Product Moment Correlation Coefficient

Definition

The product moment correlation coefficient (PMCC), also known as Pearson’s correlation coefficient (r), quantifies the strength and direction of the linear relationship between two variables.
It ranges from -1 to 1, where 1 denotes a perfect positive correlation, -1 a perfect negative, and 0 indicates no linear relationship.
A positive value suggests that as one variable increases, the other variable also increases. Conversely, a negative value suggests that as one variable increases, the other decreases.

Calculation Method

PMCC is calculated using a mathematical formula:
- r = Σ[(xi - x̄)(yi - ȳ)] / sqrt[(Σ(xi - x̄)^2)(Σ(yi - ȳ)^2)]
Here,
- xi and yi are individual data point values
- x̄ and ȳ are mean values for each variable
- The sums (“Σ”) are over all data points.
This formulation automatically compensates for changes in scale, making the PMCC a unitless measure.

Interpretation

The PMCC can be used to interpret the strength and direction of the linear relationship:
- A strong correlation is when the absolute value of r is close to 1.
- A weak correlation is when the absolute value of r is far from 1.
- A positive correlation is indicated by a positive r.
- A negative correlation is indicated by a negative r.
The PMCC, on its own, does not establish causal relationships.
Observing a PMCC of 0 does not rule out other possible types of relationships (e.g., quadratic or exponential relationships).

Limitations

The PMCC can only measure linear relationships. It’s unable to detect more complex relationship patterns.
It is sensitive to outliers, a single outlier can dramatically influence the coefficient.
It only considers pairwise relationships – cannot capture multiple variable interactions.

Remember, practice and understanding the principle behind the math are crucial while revising product moment correlation coefficient. Try to solve multiple questions and case studies to familiarise yourself with different scenarios.