In statistics, the **fraction of variance unexplained** (**FVU**) in the context of a regression task is the fraction of variance of the regressand (dependent variable) *Y* which cannot be explained, i.e., which is not correctly predicted, by the explanatory variables *X*.

## Formal definition

Suppose we are given a regression function yielding for each an estimate where is the vector of the *i*^{th} observations on all the explanatory variables.^{[1]}^{:181} We define the fraction of variance unexplained (FVU) as:

where *R*^{2} is the coefficient of determination and *VAR*_{err} and *VAR*_{tot} are the variance of the residuals and the sample variance of the dependent variable. *SS*_{err} (the sum of squared predictions errors, equivalently the residual sum of squares), *SS*_{tot} (the total sum of squares), and *SS*_{reg} (the sum of squares of the regression, equivalently the explained sum of squares) are given by

Alternatively, the fraction of variance unexplained can be defined as follows:

where MSE(*f*) is the mean squared error of the regression function *ƒ*.

## Explanation

It is useful to consider the second definition to understand FVU. When trying to predict *Y*, the most naïve regression function that we can think of is the constant function predicting the mean of *Y*, i.e., . It follows that the MSE of this function equals the variance of *Y*; that is, *SS*_{err} = *SS*_{tot}, and *SS*_{reg} = 0. In this case, no variation in *Y* can be accounted for, and the FVU then has its maximum value of 1.

More generally, the FVU will be 1 if the explanatory variables *X* tell us nothing about *Y* in the sense that the predicted values of *Y* do not covary with *Y*. But as prediction gets better and the MSE can be reduced, the FVU goes down. In the case of perfect prediction where for all *i*, the MSE is 0, *SS*_{err} = 0, *SS*_{reg} = *SS*_{tot}, and the FVU is 0.

## See also

- Coefficient of determination
- Correlation
- Explained sum of squares
- Regression analysis
- Linear regression
- Lack-of-fit sum of squares

## References

**^**Achen, C. H. (1990). "'What Does "Explained Variance" Explain?: Reply".*Political Analysis*.**2**(1): 173–184. doi:10.1093/pan/2.1.173.