Quantum Mechanics - How do we know that the observed locations of electrons are random? How do we know that the observed location of a electron (or any quantum object) is purely random (there is no way to predict it) within the probability-function instead of normal randomness (we don't know how to predict it)?
As an analogy if someone would only be able to measure the amount of people (amount of electrons) and their individual IQ (location) he would see a perfect distribution (obviously only if he has enough samples) within a specific range (probability) and each individual measurement seems purely random (QM-randomness).
He would then assume that their IQ is purely random as he has just not enough knowledge to prove otherwise (he hasn't spoken to any of them, hasn't analysed any in respect to education nor has he analysed their DNA).
Could the same be happening when we observe electrons or have we proven that their location is purely random?
 A: 
How do we know that the observed location of a electron (or any quantum object) is purely random within the probability-function?

The location of an electron is not random by any means, rather it is exactly determined by the coefficients of its expansion onto a basis. Namely let the state of the electron be described by
$$
|\psi\rangle = \sum_n c_n |a_n\rangle
$$
where $|a_n\rangle$ is a complete basis of the Hilbert space and eigenstates of an observable $\hat{A}$. This means that if we perform infinite measurements of the physical quantity associated to $\hat{A}$, we found the value $a_n$ exactly $|c_n|^2$ times each. Replace $\hat{A}$ with the position operator and the coefficients will represent the probability of finding the particle in that position if we make infinite measurements.
The fact that you get different values if you perform many different measurements of the same quantity is not due to the fact that you do not have enough information, rather it is a definition in quantum mechanics. Any observable can have in principle infinitely many different values when measured on a state (unless you are on an eigenstate of said observable).
The above assumptions have been proven right experimentally, in particular the position of an electron according to the above was one of the first precise proofs of quantum mechanics, see double-slit experiment.
