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This article describes the central limit theorem, a cornerstone theorem in the realm of statistics, in an approachable and non-mathematical way you would not expect, providing a clear understanding of how it works and its practical significance no matter you are math-savvy or not.

What Does the Central Limit Theorem Say?

Suppose you have a population with a large number of individuals (data instances) The central limit theorem says that if we picks many data samples (say, smaller subgroups) randomly among the original whole population, and calculate the average of each one, those average values will visually distribute like a bell-shaped curve (in statistics, the normal distribution). This will happen no matter what the original data distribution was: the only condition for this to occur is that picked samples are sufficiently large.

Demystifying Central Limit Theorem

Who needs dense math when we can understand the central limit theorem through a real-world example?

Suppose you are taking a nationwide survey on rural areas (villages and towns below 1K inhabitants) across a province. You don’t know the age of each person living in these areas, but it turns out ages are visually distributed based on the following histogram on the left-hand side:

Illustrating the Central Limit Theorem

Interestingly, due to government plans to repopulate rural areas, the age distribution is slightly skewed towards younger ages, which means there is a predominance of younger adults populating such areas nowadays.

Now let’s omit the right-hand side plot for now and get to the point of what the central limit theorem can do for you. Imagine you want to know the average age of this entire population, but you cannot ask so many people for their age in one go. Instead, during several days, you randomly select groups of 50 people every day, ask their ages, annotate them, and calculate the average age across that group of 50.

You repeat this process many times, asking a new random group of 50 every day (sometimes the person might get asked again), and as a result, you annotate a new average value for the age every day. You will end up with a list of many average ages.

This is where the central limit theorem shines: even though individual people’s ages may vary significantly, the average ages calculated for these groups (samples) day after day will follow a bell-shaped distribution (the normal distribution mentioned earlier, see right-hand side image above). This means most daily average values will lie near the real average for the entire population, and only a few values will lie further away to the sides. Thus, the longest bar(s) in the right-hand side histogram of sample-level average ages can be reliable indicators of what the true average age for the entire population would be.

Wrapping Up

The central limit theorem is very useful in practice because, thanks to it, we can perform inferences about an entire large population by just performing measurements across smaller parts of it, as long as we repeat this process several times. It allows us to accurately estimate the overall behavior of a large population without having to exhaustively take a certain measure or collect certain data for every single individual.

Many statistical tools in medicine, surveying, quality control, and more follow the principle underlying this theorem.