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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_r
    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...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=1n1Σ(X1i¯X1)(X2i¯X2)sx1sx2 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/Bookshelves/Probability_Theory/Applied_Probability_(Pfeiffer)/12%3A_Variance_Covariance_and_Linear_Regression/12.02%3A_Covariance_and_the_Correlation_Coefficient
    Reference to Figure 12.2.1 shows this is the average of the square of the distances of the points (r,s)=(X,Y)(ω) from the line s=r (i.e. 1ρ is proportional to the var...Reference to Figure 12.2.1 shows this is the average of the square of the distances of the points (r,s)=(X,Y)(ω) from the line s=r (i.e. 1ρ is proportional to the variance abut the ρ=1 line and 1+ρ is proportional to the variance about the ρ=1 line. ρ=0 iff the variances about both are the same.
  • 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_r
    If all the values of X1 and X2 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 X1 and X2 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/Introductory_Statistics/Statistics_Through_an_Equity_Lens_(Anthony)/01%3A_Chapters/1.06%3A_Correlation_and_Regression_Analysis
    The page discusses the impact of a U.S. Supreme Court ruling on affirmative action and its implications for diversity in higher education and healthcare. It examines statistical concepts like inferent...The page discusses the impact of a U.S. Supreme Court ruling on affirmative action and its implications for diversity in higher education and healthcare. It examines statistical concepts like inferential analysis, emphasizing the distinction between correlation and causation. Correlation measures the linear relationship between variables, while regression predicts dependent variables based on independent ones through equations.
  • https://stats.libretexts.org/Bookshelves/Applied_Statistics/Basic_Statistics_Using_R_for_Crime_Analysis_(Choi)/01%3A_Chapters/1.09%3A_Correlation
    This page provides an introduction to correlation, focusing on the Pearson product-moment correlation coefficient, which measures the linear relationship between two variables. It clarifies the miscon...This page provides an introduction to correlation, focusing on the Pearson product-moment correlation coefficient, which measures the linear relationship between two variables. It clarifies the misconception that correlation does not imply causation, explaining that while necessary, correlation alone is not sufficient for causation. The text elaborates on calculating and interpreting Pearson's r, using the USArrests dataset as an example.
  • https://stats.libretexts.org/Bookshelves/Applied_Statistics/Business_Statistics_(OpenStax)/14%3A_Apppendices/14.01%3A_B__Mathematical_Phrases_Symbols_and_Formulas/14.1.02%3A_Formulas
    This page summarizes essential concepts in probability and statistics, covering key rules, distributions, and formulas for calculating probabilities, means, and variances. It delves into hypothesis te...This page summarizes essential concepts in probability and statistics, covering key rules, distributions, and formulas for calculating probabilities, means, and variances. It delves into hypothesis testing and confidence intervals using Z, t, Chi-square, and F tests.
  • https://stats.libretexts.org/Bookshelves/Applied_Statistics/Business_Statistics_(OpenStax)/13%3A_Linear_Regression_and_Correlation/13.01%3A_The_Correlation_Coefficient_r
    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,...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/Bookshelves/Applied_Statistics/Basic_Statistics_Using_R_for_Crime_Analysis_(Choi)/01%3A_Chapters/1.10%3A_Linear_Regression
    This page provides an introduction to regression analysis, highlighting its relationship with correlation analysis. Regression is a statistical method used to understand and predict the relationship b...This page provides an introduction to regression analysis, highlighting its relationship with correlation analysis. Regression is a statistical method used to understand and predict the relationship between dependent and independent variables. The chapter focuses on simple and multiple linear regression, explained through an inmate survey study assessing the impact of low self-control and age on risky lifestyles.
  • 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_r
    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...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=1n1Σ(X1i¯X1)(X2i¯X2)sx1sx2 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/Bookshelves/Introductory_Statistics/Book%3A_Visual_Statistics_Use_R_(Shipunov)/06%3A_Two-Dimensional_Data_-_Models/6.01%3A_Analysis_of_Correlation
    A positive value of means the correlation is positive (the higher the value of one variable, the higher the value of the other), while negative values mean the correlation is negative (the higher the ...A positive value of means the correlation is positive (the higher the value of one variable, the higher the value of the other), while negative values mean the correlation is negative (the higher the value of one, the lower of the other).

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