# Estimated Variance in LinearModelFit

Suppose that lmModel is a linear model which is fit using LinearModelFit. How come lmModel["EstimatedVariance"] is not equal to Variance[lmModel["FitResiduals"]]? How is lmModel["EstimatedVariance"] actually calculated?

This documentation page spells out how EstimatedVariance is computed. It is the "squared sum of FitResiduals divided by the degrees of freedom $n-p$. A linear regression will also have an intercept term, in addition to terms corresponding to each of the predictor columns.

For example,

headings = ExampleData[{"Statistics", "FisherIris"}, "ColumnHeadings"];
data = ExampleData[{"Statistics", "FisherIris"}]; Subsetting the data so that we can see a nice linear trend:

d = ds[Select[#["PetalLength"] > 2 &]];
ListPlot[d[All, {"SepalLength", "PetalLength"}]] Fitting the model (converting the Dataset data into normal values):

lmModel = LinearModelFit[
d[All, {"SepalLength", "PetalLength"}] // Normal // Values
, x, x];


we can extract the Estimated Variance:

lmModel["EstimatedVariance"]
(* 0.21594 *)


The Fitted Residuals can be extracted:

fr = lmModel["FitResiduals"];


and compute its variance with the help of Variance:

Variance[fr]
(* 0.213759 *)


Explicitly computing the unbiased variance (same as Variance):

Total[(fr - Mean[fr])^2]/(Length[fr] - 1)
(* 0.213759 *)


Recalling that we have to subtract out two degrees of freedom $(p=2)$ due to the intercept and the slope term:

Total[(fr - Mean[fr])^2]/(Length[fr] - 2)
(* 0.21594 *)


which explains the origins of these Properties.