1. What is the Linear Correlation Coefficient?

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

2. How Does the Calculator Work?

The calculator uses Pearson's r formula:

Where:

• \( x_i \) — Individual x data point
• \( \bar{x} \) — Mean of x values
• \( y_i \) — Individual y data point
• \( \bar{y} \) — Mean of y values
• \( \sum \) — Summation across all data points

Explanation: The numerator measures how x and y vary together (covariance), while the denominator normalizes this by how x and y vary separately (standard deviations).

3. Interpretation of r Values

Guidelines:

• 0.00 to ±0.30: Negligible or very weak correlation
• ±0.30 to ±0.50: Weak correlation
• ±0.50 to ±0.70: Moderate correlation
• ±0.70 to ±0.90: Strong correlation
• ±0.90 to ±1.00: Very strong correlation

4. Using the Calculator

Tips: Enter comma-separated numerical values for both X and Y variables. Ensure both lists have the same number of values (minimum 2). Example: "1,2,3,4,5" for X and "2,4,6,8,10" for Y.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between correlation and causation?
A: Correlation measures association, but doesn't imply causation. Other factors may influence the relationship.

Q2: What are the assumptions for Pearson's r?
A: Assumes linear relationship, continuous variables, normally distributed data, and homoscedasticity.

Q3: When should I use Spearman's rank instead?
A: Use Spearman's for ordinal data or when the relationship is monotonic but not linear.

Q4: Can outliers affect the correlation coefficient?
A: Yes, outliers can dramatically influence r. Always examine your data visually.

Q5: What does r² represent?
A: r² (coefficient of determination) represents the proportion of variance in y explained by x.