Correlation Coefficient Calculator (Pearson's $r$)

Use this calculator to find the strength and direction of the linear relationship between two variables, X and Y. Enter your paired data points below. Each line should contain the X-value, followed by the Y-value, separated by a comma or space.

Pearson's Correlation Coefficient ($r$) Formula

Pearson's $r$ is calculated using the following formula, which is designed to measure the covariance of X and Y normalized by the standard deviations of X and Y:

Where:

• $n$ is the number of paired observations (data points).
• $\sum x$ and $\sum y$ are the sums of the X and Y values.
• $\sum x^2$ and $\sum y^2$ are the sums of the squared X and squared Y values.
• $\sum xy$ is the sum of the products of X and Y for each pair.

This formula always yields a value between $-1$ and $+1$.

Interpreting the Correlation Coefficient ($r$)

The value of $r$ tells you two things about the relationship between X and Y:

1. Direction (Sign):
   • **Positive ($r > 0$):** As X increases, Y tends to increase (a positive slope).
   • **Negative ($r < 0$):** As X increases, Y tends to decrease (a negative slope).
   • **Zero ($r \approx 0$):** No linear relationship exists.
2. Strength (Magnitude): The closer the value is to $\pm 1$, the stronger the linear relationship. The closer it is to 0, the weaker the relationship.

Strength Guidelines (Commonly Used)

Frequently Asked Questions (FAQ)

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

$r = \frac{n \sum xy - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}$