Correlation Calculator

About Correlation Calculator

Pearson's correlation coefficient measures linear relationship between two variables. r = 1 is perfect positive, r = -1 is perfect negative.

How to use this calculator

1. Enter the two datasets as matched pairs, with each \(x_i\) lined up to the corresponding \(y_i\).
2. Check that both lists have the same number of values, because each pair is used together in the calculation.
3. Submit the data to compute Pearson’s \(r\), which will be between \(-1\) and \(1\).
4. Read the result and the step-by-step work to see how the averages, deviations, and final ratio are formed.

The formula explained

The formula computes the correlation coefficient \(r\), which compares how the two variables vary together against how much each variable varies on its own. A value near \(1\) means a strong positive linear relationship, near \(-1\) means a strong negative linear relationship, and near \(0\) means little or no linear relationship.

• r = Pearson correlation coefficient
• \(x_i\) = the \(i\)-th value in the first dataset
• \(y_i\) = the \(i\)-th value in the second dataset
• \(\bar{x}\) = the mean of the \(x\)-values
• \(\bar{y}\) = the mean of the \(y\)-values

Step by step method

1. Find the mean of the \(x\)-values and the mean of the \(y\)-values.
2. For each pair, subtract the mean from each value to get deviations, then multiply the paired deviations and also square each deviation.
3. Add the paired products and the squared deviations, then divide the total paired product by the square root of the product of the two squared-deviation sums.

Worked example

Problem. Find the Pearson correlation coefficient for the paired data \((1,2), (2,3), (3,5), (4,4)\).

1. Compute the means: \(\bar{x} = (1+2+3+4)/4 = 2.5\), \(\bar{y} = (2+3+5+4)/4 = 3.5\).
2. Find deviations and products: \((-1.5)(-1.5)=2.25\), \((-0.5)(-0.5)=0.25\), \((0.5)(1.5)=0.75\), \((1.5)(0.5)=0.75\). The sum of products is \(4.0\). Also, \(\sum(x_i-\bar{x})^2 = 2.25+0.25+0.25+2.25 = 5\), and \(\sum(y_i-\bar{y})^2 = 2.25+0.25+2.25+0.25 = 5\).
3. Compute \(r = 4 / \sqrt{5\cdot 5} = 4/5 = 0.8\).

Tips and common mistakes

• Correlation only measures linear relationship, so a curved pattern can have a low \(r\) even if the variables are related.
• Make sure the data are paired correctly, because switching one value to the wrong partner changes the result completely.

Frequently asked questions