Question
What is the formula for calculating the correlation coefficient?
Answers
GPT-4o mini
The correlation coefficient, often denoted as \( r \), measures the strength and direction of the linear relationship between two variables. The most commonly used correlation coefficient is Pearson's correlation coefficient. The formula for calculating \( r \) is given by:
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]
Where:
- \( n \) is the number of data points,
- \( \sum xy \) is the sum of the product of each pair of scores,
- \( \sum x \) is the sum of the \( x \) values,
- \( \sum y \) is the sum of the \( y \) values,
- \( \sum x^2 \) is the sum of the squares of the \( x \) values,
- \( \sum y^2 \) is the sum of the squares of the \( y \) values.
The result \( r \) ranges from -1 to 1:
- \( r = 1 \) indicates a perfect positive linear relationship,
- \( r = -1 \) indicates a perfect negative linear relationship,
- \( r = 0 \) indicates no linear relationship.
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]
Where:
- \( n \) is the number of data points,
- \( \sum xy \) is the sum of the product of each pair of scores,
- \( \sum x \) is the sum of the \( x \) values,
- \( \sum y \) is the sum of the \( y \) values,
- \( \sum x^2 \) is the sum of the squares of the \( x \) values,
- \( \sum y^2 \) is the sum of the squares of the \( y \) values.
The result \( r \) ranges from -1 to 1:
- \( r = 1 \) indicates a perfect positive linear relationship,
- \( r = -1 \) indicates a perfect negative linear relationship,
- \( r = 0 \) indicates no linear relationship.