Page 296 - DMTH404_STATISTICS

Statistics

Notes can compute mean, variance, skewness etc. for each individual characteristics. In addition to
this, in the study of a bivariate distribution, we are also interested in knowing whether there
exists some relationship between two characteristics or in other words, how far the two variables,
corresponding to two characteristics, tend to move together in same or opposite directions i.e.
how far they are associated.

The knowledge of this type of relationship is useful for predicting the value of one variable
given the value of the other. It also helps in understanding and analysis of various economic and
business problems. It should be noted here that statistical relations are different from the exact
mathematical relations. Given a statistical relation Y = a + bX, between two variables X and Y,
we can only get a value of Y that we expect on the average for a given value of X. The study of
relationship between two or more variables can be divided into two broad categories:
(i) To determine whether there exists some sort of association between the variables. If so,
what is the degree of association or the magnitude of correlation between the two.
(ii) To determine the most suitable form of the relationship between the variables given that
they are correlated.
The first category relates to the study of 'Correlation' which will be discussed in this chapter and
the second relates to the study of 'Regression', to be discussed in next chapter.
22.1 Definition of Correlation

Various experts have defined correlation in their own words and their definitions, broadly
speaking, imply that correlation is the degree of association between two or more variables.
Some important definitions of correlation are given below:
(i) "If two or more quantities vary in sympathy so that movements in one tend to be accompanied by
corresponding movements in other(s) then they are said to be correlated.”
— L.R. Connor
(ii) "Correlation is an analysis of covariation between two or more variables.”
— A.M. Tuttle

(iii) "When the relationship is of a quantitative nature, the appropriate statistical tool for discovering and
measuring the relationship and expressing it in a brief formula is known as correlation.”
— Croxton and Cowden

(iv) "Correlation analysis attempts to determine the 'degree of relationship' between variables".
— Ya Lun Chou
Correlation Coefficient: It is a numerical measure of the degree of association between two or
more variables.
22.1.1 Scope of Correlation Analysis

The existence of correlation between two (or more) variables only implies that these variables
(i) either tend to increase or decrease together or (ii) an increase (or decrease) in one is accompanied
by the corresponding decrease (or increase) in the other. The questions of the type, whether

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