Reviewed by Jan 29, 2021| Updated on
In probability and statistics, the law of large numbers states that as a sample size starts to grow its mean comes closer to the average of the entire population. Mathematician, Girolamo Cardano, acknowledged the Law of Large Numbers in the 16th century but was never able to prove it.
In 1713, this theorem was proved by the Swiss mathematician, Jakob Bernoulli in his book, Ars Conjectandi. This was later refined by other prominent mathematicians, such as Pafnuty Chebyshev, who was the founder of the St. Petersburg Mathematical School.
In a financial sense, the law of large numbers shows that a large company that is rapidly expanding cannot sustain the rate of growth forever. The biggest of the blue chips are frequently cited as indicators of this trend, with market rates in the hundreds of billions.
The law of large numbers can be used in statistical analysis on a variety of subjects. It may not be possible to poll each person within a given population to gather the required amount of data, but each additional gathered data point has the potential to increase the probability that the result is a true measure of the mean.
The law of large numbers doesn't mean that a given sample or group of successive samples will always represent the true characteristics of the population, especially for small samples. It also implies that if a given sample or sequence of sample deviates from the true average population, the rule of large numbers will not guarantee that subsequent samples will shift the observed average toward the mean population (as implied by the Gambler's Fallacy).
The Law of Large Numbers should not be confused with the Law of Averages, which mentions that the distribution of results within a sample (big or small) represents the distribution of results of the population.