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I know that:

$Variance_{Total}=Variance_{Explained} + Variance_{Unexplained}$,

but I am wondering how the $Variance_{Total}$ relates to the $Variance_{Unexplained}$ and the $Variance_{Explained}$ if the $Variance_{Explained}+Variance_{Unexplained}$ is larger than the $Variance_{Total}$, i.e. if the model we're checking against our null-model lies higher than the point we're checking against. Example of the situation

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    $\begingroup$ Welcome to CV Laurits. I've edited your notation so it is no longer using a minus sign which is confusing. Your figure describes the elements of variance incorrectly. Total variance is the sum of the red dots. Explained variance is the sum of the points on the blue line. Unexplained variance is the difference between the red and blue at each point. Hopefully that will help you work it out from there. $\endgroup$
    – ReneBt
    Nov 8, 2021 at 9:26
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    $\begingroup$ How can total variance equal the sum of explained and unexplained variance yet also be less than the sum of explained and unexplained variance? $\endgroup$
    – Dave
    Nov 8, 2021 at 10:39

1 Answer 1

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The part of unexplained variance is the variance of the residuals ($\hat y - y$) of your regression / total variance - that's $1-R^2$.

If you know for sure that your hypothesis function is relevant and well-fit to your data (eg in the case where you generated the data yourself), the unexplained variance will be noise only.

On your figure, for each point, the explained variance is represented by the distance from $\bar y$ (horizontal line, $\bar y$ is the mean of all data points) to $\hat y$ (the sinusoid). While the unexplained is the distance from $\hat y$ to $y$ (the data point).

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