chart.
5. Correlation
Karl Pearson’s Co-Efficient of Correlation:
r = (∑ xy) / (N σxσ y )
where
x
x = (x-¯)
y
y= (y- ¯)
σx =Standard Deviation of Series X.
σ y =Standard Deviation of Series Y.
r= The correlation Coefficient
N=Number of Pairs of Observation.
6. Demerits
• By applying Scatter diagram method we can get
an idea about the direction of correlation and also
whether it is high or low. But we cannot establish the
exact degree of correlation between the variables as it is
possible by applying mathematical methods.
• The Karl Pearson’s method is based on the
assumption that the population being studied is normally
distributed. When it is known that the population is not
normal or when he shape the distribution is not known,
there is need for a measure of correlation that involves
no assumption about the parameter of the population.
7. So Why Rank Correlation????
It is possible to avoid making any assumptions
about the populations being studied by ranking the
observations according to size and basing the
calculations on the ranks rather than upon the original
observations. It does not matter which way the items are
ranked, item number one may be the largest or it may be
the smallest. Using ranks rather than actual observations
gives the coefficient of rank correlations.
This method of finding out co variability or the
lack of it between two variables was developed by the
British Psychologist Charles Edward Spearman in 1904.
8. Ranking
A ranking is a relationship between a set of items
such that, for any two items, the first is either 'ranked
higher than', 'ranked lower than' or 'ranked equal to'
the second. In mathematics, this is known as a weak
order or total preorder of objects. It is not necessarily
a total order of objects because two different objects
can have the same ranking. The rankings themselves
are totally ordered. For example, materials are totally
preordered by hardness, while degrees of hardness
are totally ordered.
9. Rank Correlation
“Rank correlation” is the study of relationships
between different rankings on the same set of items. It
deals with measuring correspondence between two
rankings, and assessing the significance of this
correspondence. Spearman’s correlation coefficient is
defined as:
r = 1-((6∑D2)/(N(N-1)2))
Where r , denotes rank coefficient of correlation and D
refers to the difference of rank relation between paired I
tems in two series.
10. Features
The rank method has principal uses:
• The sum of the differences between two variables is
zero.
• Spearman’s rank correlation coefficient ρ is the
Pearsonian correlation coefficient between the ranks.
• The rank correlation can be interpreted in the same
way as Karl Pearson’s correlation coefficient.
11. Features
• Karl Pearson correlation coefficient assumes that the
sample observations are drawn from a normal
population. Rank correlation coefficient is a distribution-
free measure since no strict assumption is made about
the population from which it is drawn.
• The values obtained for two formulae are different due
to the fact that when ranking is used some information
is hidden.
• Spearman’s formula is the only formulae available to
find the correlation between qualitative characters.
12. Types of Rank Methods
In the rank correlation we may have two types of
problems:
• Where ranks are given
• Where ranks are not given
• Where repeated ranks occur
Note:
If r = 1 then there is a perfect Positive correlation
If r = 0 then the variables are uncorrelated
If r=-1 then there is a perfect Negative Correlation
13. Where ranks are given :
Where actual ranks are given to us the steps required for
computing rank correlation are:
• Take the difference of the two ranks, i.e., (R1-R2)
and denotes these differences by D.
• Square these difference and obtain the total
• Apply the formula
14. Where ranks are not given
• When we are given the actual data and not the
ranks, it will be necessary to assign the ranks. Ranks
can be assigned by taking either highest values as 1 or
the lowest value as 1. But whether we start with the
lowest value or the highest value we must follow the
same method in case of both the variables.
