The coefficient of correlation r(X , Y) between two variables X and Y, due to Kearl Pearson defined by
r(X , Y) = Covariance (X , Y)/{SD(X).SD(Y)} ,
measures the degree of linearity of both way dependence of X and Y (i.e. the degree of linear correlation between X and Y).
(Here, SD means standard deviation.)
Now, Covariance (X , Y) is not based on the deviations between all pairs of observations on each of the two variables X and Y.
Therefore, logically it can not be regarded as the proper measure of co-variation of X and Y..
Similar is the case of SD(X) and of SD(Y).
Thus it is point to think of whether he coefficient of correlation r(X , Y) between X and Y due to Kearl Pearson, as defined above, can give the actual value of the degree of linearity of both way dependence (i.e. correlation) of X and Y.
Thus one question is
" Can the coefficient of correlation defined by Kearl Pearson be a proper measure of the degree of linear correlation between two variables ? "
In my view it does, although i did not understand what you meant by 'proper' measure. At the end of the day, where and how the measure is used needs consideration.
@ Subrata Chakraborty ,
A measure measure is proper if it can actually measure what is wanted to measure.
There are various measures of dispersion. Logically, there can be only one proper measure of dispersion if exists. A proper measure of dispersion must be based on the deviations between all pairs of observations. Since variance is not based on the deviations between all pairs of observations, how can variance be a proper measure of dispersion ? Similar is the case of co-variance. So how can the coefficient of correlation defined by Kearl Pearson, which is a function of co-variance and variance, be a proper measure of the degree of linear correlation between two variables ?
I think in order to obtain a valid proper measure of correlation, valid proper measures of variation and co-variation are to be used for which
valid proper measures of variation and co-variation are to be searched and defined first.
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