Pearson's productmoment correlation coeffecient
Pearson’s productmoment correlation coeffecient
Pearson’s ProductMoment Correlation Coefficient

Pearson’s productmoment correlation coefficient (often just referred to as Pearson’s correlation coefficient or simply Pearson’s r) is a statistic that measures the strength and direction of the linear relationship between two quantitative variables.

This coefficient, denoted by
r
, can take any value from 1 to 1. The closerr
is to either 1 or 1, the stronger is the linear relationship between the variables. 
A Pearson’s coefficient of
r = +1
implies a perfect positive linear relationship,r = 1
implies a perfect negative linear relationship, andr = 0
implies no linear relationship.
Calculation of Pearson’s Correlation Coefficient

The formula to calculate Pearson’s correlation coefficient is given by
r = Σ[(xi  x̄)(yi  ȳ)] / [(n1)sx sy]
wherexi
andyi
are the individual sample points indexed with i,x̄
andȳ
are the sample means,sx
andsy
are the sample standard deviations, and n is the number of samples. 
This formula represents a covariance normalization: it divides the covariance of the two variables by the product of their standard deviations.
Properties of Pearson’s Correlation Coefficient

Pearson’s correlation coefficient is symmetric: correlating
X
withY
is the same as correlatingY
withX
. 
The value of Pearson’s correlation is unaffected by linear transformations (i.e., multiplying a variable by a constant or adding a constant).

However, it is important to note that Pearson’s correlation requires that the variables are measured on at least interval scales, and also assumes linearity and homoscedasticity of the variables.
Interpretation and Application of Pearson’s Correlation Coefficient

The sign of
r
indicates the direction of the correlation: positive values indicate a positive correlation, negative values a negative correlation. 
The magnitude tells us about the strength of the relationship: the closer the coefficient is to either ±1, the stronger the correlation.

Pearson’s coefficient is widely used in scientific research to quantify the degree of linear association between variables.

Even though correlation does not imply causation, this statistic is nonetheless valuable as it can highlight trends in data and suggest potential hypotheses to test.
Limitations of Pearson’s Correlation Coefficient

Pearson’s r only measures linear relationships. Therefore, variables that have a nonlinear relationship may have a correlation coefficient close to zero – even if there is a clear and predictable pattern between them.

Another limitation of Pearson’s r is that it is sensitive to outliers. An unusual observation may greatly increase or decrease the correlation coefficient.

It does not take into account the slope or shape of the distribution, only the strength and direction of the linear relationship. Therefore, it is always recommended to pair it with scatter plots for getting a deeper insight about the nature of association.