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In statistics, the correlation coefficient, often represented by the symbol \(r\), indicates the strength and direction of a linear relationship between two variables. The values of \(r\) range from -1 to 1. A value of 1 means there's a perfect positive linear relationship, while -1 indicates a perfect negative linear relationship.
It's essential to grasp that the correlation coefficient alone tells us about the direction and strength, but not necessarily the extent, of the association between variables.

• If \(r = 0\), it means there is no linear relationship present.
• Values closer to 1 or -1 show a stronger relationship.
• A positive \(r\) implies that as one variable increases, so does the other, while a negative \(r\) suggests that as one variable increases, the other decreases.

Identifying the correlation coefficient is only the first step in understanding a relationship's nature. More can be derived when we delve deeper and explore its squares, leading us to the concept of the Coefficient of Determination.

A linear relationship between two variables suggests that the rate of change in one variable is constant concerning the other. Such relationships, depicted as straight lines on graphs, are foundational in many analytical processes.
The concept is simple: if you plot the variables on a graph, and they align to form a straight line, they are linearly related.

• The slope of this line determines the rate at which the variables change in relation to each other.
• In a perfect linear relationship, all data points will lie exactly on a single line.
• However, real-world data often just "approximates" a line, indicating linear tendencies.

Understanding whether a relationship is linear is vital for predicting or explaining data trends. This understanding is crucial when determining how much variation in one variable can be "determined" or predicted by another, which ties directly into studying the Variance Explained.

Variance Explained, often represented as the Coefficient of Determination, or \(R^2\), provides insight into how well data fit a statistical model, commonly a regression model. The value, expressed as a percentage, describes the proportion of the variance in the dependent variable that is predictable from the independent variable(s).
To compute \(R^2\), simply square the correlation coefficient (\(r\)) of a linear relationship.

• An \(R^2\) value of 0 means none of the variance is explained by the model.
• An \(R^2\) value of 1 indicates complete explanatory power from the independent variables over the dependent variable.
• The closer the \(R^2\) is to 1, the stronger the predictive power of the model.

Returning to our original exercise, we determine which correlation coefficient indicates a stronger correlation by squaring them to find \(R^2\). Hence, a correlation of -0.70, when squared, yields an \(R^2\) of 0.49, while +0.50 results in an \(R^2\) of only 0.25.
This illustrates that even though -0.70 is negative, it exhibits a stronger linear relationship due to its higher \(R^2\) value, explaining more variance than +0.50.