Pearson Correlation Calculation

Pearson Correlation Formula:

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

Pearson Correlation Coefficient (r):

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1. What is Pearson Correlation?

The Pearson correlation coefficient (r) measures the linear relationship between two continuous 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 the Pearson correlation formula:

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

Where:

\( x_i, y_i \) — Individual data points
\( \bar{x}, \bar{y} \) — Means of x and y variables
\( \sum \) — Summation across all data points

Explanation: The formula calculates how much two variables change together relative to how much they vary individually.

3. Interpretation of Correlation Coefficient

Details:

r = +1: Perfect positive linear relationship
0.7 ≤ r < 1: Strong positive correlation
0.3 ≤ r < 0.7: Moderate positive correlation
0 < r < 0.3: Weak positive correlation
r = 0: No linear correlation
-0.3 < r < 0: Weak negative correlation
-0.7 < r ≤ -0.3: Moderate negative correlation
-1 < r ≤ -0.7: Strong negative correlation
r = -1: Perfect negative linear relationship

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 Pearson correlation coefficient.

5. Frequently Asked Questions (FAQ)

Q1: What does Pearson correlation measure?
A: Pearson correlation measures the strength and direction of the linear relationship between two continuous variables.

Q2: What are the assumptions for Pearson correlation?
A: Variables should be continuous, normally distributed, have a linear relationship, and show homoscedasticity (constant variance).

Q3: How is Pearson correlation different from Spearman correlation?
A: Pearson measures linear relationships, while Spearman measures monotonic relationships (both are rank-based and don't require normality).

Q4: Can correlation imply causation?
A: No, correlation only indicates association. Causation requires additional evidence from controlled experiments or theoretical justification.

Q5: What sample size is needed for reliable correlation?
A: Generally, larger samples provide more reliable estimates. A minimum of 30 pairs is often recommended for reasonable accuracy.