Define correlation coefficient. What are its properties?

Correlation Coefficient: Definition and Properties

The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It provides a numerical value that indicates how closely the values of two variables are related. The correlation coefficient is commonly denoted by the symbol "r" and can range from -1 to +1.

Calculation of Correlation Coefficient:

The correlation coefficient is calculated using the following formula:

\[ r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \sum{(y_i - \bar{y})^2}}}\]

Where:
- \(x_i\) and \(y_i\) are individual data points of the two variables.
- \(\bar{x}\) and \(\bar{y}\) are the means of the two variables.

Properties of Correlation Coefficient:

1. Range and Sign: The correlation coefficient always ranges between -1 and +1. A value of -1 indicates a perfect negative correlation, +1 indicates __________ _____ _______ _______ _____ _________ ________ _____.
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