The correlation coefficient (denoted as ‘r’) describes the relationship between two variables X and Y. If the value of r is positive it means that the variables are positively correlated and if the value of r is negative it means that the variables are negatively correlated.

Some of the properties of Karl Pearson’s coefficient of correlation are as follows:

1) The value of the correlation coefficient between two variables X and Y can be calculated using the formula,

r= Cov(X,Y)/SD(X)*SD(Y).

Here, Cov(X,Y) is the covariance and SD is the standard deviation.

2. The correlation coefficient is symmetric between the variables X and Y. That is, r_{xy} = r_{yx}.

3. The value of the correlation coefficient always lies between -1 and +1.

-1 ≤ r_{xy} ≤ +1.

4. The correlation coefficient remains unaffected by a change of scale or change of origin. For example, if u = x-a/h and v=y-b/k then,

r_{xy} = r_{uv}.

5. The correlation coefficient is a unitless quantity.

6. The correlation coefficient is the geometric mean of the two regression coefficients,

r^{2} = b_{xy}*b_{yx}.

7. If the correlation coefficient is zero, it does not mean that the two variables are not dependent. It only shows that there is no** linear** dependence between the two variables.

The two variables may still be dependent on each other via a non-linear relationship. For example, suppose we calculate the correlation coefficient for the following data values:

X | Y |

-1 | 1 |

0 | 0 |

1 | 1 |

We can calculate that r=0 for the given data, but the variables X and Y are dependent on each other via the relationship y = x^{2}.

8. We can calculate the coefficient of determination r^{2} which tells us about the explanatory power of our regression model.

Suppose we construct a regression model of the dependent variable Y on the independent variable X. If the value of r^{2} is 0.87, then it means that 87% of the variation in the variable Y is explained by the independent variable X.