1. What is Pearson's Correlation Coefficient?

Pearson's correlation coefficient (r) measures the strength and direction of the linear relationship between two variables. It ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation), with 0 indicating no linear correlation.

2. How Does the Calculator Work?

The calculator uses Pearson's correlation formula:

Where:

• \( r \) — Pearson's correlation coefficient (dimensionless)
• \( Cov(X,Y) \) — Covariance between variables X and Y
• \( \sigma_x \) — Standard deviation of variable X
• \( \sigma_y \) — Standard deviation of variable Y

Explanation: The formula standardizes the covariance by dividing it by the product of the standard deviations, resulting in a dimensionless measure between -1 and 1.

3. Importance of Correlation Analysis

Details: Correlation analysis is fundamental in statistics for understanding relationships between variables, identifying patterns, and guiding further statistical modeling. It's widely used in research, finance, social sciences, and data analysis.

4. Using the Calculator

Tips: Enter the covariance between X and Y, and the standard deviations for both variables. All values must be valid (standard deviations > 0). The result is a dimensionless coefficient between -1 and 1.

5. Frequently Asked Questions (FAQ)

Q1: What does the correlation coefficient value mean?
A: Values close to +1 indicate strong positive correlation, close to -1 indicate strong negative correlation, and values near 0 indicate weak or no linear correlation.

Q2: Can correlation imply causation?
A: No, correlation only measures association. Causation requires additional evidence from controlled experiments or established theoretical frameworks.

Q3: What are the assumptions for Pearson's correlation?
A: Variables should be continuous, linearly related, approximately normally distributed, and have homoscedasticity (constant variance).

Q4: When should I use other correlation measures?
A: Use Spearman's rank correlation for ordinal data or when assumptions of Pearson's correlation are violated. Use point-biserial correlation for one continuous and one dichotomous variable.

Q5: How do I interpret the strength of correlation?
A: Generally: ±0.9-1.0 (very strong), ±0.7-0.9 (strong), ±0.5-0.7 (moderate), ±0.3-0.5 (weak), ±0.0-0.3 (very weak or none).