**Instructions:** Use this Multiple Correlation Coefficient Calculator for a multiple linear regression. Please input the data for the independent variables \((X_i's)\) and the dependent variable (\(Y\)), in form below, and the step-by-step calculations will be shown:

## Multiple Correlation Coefficient

The multiple correlation coefficient is a numerical measure of how well a linear regression model fits a set of data \(Y_i\).

Technically speaking, it is the simple correlation coefficient for dependent variable values \(Y_i\) and the predicted values \(\hat Y_i\) that are obtained with the least squares multiple linear regression

Mathematically,

\[R_{mult} =\frac{n \sum_{i=1}^n hat Y_i Y_i - \left(\sum_{i=1}^n \hat Y_i \right) \left(\sum_{i=1}^n Y_i \right) }{\sqrt{n \sum_{i=1}^n \hat Y_i^2 - \left( \sum_{i=1}^n \hat Y_i \right)^2} \sqrt{n \sum_{i=1}^n Y_i^2 - \left( \sum_{i=1}^n Y_i \right)^2} }\]but it can also be computed \(\sqrt{\frac{SSR}{SST}}\), where \(SSR\) is the sum of regression squares and \(SST\) is the total sum of squares, because that way is a bit simpler by following some (intensive) matrix calculations .

### What are the limits of multiple correlation coefficient?

For the case of a simple linear regression, the correlation coefficient may range from -1 to 1. For the case of the multiple correlation coefficient, it ranges from 0 to 1.

### Other associated calculators

If you need to estimate the regression model instead, you can use this multiple linear regression calculator.

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