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Taking your first steps in statistics or data science is, no doubt, an enthralling experience. One of the conceptual pillars that will give you confidence when working with data is the law of large numbers. Let’s uncover what this is about.

What Is the Law of Large Numbers?

Simply put, the law of large numbers is a mathematical rule that states: the more times an experiment is repeated, the closer the average of its results will be to the expected theoretical result. In other words, the impact of “luck” or “randomness” is diluted as the volume of data collected from repeated experiments grows.

A classic example of this law is tossing a coin. We know in advance that the theoretical probability of getting “heads” is 50%. Suppose you toss the coin just 10 times. The result could be “heads” on eight of those trials and “tails” on only the other two. That’s an 80% “heads” rate: a result that seems pretty far from the theoretical value.

But if you tossed the same coin a much larger number of times, say 10,000 times, the law of large numbers would reveal itself more clearly. The percentage of heads relative to tails would likely be very close to 50%, meaning you would expect to see a “heads” outcome about 5,000 times.

In summary: the more data you collect from repeated trials, the more clearly the underlying pattern emerges.

The law of large numbers, illustrated | Image by Author

Why Is This Law Key in Data Science?

The law of large numbers can be a powerful ally in a variety of data science projects. For example, suppose you are training a predictive model or analyzing trends. This law helps explain why, if you collect a sufficiently large sample of data, its calculated averages or predictions are more likely to reflect reality. By contrast, if you have a small dataset, your models may be overly influenced by noise.

Real-world situations where this law applies are diverse. In insurance, thousands of driver histories may be analyzed to calculate an insurance policy’s price more reliably. In marketing A/B testing, thousands of data points from website visitors may be needed to confirm which design attracts more clicks. And in medical research, scientists evaluate the effectiveness of new treatments by testing them on large groups of patients.

As you can see, the law of large numbers appears in many data-intensive situations and application domains. But understanding it is much simpler than it may seem at first!