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Correlation in statistics

Correlation describes the statistical relationship between two or more variables. It indicates the extent to which changes in one variable are associated with changes in another variable. A positive correlation means that increasing values in one variable are associated with increasing values in the other variable, while a negative correlation suggests that increasing values in one variable are associated with decreasing values in the other variable.

Measurement of Correlation:

There are various methods to measure correlation, with the Pearson correlation coefficient being one of the most common. The Pearson correlation coefficient (\(r\)) ranges from -1 to 1:

• Positive Correlation (\(r = 1\)): A perfect positive linear relationship.
• No Correlation (\(r = 0\)): No linear relationship between the variables.
• Negative Correlation (\(r = -1\)): A perfect negative linear relationship.

Formula for Pearson Correlation Coefficient:

\[ r = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{\sqrt{\sum{(X_i - \bar{X})^2} \cdot \sum{(Y_i - \bar{Y})^2}}} \]

where \(X_i\) and \(Y_i\) are individual data points, \(\bar{X}\) and \(\bar{Y}\) are the means of the variables.

Application Example:

Suppose we are examining the relationship between the time spent studying and the grades achieved. A positive Pearson correlation coefficient would indicate that more study time is associated with higher grades.