If I plot your two integrands on the same sheet, each with your selected value for Sigma and both with the same plausible value of Mu and over a range large enough to capture and display the interesting part of both graphs and with PlotRange->All to display the full vertical extent of your functions:
Plot[{
((Pi^2/0.961794)*(z-\[Sigma])^2/((1+0.25 \[Mu]^2)*Sqrt[(1+0.25 \[Mu]^2)*2*Pi])*
Exp[-(z-\[Sigma])^2/((1+0.25 \[Mu]^2))*Sqrt[(1+0.25 \[Mu]^2)*2*Pi]*Erf[(z-\[Sigma])^2/
((1+0.25 \[Mu]^2)) Sqrt[(1+0.25 \[Mu]^2)*2*Pi]]])/.{\[Sigma]->-10,\[Mu]->1},
((Pi^2/0.961794)*(z-\[Sigma])^2/((1+0.25 \[Mu]^2)*Sqrt[(1+0.25 \[Mu]^2)*2*Pi])*
Exp[-(z-\[Sigma])^2/((1+0.25 \[Mu]^2))*Sqrt[(1+0.25 \[Mu]^2)*2*Pi]*Erf[(z-\[Sigma])^2/
((1+0.25 \[Mu]^2)) Sqrt[(1+0.25 \[Mu]^2)*2*Pi]]])/.{\[Sigma]->10,\[Mu]->1}},
{z,-12,12},PlotRange->All]
then the interpretation is
Your Sigma->-10 shifted the curve off to the left of z==zero and thus your integral over z from zero to infinity was almost exactly zero
Your Sigma->10 shifted the curve off to the right of z==zero and thus your integral over z from 0 to infinity was substantially greater than zero
Now at a much higher level. I understand you are really struggling trying to find something. I see that the questions you are asking and the replies you are getting are not giving you what you need. Can you put all that aside for a few minutes and think at a very high level? Forget for a few moments Mean and Variance and which variable you are integrating over and what a Plot looks like and what Mathematica syntax is.
What is it that you are really really trying to get at the very highest level?
Perhaps when that is very clear it might be possible to move towards what you really need.