Is anything inherently random? Is anything inherently random? Or is all randomness observed in data either "errors in measurement" or "lack of understanding"? Assume we could measure everything with infinite precision and had a complete understanding of all the deterministic processes. Would there be any process that would appear "random" in the data or would every dataset be deterministic at that point?
 A: In some ways this may be a philosophical question.
My view is that the randomness a statistician sees in
data analysis is often real. Randomness might result from random sampling or from inherent randomness or instability.
Either way, statistical procedures of estimation might be useful.
Let's look at two
entirely different situations.
(1) We wonder whether the true average height of women high school
seniors in a US state is 65 inches. At noon on a particular day, one might carefully measure each
of them (thousands or hundreds of thousands, depending on the state), round to the nearest tenth of an inch, and take the mean.
Even
with perfectly accurate measuring and record keeping,
the result would likely be different on another day. Some students
might have grown just a bit. Also, there is some evidence that
a person's height may depend (slightly) on how much sleep they got the previous night. Does it make sense to say that the true statewide average height fluctuates constantly in some random way?
More realistically, one might take a random sample of $n = 400$ women for the measurements and averaging. In this case,
random sampling has clearly induced randomness. It would not
be surprising if another random sample of $400$ resulted in a different mean. Using such a random sample we could
make a 95% confidence interval for heights of senior women.
Perhaps this CI would be about $0.7$ inches wide, but we
could be fairly sure that the interval contains the right
answer for the heights of senior women in the state. For
practical purposes that might be sufficiently close to the 'correct' answer. Perhaps some probability model such as
$\mathsf{Norm}(\mu = 65, \sigma=3.5)$ is approximately correct.
If so, then our two samples of size $400$ might (at random) have been
$65.2$ and $65.0.$
round(rnorm(2, 65, 3.5/20) ,1)
[1] 65.2 65.0

(2) You have a small specimen of some radioactive material
with a long half-life (several thousand years). You use an appropriate counter to
count particles emitted in one particular minute to be $746.$
Using exactly the same experimental set-up several minutes later the count is $721.$
Monitoring the specimen over
several hours, you believe the hourly rate of particle emission
in your experiment is distributed $\mathsf{Pois}(\lambda=750).$ As far as is known, the number of particles
emitted into our counter per minute is truly random, and $\mathsf{Pois}(750)$ is a reasonable model. Both of our one-minute measurements could be exactly correct, but it would be wrong to
expect them to be exactly the same. (But not often fewer than 697 or more than 804.)
set.seed(1226); rpois(2, 750)  
[1] 746 721
qpois(c(.025,.975), 750)
[1] 697 804

A: Independently of the question of existence of randomness, you simply cannot measure anything to arbitrarily high precision, because your measurement will necessarily disturb it by a greater amount if you wish to have greater precision. This is already true in classical mechanics and remains true in quantum mechanics.
But your real question is whether there is anything inherently random in the world. Although this is not a mathematical question, it suffices to say that till today there is simply zero evidence for any truly random process. Quantum mechanics does not at all suggest, not to say imply, randomness in anything in the real world. Given this, it is reasonable to ask why so many real-world processes do appear to be probabilistic.
Well, the answer is that purely deterministic processes can amplify very small perturbations, so if there is an amplification process followed by a folding process then the result can appear chaotic even if it is fully determined by the initial state. Furthermore, symmetries in the folding process may strongly dictate the limiting distribution of the result as the amplification is increased.
For example, the logistic map has chaotic behaviour for certain parameters, for which just a slight perturbation in the input results in exponential divergence of the output. Essentially, the logistic map shows that repeated application of a continuous stretch-and-fold operation on a metric space can result in unpredictability in the limit. This has implications for the real-world, in that almost all measurements are of quantities arising from innumerable iterates of some physical process (e.g. particle-particle interactions).
A: Yes, there are "inherently" random processes in the universe.  As far as I know, it's impossible to determine in advance the collapse of a wave function.  For example, which slit a particle will go through in the two slit experiment.
This question should really be asked on physics.stackexchange.
A: We do not know and may never know.  (and possibly we can not know*)
Statistics is not about the nature of reality. Statistics describes observations and these observations happen to have a random appearance due to unknown variable factors that are involved in the model descriptions.
The question whether the reality is intrinsically non-deterministic is more like philosophy than statistics.

*One may argue that the question about the deterministic nature of the world is actually not something that can be answered by a consciousness within this reality, and it may even not have any meaning at all. If the physical processes occur with some random nature, that means that the future state of the world can not be determined based on the present state, then still something makes the current state evolve into the future state and it is in some way determined. It is only not determined by an entity that can be known by the objects in the observable world. Whether or not something is deterministic is a matter of perspective.
