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 relationship.

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 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 and hypothesis testing.

4. Using The Calculator

Tips: Enter the covariance between your two variables and their respective standard deviations. All values must be valid (standard deviations > 0).

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 relationship.

Q2: What is the range of Pearson's r?
A: Pearson's r always ranges from -1 to +1, inclusive. This bounded range makes it easy to interpret the strength of relationships.

Q3: How is covariance calculated?
A: Covariance = Σ[(Xᵢ - X̄)(Yᵢ - Ȳ)] / (n-1) for sample data, where X̄ and Ȳ are the means of X and Y respectively.

Q4: Does correlation imply causation?
A: No, correlation only measures association. A strong correlation does not necessarily mean one variable causes changes in the other.

Q5: What are the assumptions for Pearson's correlation?
A: Assumes linear relationship, continuous variables, normally distributed data, and homoscedasticity (constant variance).