Formula of Karl Pearson

Pearson Correlation Coefficient:

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

Pearson Correlation Coefficient (r):

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1. What is the Pearson Correlation Coefficient?

The Pearson correlation coefficient (r) measures the linear relationship between two continuous variables. Developed by Karl Pearson, it ranges from -1 to +1, where -1 indicates perfect negative correlation, +1 indicates perfect positive correlation, and 0 indicates no linear correlation.

2. How Does the Calculator Work?

The calculator uses the Pearson correlation formula:

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

Where:

\( r \) — Pearson correlation coefficient (dimensionless)
\( x_i, y_i \) — Individual data points
\( \bar{x}, \bar{y} \) — Means of x and y variables
\( \Sigma \) — Summation operator

Explanation: The formula calculates the covariance of the two variables divided by the product of their standard deviations, providing a standardized measure of linear relationship.

3. Importance of Pearson Correlation

Details: Pearson correlation is widely used in statistics, research, and data analysis to quantify the strength and direction of linear relationships between variables. It's fundamental in fields like psychology, economics, and biomedical research.

4. Using the Calculator

Tips: Enter comma-separated values for both X and Y variables. Ensure both arrays have the same number of values. The calculator will compute the correlation coefficient and display the result.

5. Frequently Asked Questions (FAQ)

Q1: What does the value of r indicate?
A: r values: ±0.8-1.0 (very strong), ±0.6-0.8 (strong), ±0.4-0.6 (moderate), ±0.2-0.4 (weak), 0-0.2 (very weak/no correlation).

Q2: What are the assumptions for Pearson correlation?
A: Continuous variables, linear relationship, normality, homoscedasticity, and no significant outliers.

Q3: Can Pearson correlation detect non-linear relationships?
A: No, it only measures linear relationships. For non-linear relationships, consider Spearman's rank correlation.

Q4: What is the difference between correlation and causation?
A: Correlation indicates relationship, not causation. A high correlation doesn't mean one variable causes changes in the other.

Q5: When should I use Pearson vs Spearman correlation?
A: Use Pearson for linear relationships with normally distributed data. Use Spearman for monotonic relationships or when data doesn't meet normality assumptions.