# Coefficient of Correlation

The degree or level of correlation is measured with the help of correlation coefficient or coefficient of correlation. For population data, the correlation coefficient is denoted by $\rho$. The joint variation of $X$ and $Y$ is measured by the covariance of $X$ and $Y$. The covariance of $X$ and $Y$ denoted by $Cov\left( {X,Y} \right)$is defined as:

The $Cov\left( {X,Y} \right)$ may be positive, negative or zero. The covariance has the same units in which $X$ and $Y$ are measured. When $Cov\left( {X,Y} \right)$ is divided by ${\sigma _X}$ and${\sigma _Y}$, we get the correlation coefficient $\rho$. Thus $\rho = \frac{{Cov\left( {X,Y} \right)}}{{{\sigma _X}{\sigma _Y}}}$, $\rho$ is free of the units of measurement.

It is a pure number and lies between $- 1$ and $+ 1$. If$\rho = \pm 1$, it is called perfect correlation. If $\rho = - 1$, it is called perfect negative correlation. If there is no correlation between $X$ and $Y$, then $X$ and $Y$ are independent and $\rho = 0$. For sample data the correlation coefficient denoted by “$r$” is a measure of strength of the linear relation between $X$ and $Y$ variables, where “$r$” is a pure number and lies between $- 1$ and $+ 1$. On the other hand Karl Pearson’s coefficient of correlation is: