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- https://stats.libretexts.org/Courses/Fresno_City_College/Book%3A_Business_Statistics_Customized_(OpenStax)/13%3A_Linear_Regression_and_Correlation/13.02%3A_The_Correlation_Coefficient_rFor a sample of data, the statistic, r, developed by Karl Pearson in the early 1900s, is an estimate of the population correlation and is defined mathematically as: \[r=\frac{\frac{1}{n-1} \Sigma\left...For a sample of data, the statistic, r, developed by Karl Pearson in the early 1900s, is an estimate of the population correlation and is defined mathematically as: \[r=\frac{\frac{1}{n-1} \Sigma\left(X_{1 i}-\overline{X}_{1}\right)\left(X_{2 i}-\overline{X}_{2}\right)}{s_{x_{1}} s_{x_{2}}}\nonumber\] OR \[r=\frac{\sum X_{1 i} X_{2 i}-n \overline{X}_{1}-\overline{X}_{2}}{\sqrt{\left(\Sigma X_{1 i}^{2}-n \overline{X}_{1}^{2}\right)\left(\Sigma X_{2 i}^{2}-n \overline{X}_{2}^{2}\right)}}\nonumber…
- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/BFE_1201_Statistical_Methods_for_Finance_(Kuter)/08%3A_Linear_Regression_and_Correlation/8.02%3A_The_Correlation_Coefficient_rIf all the values of \(X_1\) and \(X_2\) are on a straight line the correlation coefficient will be either \(1\) or \(-1\) depending on whether the line has a positive or negative slope and the closer...If all the values of \(X_1\) and \(X_2\) are on a straight line the correlation coefficient will be either \(1\) or \(-1\) depending on whether the line has a positive or negative slope and the closer to one or negative one the stronger the relationship between the two variables.
- https://stats.libretexts.org/Bookshelves/Applied_Statistics/Business_Statistics_(OpenStax)/13%3A_Linear_Regression_and_Correlation/13.01%3A_The_Correlation_Coefficient_rThis page explains univariate, bivariate, and multivariate data types, with a focus on bivariate data analysis using time series, cross-section, and panel data. It defines the correlation coefficient,...This page explains univariate, bivariate, and multivariate data types, with a focus on bivariate data analysis using time series, cross-section, and panel data. It defines the correlation coefficient, which measures the strength and direction of linear relationships between two variables, ranging from -1 to 1. It clarifies that correlation does not imply causation and emphasizes the use of software for complex analyses along with the importance of visualizing relationships through scatter plots.
- https://stats.libretexts.org/Courses/Fresno_City_College/Introduction_to_Business_Statistics_-_OER_-_Spring_2023/13%3A_Linear_Regression_and_Correlation/13.02%3A_The_Correlation_Coefficient_rFor a sample of data, the statistic, r, developed by Karl Pearson in the early 1900s, is an estimate of the population correlation and is defined mathematically as: \[r=\frac{\frac{1}{n-1} \Sigma\left...For a sample of data, the statistic, r, developed by Karl Pearson in the early 1900s, is an estimate of the population correlation and is defined mathematically as: \[r=\frac{\frac{1}{n-1} \Sigma\left(X_{1 i}-\overline{X}_{1}\right)\left(X_{2 i}-\overline{X}_{2}\right)}{s_{x_{1}} s_{x_{2}}}\nonumber\] OR \[r=\frac{\sum X_{1 i} X_{2 i}-n \overline{X}_{1}-\overline{X}_{2}}{\sqrt{\left(\Sigma X_{1 i}^{2}-n \overline{X}_{1}^{2}\right)\left(\Sigma X_{2 i}^{2}-n \overline{X}_{2}^{2}\right)}}\nonumber…
