Law of Large Numbers

Summary: The Law of Large Numbers is a statistical theory related to the probability of an event. This theory states that the greater number of times an event is carried out in real life, the closer the real-life results will compare to the statistical or mathematically proven results. In research studies, this means that large sample sizes average out to be more reflective of reality than small sample sizes.

Originators: Gerolama Cardano (1501-1576), Jacob Bernoulli (1654-1705)

Key Words: Probability, mathematics, sample size, anomalies, statistics, percentage, average, mean

The Law of Large Numbers was first observed by the mathematician Gerolama Cardano in the 16th century. Cardano noticed the theoretical presence of The Law of Large Numbers, but he never took the time to prove it mathematically. Another mathematician, Jacob Bernoulli, figured out the equations behind The Law of Large Numbers in 1713.[i]

A simple way to understand The Law of Large Numbers is to consider the probability of a coin toss. When a coin is tossed, there is a 50% chance that the coin will land on heads and a 50% chance that the coin will land on tails. This is a statistically proven fact. However, if a person tossed a coin in the air 5 times, there is a chance that the coin would land on heads every single time. This event would not seem to align with the mathematically proven probability of landing on tails 50% of the time.

How can we explain this? These real-life results don’t mean the math is wrong. They simply mean that the coin toss has to be carried out more times to accurately reflect what math says is true. If the same person tossed the coin in the air 500 times, by the end of all the tosses, the coin would have landed on heads an average of 250 times and on tails an average of 250 times. The real life coin toss is now more reflective of what math says to be true because it has been carried out a larger number of times.

Sample Sizes

The Law of Large Numbers is most applicable to scientific research and sample sizes.[ii] When scientists complete research studies, they make decisions about how many people will be in the study. This is an important decision because small sample sizes can greatly skew results due to the presence of anomalies. The larger the sample size, the more the results will reflect the true nature of the population that is being studied.

Consumers trying to understand scientific research should take sample size into consideration when determining the validity of a study. Scientists should do everything within in their power to work with large sample sizes, as this makes their work more accurate and thus more beneficial to society.

Personal Decisions

The Law of Large Numbers is also an important reminder that individual instances don’t provide the whole story. There are times when people make decisions based on one event or instance they have experienced or heard about. This is often a bad way to make decisions.

For example, someone might hear a story about how their friend had a terrible reaction to a medication and refuse to take that medication based on that one example. However, this is a bad way to make a choice about medication, as one experience or story is often not reflective of the way things typically work. The medication may be extremely safe, and the one story simply reflects an anomaly. When making personal decisions it is important to gather a range of information. The Law of Large Numbers explains the theory and mathematics behind this important concept.


[i] Seneta, E. (2013). A tricentenary history of the law of large numbers. Bernoulli, 19(4), 1088-1121

[ii] Dinov, I. D., Christou, N., & Gould R. (2009). The law of large numbers: The theory, applications, and technology-based education. Journal of Statistics Education, 17(1), 1-19.