Sample Correlation Coefficient R Calculator

The Sample Correlation Coefficient (R) is a statistical measure used to determine the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no correlation.

Historical Background

The concept of correlation was first introduced by Sir Francis Galton in the late 19th century. He sought to measure the relationship between different variables, which eventually led to the development of the correlation coefficient by Karl Pearson, his student. Pearson’s product-moment correlation coefficient is widely used today as the "sample correlation coefficient."

Calculation Formula

The sample correlation coefficient (R) between two sets of data \(X\) and \(Y\) is calculated using the following formula:

\[ R = \frac{n\sum{XY} - \sum{X}\sum{Y}}{\sqrt{\left(n\sum{X^2} - (\sum{X})^2\right) \left(n\sum{Y^2} - (\sum{Y})^2\right)}} \]

Where:

• \(n\) is the number of data points.
• \(\sum{X}\) and \(\sum{Y}\) are the sums of the \(X\) and \(Y\) values, respectively.
• \(\sum{XY}\) is the sum of the products of corresponding \(X\) and \(Y\) values.
• \(\sum{X^2}\) and \(\sum{Y^2}\) are the sums of the squares of \(X\) and \(Y\) values.

Example Calculation

Consider two datasets:

• \(X = [1, 2, 3, 4]\)
• \(Y = [2, 4, 6, 8]\)

Step-by-step calculation:

1. \(\sum{X} = 1 + 2 + 3 + 4 = 10\)
2. \(\sum{Y} = 2 + 4 + 6 + 8 = 20\)
3. \(\sum{XY} = (1 \times 2) + (2 \times 4) + (3 \times 6) + (4 \times 8) = 60\)
4. \(\sum{X^2} = 1^2 + 2^2 + 3^2 + 4^2 = 30\)
5. \(\sum{Y^2} = 2^2 + 4^2 + 6^2 + 8^2 = 120\)
6. Apply the formula:

\[ R = \frac{4(60) - (10)(20)}{\sqrt{(4)(30) - (10)^2) \cdot ((4)(120) - (20)^2)}} = 1 \]

This indicates a perfect positive correlation.

Importance and Usage Scenarios

• Data Analysis: It is commonly used to measure relationships between different variables in fields such as finance, biology, and social sciences.
• Predictive Modeling: In predictive analytics, correlation helps assess which variables might have predictive power over others.
• Experimental Research: Helps scientists determine the strength of relationships between variables in controlled experiments.

Common FAQs

1. What is a good value for R?
   A value close to 1 or -1 indicates a strong correlation, while values near 0 suggest a weak or no correlation.

2. Can R be greater than 1 or less than -1?
   No, R is always between -1 and 1.

3. Does a high correlation imply causation?
   No, correlation does not imply causation. It only indicates that two variables move together.