Abstract
This paper investigates the mathematical foundations and properties of the correlation coefficient, specifically focusing on the validation and proof of the interval [-1, 1]. The research addresses the fundamental statistical concept that the correlation coefficient, a measure of the strength and direction of a linear relationship between two variables, must strictly reside within these bounds. By utilizing algebraic derivations and geometric interpretations, the author provides a rigorous "bases proof" to demonstrate why the coefficient cannot exceed 1 or fall below -1. The study explores the Cauchy-Schwarz inequality as a primary mathematical framework for this proof, ensuring that the ratio of covariance to the product of standard deviations remains normalized. Furthermore, the article discusses the implications of these boundaries in practical data analysis, highlighting how values approaching the extremes indicate perfect linear dependence, while a value of zero suggests a lack of linear correlation. This comprehensive proof serves to reinforce the theoretical reliability of the Pearson correlation coefficient in statistical inference and helps researchers better understand the constraints of bivariate data analysis. Through this formal verification, the paper contributes to the academic clarity of statistical periods and the foundational properties of association measures.
DOI
10.33095/jeas.v17i63.979
Subject Area
Statistical
First Page
254
Last Page
259
Recommended Citation
AlGharabi, S. I. (2011). Bases Proof for Period (1.1) for Correlation Coefficient. Journal of Economics and Administrative Sciences, 17(63), 254-259. https://doi.org/10.33095/jeas.v17i63.979
