• Correlation measures the strength and direction of the relationship between two variables
• Positive correlation: As one variable increases, the other variable also tends to increase
• Negative correlation: As one variable increases, the other variable tends to decrease
• Types of correlation: Pearson, Spearman, Kendall

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

• Measures the strength and direction of the linear relationship between two variables
• Utilizes the covariance of the variables divided by the product of their standard deviations
• Widely used for assessing linear relationships between continuous variables in many fields, including statistics, economics, and psychology

\[ rho = \frac{\sum{(x' - m_{x'})(y'_i - m_{y'})}}{\sqrt{\sum{(x' - m_{x'})^2\sum{(y' - m_{y'})^2}}}}\]

• Measures the strength and direction of monotonic relationships between two variables
• Utilizes ranks of the data rather than their actual values, offering robustness against outliers and non-linear associations
• Useful when the data violates assumptions of linearity or normality

\[ tau = \frac{n_{c} -n_{d}}{\frac{1}{2}n(n-1)}\]

• Quantifies the degree of association between variables by comparing concordant and discordant pairs in their ranks
• Assesses monotonic relationships without assuming linearity or normality in the data
• Well-suited for situations where the data lacks linearity or when there’s a need for a non-parametric correlation measure
• The following slides use the mtcars dataset to provide clearer examples of correlation.

# Load the mtcars dataset
data(mtcars)

# Calculate correlation between 'cyl' and 'qsec' for the entire dataset
correlation <- cor(mtcars$cyl, mtcars$qsec)

# Print the correlation value
print(paste("Correlation between 'cyl' and 'qsec':", correlation))

## [1] "Correlation between 'cyl' and 'qsec': -0.591242073768869"

• Moderate negative correlation (-0.59) implies that as the number of cylinders increases, the quarter-mile time tends to decrease.

• We explored the concept of correlation and its significance in statistical analysis
• Key findings include the identification of different types of correlation coefficients—Pearson, Spearman, and Kendall—and their respective applications
• Through visualizations and mathematical formulations, we illustrated how correlation analysis can provide insights into relationships within data sets
• Understanding correlations enables us to make informed decisions in various fields such as economics, finance, and healthcare