**Instructions:** You can use this step-by-step Correlation Coefficient Calculator for two variables X and Y. All you have to do is type your X and Y data, either in comma or space separated format (For example: "2, 3, 4, 5", or "3 4 5 6 7").

## Correlation Coefficient Calculator

The correlation coefficient calculated above corresponds to Pearson's correlation coefficient. The requirements for computing it is that the two variables X and Y are measured at least at the interval level (which means that it does not work with nominal or ordinal variables).

The formula for Pearson's correlation coefficient is:

\[r =\frac{n \sum_{i=1}^n x_i y_i - \left(\sum_{i=1}^n x_i \right) \left(\sum_{i=1}^n y_i \right) }{\sqrt{n \sum_{i=1}^n x_i^2 - \left( \sum_{i=1}^n x_i \right)^2} \sqrt{n \sum_{i=1}^n y_i^2 - \left( \sum_{i=1}^n y_i \right)^2} }\]or equivalently

\[r = \frac{\sum_{i=1}^n x_i y_i - \frac{1}{n}\left(\sum_{i=1}^n x_i \right) \left(\sum_{i=1}^n y_i \right) }{\sqrt{\sum_{i=1}^n x_i^2 - \frac{1}{n}\left( \sum_{i=1}^n x_i \right)^2} \sqrt{\sum_{i=1}^n y_i^2 - \frac{1}{n}\left( \sum_{i=1}^n y_i \right)^2}} = \frac{SS_{XY}}{\sqrt{SS_{XX}\cdot SS_{YY} }}\]If you have two or more variables, you could use our correlation matrix calculator. Also, if the data for the variables \(X\) and \(Y\) does not meet the parametric assumptions for Pearson's correlation, then you should use this Spearman's correlation calculator instead.

### Can I use z-scores to compute the correlation coefficient

Certainly! You have seen z-scores everywhere in Statistics and naturally, you wonder if you can compute the correlation with z-scores. You can definitely do it, and in fact, it is the customary way of doing it in Social Sciences stats.

### Other calculators similar to this correlation calculator

Also, there is the concept of multiple correlation coefficient, when you have more than one predictor, which is obtained by computing the correlation between the observed \(Y\) values and the predicted values \(\hat Y\) by the regression.

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