Saying that quantum mechanics is an inherently statistical theory is a blatant misrepresentation. Precisely the point that makes QM so weird is that it is not caused by statistics. In a (properly set up) double slit experiment, a single electron is simultaneously travelling through both slits and causing an interference pattern.
Actually, that's an interpretation of quantum mechanics, not quantum mechanics itself. QM itself is in fact a statistical theory: the result of a calculation is a probability distribution of possible outcomes (even in the cases where the outcome might be deterministic). Taking your example of the double slit experiment, the calculation yields, in the case of two slits, a probability distribution with peaks and nulls corresponding to interference fringes. Close one slit and you get a different distribution without said interference.
What you describe is an interpretation of those probability distributions, and there are many conflicting views on the interpretation of QM (though, to date, these interpretations have not led to conflicting numerical results in the calculations). Wiki has a reasonable list of the extant interpretations [1] but I'll also draw your attention to an interpretation that is inherently statistical: Quantum Bayesianism (Nature has a nice overview [2]). To be crude, the idea is to look at QM from the point of view of the experimentalist: what information does she have available to her? In the case of the double slit experiment, it's impossible to say from a measurement of the electron at the screen which slit it passed through, so you cannot condition the measurement on either one individually, but both together.
QuBism is only one of the many interpretations of QM; others include the many-worlds interpretation (only deterministic in the crudest sense; any collapse of a wavefunction is thought to cause as many divergent universes as there are possible outcomes, but the one the experimentalist observes is still a statistical result), the copenhagen interpretation (also statistical, while the wavefunction is thought to represent the "true" state of a particle, its evolution under measurement is still non-deterministic), etc.
Finally, in case anyone is interested in more in-depth reading, the Stanford Encyclopedia of Philosophy has some fantastic readings on these, and related, topics. Articles worth checking out are that on measurement in QM [3] and those on the various interpretations (e.g. [4-6]).
