Question

Explain entire concept of Correlation coefficient using 5 real life situations which explains the concept of correlation coefficient in detail.

Answer 1

Let us have a situation to consider two variables at a time. For example,income (x) and expenditure (y) of several families of a village. Then we will collect data on these two variables from the sample selected. Statistical data relating to the simultaneous measurement of two variables are called bivariate data. From the data, if we feel some association, may be linear or non-linear, between these variables x and y then we may be interested to measure numerically how strong the association between variables. Such a measure of the degree of linear or non-linear association between the variables is coming under correlation analysis.

If there is a linear relation between the variables x and y, the degree of linear relation is measured by the coefficient of correlation. If all they given (xi,yi) points are almost satisfying a linear relation, then we are saying that there is a high degree of linear relation between the variables. If the linear relation fitted for the variables is in such a way that the increment in one variable results in the increment of the other also, then there is a direct (or positive) correlation existing between the variables. On the other hand, if the linear relation fitted for the variables is in such a way that the increment in one variable results in the decrease of the other, and then there is an inverse (or negative) correlation existing between the variables. If there is no linear relation existing between the variables, the correlation is zero.

A famous British Statistician, Karl Pearson (1857-1936) suggested a coefficient measure of the degree of linear correlation between two variables x and y, known as Pearson’s coefficient of correlation is denoted by r_xy, where

_{$r_{xy}=\frac{Cov(x,y))}{SD(x)SD(y)}$}

_{Maximum value of} r_xy, is +1 and minimum value is -1.

_{Case 1: Consider X as the sale volume and Y as the profit. Usually Y increases as X increases. If Y is appoximately expressed as alinear function of X in the form aX + b . Then there is a positive correlation exists between X and Y. Here the corefficienf of correlation is between 0 and +1}

_{case 2}

_{If X and Y are perfectly related as Y = aX + b. Then the value of} r_xy, is +1

Case 3:

If X denote the Total production volume of a tea packets of a tea factory and Y is the taverage salary of employees.   Here if we can't identify any linear relation between these variables then r_xy, is 0.

Case 4:

Consoider X as the number of working days of a factory and Y as the number of covid cases reported in the place of factory. Here we know as X increases, Y decresse. If we can identify a relation between X and Y in linear form, and if that relation is not perfect, r_xy, is between 0 and -1

Case 5:

In the above case if the linear relation between X and Y is perfect, then r_xy, is - 1