The Limitless Central Limit Theorem
By: Sahaj Shukla
Tags: Statistics, Storytelling, Data Analytics, Central Limit Theorem, Data Science

As a Data Analyst, I often start my day sipping on my cup of strong coffee and cribbing about the data quality. Cleaning the data before getting to play with it and finding insights can be insanely challenging. Almost like spreading the dominoes with utmost patience just to watch them fall a satisfying fall. It tests your limits.

Speaking of limits, I remember one of the best theorems that exist in the world of Statistics. Something that makes me hope everyday. The Central Limit theorem. This is such a wonderful piece of mathematics that almost sounds like a strong top-10 contender in the book of Mathemagics (sorry for that). On giving it a nice intuitive approach though, it seems to make perfect sense. Let us start with the basic understanding of this theorem and give it the Wikipedia definition.

In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution(informally a bell curve) even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.

After a daunting read through possibly the most apt single paragraph explanation , let us try to understand the intuition behind the theorem and why it is so easy to understand and yet so cool!

For any distribution that shows the probability of the population in any form, the samples will yield a normal distribution! (guess that explains why all statistical geeks are portrayed spending their lives around bell curves in movies).

This seems particularly counterintuitive to hear initially. Almost sounds like a theory of unification of every data in just one bell shaped model! So how do we make sense out of this? Let’s just dive right in!

First, let us start by understanding the difference between the population modeling and sample modeling. The population modeling is generally about the probability of an event occuring. For example, getting 3 heads out of 5 tosses has a certain possibility of happening. The expected time between 911 calls could be also has a probability value attached to it. However, the coin toss in case 1, uses something called binomial distribution and the time between two 911 calls(case 2) uses Poisson distribution. I will attach the graphs and one can clearly see how both of these distributions are so different!

Fig. 2 Poisson distribution for expected 911 calls with population mean of 4 minutes.

These distributions give you the probability of what a value would be before the event occurs. Example, there is a 25 percent probability of the calls being made within a minute after the first call (in Fig. 2). Clearly enough, these two distributions are extremely different looking.

Let us now try and understand what the samples are, shall we? The sample distributions occur after the occurrence of the event. The samples of output are collected and plotted. The more outputs we have, the better they fit to the normal curve ( according to the central limit theorem). This is really intuitive to understand now, isn’t it? The actual sample standard deviation is inversely proportional to the number of samples throughout the sample space! Infact, it is equal to the population standard deviation divided by the root of number of samples.

Source: Byjus

Standard deviation can also be seen as a measure of uncertainty boundaries if I may. The more values you have, the cleaner the model becomes. It shouldn’t be difficult to understand now, that every type of data with the count of samples for distinct values can be fit inside a normal curve. For example, the 911 calls are centered around 0 will follow a beautiful normal distribution for large samples with mean nearing to the population mean. The population data was not normally distributed. However, the samples are! Thus, we get an idea about the central limit theorem.

That is how statements that are true yet simple, can be made with such a great power and yet have a simple explanation. The true beauty of CLT lies in its simplicity and the art of making a bold statement itself.

No theorem can be revered for its simplicity or beauty alone. For all practical purposes, its implementation and applications are highly essential to understand. Let us look at some wonderful applications of the CLT!

It is used literally everywhere! From predicting election results, to predicting how many teeth a species of dinosaurs could have had based on just 40 odd samples. It finds an application everywhere that we need to find a basic idea about the larger picture based on just small sample. Thanks to the central limit theorem, we have now found a way to predict the iceberg looking at the tip of it!

At a personal level, this morning was a happy one. I finished my cup of coffee without cribbing about bad data but rather being grateful to the likes of CLT that sit at the heart of statistical maths and make all of our lives easier!