Pearson product-moment correlation coefficient

The Pearson product-moment correlation coefficient (PPMCC), also known simply as Pearson's correlation coefficient or the Pearson correlation, is a measure of the strength and direction of association that exists between two continuous variables. This coefficient, denoted as r, quantifies the degree to which a relationship between two variables can be described by a line.

Definition

The Pearson correlation coefficient is calculated as the covariance of the two variables divided by the product of their standard deviations. Mathematically, it is represented as:

\[ r = \frac{\sum (x_i - \overline{x})(y_i - \overline{y})}{\sqrt{\sum (x_i - \overline{x})^2 \sum (y_i - \overline{y})^2}} \]

where:

• \( x_i \) and \( y_i \) are the values of the two variables,
• \( \overline{x} \) and \( \overline{y} \) are the means of the variables,
• \( \sigma_X \) and \( \sigma_Y \) are the standard deviations of the variables.

Interpretation

The value of r ranges from -1 to +1. A value of +1 indicates a perfect positive correlation, meaning that as one variable increases, the other variable also increases. A value of -1 indicates a perfect negative correlation, meaning that as one variable increases, the other decreases. A value of 0 indicates no correlation between the variables.

Applications

Pearson's correlation coefficient is widely used in the fields of statistics, economics, psychology, medicine, and more. It helps researchers to determine the strength of the relationship between variables, which can be crucial for making predictions and for the scientific understanding of relationships.

Limitations

While Pearson's correlation is a powerful tool for statistical analysis, it has its limitations. It is only appropriate for quantifying linear relationships and is sensitive to outliers. Non-linear relationships require different types of analysis, such as Spearman's rank correlation coefficient or Kendall rank correlation coefficient.

See also

• Covariance
• Standard deviation
• Spearman's rank correlation coefficient
• Kendall rank correlation coefficient