Correlation Coefficient Calculator

Correlation Coefficient Formula:

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

Correlation Coefficient (r):

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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 perfect positive correlation, -1 indicates perfect negative correlation, and 0 indicates no linear correlation.

2. How Does the Calculator Work?

The calculator uses the Pearson correlation coefficient formula:

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

Where:

\( x \) — Values of the first variable
\( y \) — Values of the second variable
\( \bar{x} \) — Mean of x values
\( \bar{y} \) — Mean of y values
\( \sum \) — Summation operator

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

3. Importance of Correlation Coefficient

Details: Correlation coefficient is fundamental in statistics for identifying relationships between variables, but it's important to remember that correlation does not imply causation.

4. Using the Calculator

Tips: Enter comma-separated values for both X and Y variables. Both datasets must have the same number of values. The calculator will compute the Pearson correlation coefficient.

5. Frequently Asked Questions (FAQ)

Q1: What does a correlation coefficient of 0.8 mean?
A: A correlation coefficient of 0.8 indicates a strong positive linear relationship between the two variables.

Q2: Can correlation coefficient be greater than 1?
A: No, the Pearson correlation coefficient always ranges between -1 and +1.

Q3: What's the difference between correlation and causation?
A: Correlation measures association, while causation implies that one variable directly affects the other. Correlation does not prove causation.

Q4: When is correlation coefficient most useful?
A: It's most useful when examining linear relationships between continuous variables with normal distributions.

Q5: What are the limitations of correlation coefficient?
A: It only measures linear relationships, is sensitive to outliers, and doesn't work well with non-linear relationships or categorical data.