There are two things I don't understand here:
Firstly, why in the standard deviation expression we use the Q operator instead of the Q value itself? Writting Q seemed more natural and then in the next expression I would just stick the corresponding operator. Which would lead to the same result or is it just two errors cancelling each other?
Secondly, why in the last expression we are not using an operator of the quantity that was in the previous expression under the "average" brackets? Is it an operator of itself like coordinate x is for example?

Is there a difference? I think there is none. Feel free to correct me

Let me clarify. First of all I don't understand why in this expression
[tex]\sigma^{2}_{Q}=\langle(\hat{Q}-{\langle}Q{\rangle})^2\rangle[/tex]
we use the operator Q instead of the value Q.
For example if we make repeated measurements on identically prepared systems we may get a bunch of different Q values, we take their average and hence we can calculate the standard deviation. So why use an operator?

Notice I didn't write the hat on H in the second expression ( the part I don't understand )

Then I would stick in the wave equation you gave me and start multiplying the members. The orthogonal eigenstates would give zero while others would sum up to a number thus I will end up with a certain value. Is that correct?

So what you are saying is that I can already write the operator within the "average" brackets like this [tex]
< \hat{Q} >
[/tex]?
I thought I was not allowed to do that and had to write [tex]
< Q >
[/tex]
instead and only once I am trying to compute the expectation value I sandwich the corresponding operator between the wave functions.