Pearson Product-Moment Correlation is one of the measures of correlation which quantifies the strength as well as direction of such relationship. It is usually denoted by Greek letter ρ.
This coefficient is used if two conditions are satisfied
The coefficient (ρ) is computed as the ratio of covariance between the variables to the product of their standard deviations. This formulation is advantageous.
First, it tells us the direction of relationship. Once the coefficient is computed, ρ > 0 will indicate positive relationship, ρ < 0 will indicate negative relationship while ρ = 0 indicates non existence of any relationship.
Second, it ensures (mathematically) that the numerical value of ρ range from -1.0 to +1.0. This enables us to get an idea of the strength of relationship - or rather the strength of linear relationship between the variables. Closer the coefficients are to +1.0 or -1.0, greater is the strength of the linear relationship.
As a rule of thumb, the following guidelines are often useful (though many experts could somewhat disagree on the choice of boundaries).
|Value of ρ||Strength of relationship|
|-1.0 to -0.5 or 1.0 to 0.5||Strong|
|-0.5 to -0.3 or 0.3 to 0.5||Moderate|
|-0.3 to -0.1 or 0.1 to 0.3||Weak|
|-0.1 to 0.1||None or very weak|
This measure of correlation has interesting properties, some of which are enunciated below:
While ρ is a powerful tool, it is a much abused one and hence has to be handled carefully.
Under such circumstances it is possible that a non linear relationship exists.
A scatter diagram can reveal the same and one is well advised to observe the same before firmly concluding non existence of a relationship. If the scatter diagram points to a non linear relationship, an appropriate transformation can often attain linearity in which case ρ can be recomputed.
For example, one could compute ρ between size of a shoe and intelligence of individuals, heights and income. Irrespective of the value of ρ, such a correlation makes no sense and is hence termed chance or non-sense correlation.
However the same value of ρ does not tell us if X influences Y or the other way round - a fact that is of grave import in regression analysis.