# Philosophical implications of central limit theorem (CLT)

Central limit theorem (CLT) establishes that, for the most commonly studied scenarios, when independent random variables are added, their sum tends toward a normal distribution (commonly known as a bell curve) even if the original variables themselves are not normally distributed.

According to this theorem, the probability distribution for total distance covered in a random walk or even for flipping a large number of coins for the total number of heads (or tails), tend towards a normal distribution.

I'm curious to know the implications CLT on the nature of knowledge, reality, and existence?

• None whatsoever? Or rather it might imply something about bayesian inference maybe. But considering it seems to mostly look at random events, it's hard to see what it implies about the `nature of knowledge,reality, and existence` May 25, 2017 at 12:11
• I agree with virmaior here. It's just a theorem of maths - it actually makes a lot of sense when you think about it. It actually says that if you have any random probability distribution, the distribution of sample means will make a normal distribution (which makes 100% sense if you think about it). onlinestatbook.com/stat_sim/sampling_dist is a good way of seeing it May 25, 2017 at 15:24
• This is a bit broad don't you think? Can you narrow down what it is about CLT that you're curious? Do you care about the meaning of the CLT within the philosophy of statistics? Probability? Technical underpinnings of hypothesis testing? How it informs the frequentist vs bayesian debate? Any of the narrowest interpretations of these could fill a book (and has). May 25, 2017 at 19:30
• For fun though, CLT = the mean of the mean is the mean or slightly less telegraphic, the mean of a set of means of random samples from any kind of distribution is, in the limit, a normal distribution whose mean is that of the original distribution. May 25, 2017 at 19:33
• The Stanford Encyclopedia of Philosophy hosts a massive amount of survey essays about topics in philosophy with extensive bibliographies to lead towards further research. You should check out the essays on chance vs. randomness, Bayesian epistemology, as well as interpretations of probability. Some essays tend to be a lot more technical than others but most are fairly readable at a university education level. May 26, 2017 at 11:14

The CLT has no implications for knowledge or reality or existence unless you make the assumption that the universe consists of random variables. If you make that assumption, it says a lot about what can and cannot be. However, that assumption is not an easy one. As far as I know, it is generally assumed that all macroscopic "random" events are not random variables, but the interaction of non-random events that we cannot see. For such cases, the CLT can only be used on models of the universe, not the universe itself.

In quantum mechanics, we currently believe that the universe is well modeled as random variables, under the Copenhagen Interpretation. As such, many scientists currently believe that you can draw conclusions about the universe using the CLT. It is used, for example, to explain wavefunction collapse. However, that's all science. We don't actually know. The next major advancement in particle physics may uproot the idea that the world is well modeled as a statistical system.

The real power of the CLT in our lives is that in many cases, we are comfortable making assumptions that some of the universe is well modeled as random variables. We can make very good approximations and predictions using it. However, philosophically, those would not be considered knowledge. They're just really convenient shortcuts.

The CLT gives us a way to understand the universe. It gives rise to certain contradictions. On its own, it is pretty meaningless outside mathematics, but in combination with some beliefs, it can lead to other beliefs.

For example, we can say, "human heights for people in this country are roughly independent (if we consider 30 year olds alive today) and identically distributed (if we choose people at random and we don't have a bias, say, for choosing people in a particular suburb), and that the variance of heights is finite. The 300 people on this plane (and that's probably not great for my assumptions) have an average height of 5 feet and 6 inches. Therefore, the average height for people in this country is therefore near 5 feet and 6 inches or more, or my assumptions are wrong, or this plane is full of unusually tall people. I don't think my assumptions are very wrong, so I think that either the average height is around 5 feet 6 inches or more, or this plane is full of unusually tall people."

Without the CLT, we wouldn't be able to make that inference. It would be perfectly possible that the people on the plane weren't unusually tall, even though the average of their heights is more than the average height for the population. (Obviously, it's hard to imagine a counterfactual to a rule of mathematics, but imagine there were a small number of really, really tall people, none of whom were on the plane.) The CLT removes that possibility, leaving only the possibilities of bad assumptions, fact about the world, or unlikely data.

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