I know that when you perform a measurement of position in quantum mechanics, the wave function collapses to something proportional to it, but in a small range of values of positions, depending on the precision of the instrument and I guess one of the values obtained from the measurement belongs to that range. If we measure very quickly a second time, we should get a value that belongs to that range again (I am not sure about this, please confirm).If we continue doing this process indefinitely and take the average of the measurements, will we get the so called "true value" of the measurements?. So are we generating "true values" when we measured for the first time? Do the subsequent measurements will be just limited by the precision of the instrument or also by the collapsed state?
There are different posts about the nature of the wave function collapse on PhysicsSE, but your question seems slightly different. We'll start from the beginning.
In the case you are interested in, the wave function contains the information about all the possible locations of the object you want to measure. There is a certain probability assigned to each position. When you perform the measurement however, you know where the object of interest is located - mathematically speaking you obtain an eigenvalue of the position operator. All the other probabilities become zero and the result of your measurement tells you where your object is located. If you measure it again, you will obtain the exact same result over and over again. This is what's called the collapse of the wave function.
You are now throwing another aspect into the mix - statistics and measurement precision. This is a concept which is not connected to wave functions but applies to every real-life situation. It doesn't matter what you are measuring - your instrument can never be 100% precise. You are always operating within a certain range of precision. And in order to minimize the error, you can measure identically prepared systems multiple times. However, you cannot measure the same thing over and over again and expect the error to be reduced. If you measure the length of a piece of paper with a ruler for example, you will not obtain a different result when you measure it again.
When you now measure a quantum system, you have two types of statistics in there. First, your particle exhibits certain probabilities to be at a certain location. Secondly, your measurement apparatus is never 100% precise. The best you can do is perform lots of different measurements of identically prepared systems which is also a source of uncertainty because identical preparation might be difficult in real life. You will obtain the probability of presence of the particle at a certain position. But since the position is a continuous variable, there will always be an error. You can never measure "true values" but you will not be "limited by the collapsed state" as you formulated it.