Final answer:
The correlation coefficient (r) measures the strength and direction of the linear relationship between two variables, ranging from -1 to +1. Developed by Karl Pearson, this statistic is crucial for understanding how variables are related, though it does not indicate causation. The coefficient of determination (r²) represents the variance explained by the relationship.
Step-by-step explanation:
The correlation coefficient, represented by the letter r, measures the degree to which two variables are related. It is a statistical index that ranges from -1 to +1, with -1 indicating a perfect inverse correlation, +1 indicating a perfect direct correlation, and 0 indicating no correlation at all. The strength and direction of a linear relationship between an independent variable (x) and a dependent variable (y) are measured by this coefficient.
Karl Pearson developed this measure in the early 1900s, and it is calculated using a formula that incorporates the sum of the products of the x-coordinates and y-coordinates, the sum of the x-coordinates, the sum of the y-coordinates, and the squares of these sums.
The coefficient of determination, r², is another important term related to the correlation coefficient. It is the square of the correlation coefficient and represents the proportion of the variance in the dependent variable that can be predicted from the independent variable. In other words, it quantifies how well the regression line approximates the real data points.
When analyzing data, it is essential to note that correlation does not imply causation; a high correlation between two variables does not mean that one causes the other. It is also important to consider the sample size (n), as it affects the reliability of the correlation.
The correlation coefficient can readily be calculated using statistical software, but it is essential to understand what the value signifies and its limitations. As such, r provides critical insight into the nature of relationships between variables but must be interpreted within the context of the entire dataset.