1. What is the Correlation Coefficient?

The correlation coefficient (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 the correlation coefficient formula:

Where:

• \( cov(x,y) \) — Covariance between variables x and y
• \( sd_x \) — Standard deviation of variable x
• \( sd_y \) — 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 value between -1 and 1.

3. Interpretation of Results

Details:

• 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 the covariance between your variables and their standard deviations. All values must be valid (standard deviations > 0).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between covariance and correlation?
A: Covariance measures the joint variability but isn't standardized, while correlation is a standardized measure between -1 and 1.

Q2: Can correlation imply causation?
A: No, correlation only measures association. Other factors may be responsible for the observed relationship.

Q3: What are the assumptions for Pearson's r?
A: Variables should be continuous, linearly related, and approximately normally distributed.

Q4: When should I use other correlation measures?
A: Use Spearman's rho for ordinal data or when the relationship is monotonic but not linear.

Q5: How many data points are needed for reliable r?
A: Generally, at least 30 paired observations are recommended for stable estimates.