Law Of Large Numbers

SEC Release IA-1092

What is the law of large numbers?

The law of large numbers, in probability and statistics, stipulates that as the size of a sample increases, its average approaches the average of the whole population. In the 16th century, the mathematician Gerolama Cardano recognized the law of large numbers but never proved it. In 1713, the Swiss mathematician Jakob Bernoulli proved this theorem in his book, Ars Conjectandi. It was then refined by other renowned mathematicians, such as Pafnuty Chebyshev, founder of the Saint Petersburg school of mathematics.

In a financial context, the law of large numbers indicates that a large, rapidly developing entity cannot maintain this growth rate indefinitely. The largest tokens, whose market value is in the hundreds of billions, are frequently cited as examples of this phenomenon.

Key points to remember

  • The law of large numbers stipulates that a sample mean observed from a large sample will be close to the true population mean and that it will get closer the larger the sample.
  • The law of large numbers does not guarantee that a given sample, especially a small sample, will reflect the true characteristics of the population or that a sample that does not reflect the true population will be balanced by a subsequent sample.
  • In business, the term “law of large numbers” is sometimes used in a different sense to express the relationship between scale and growth rates.

Understanding the law of large numbers

In statistical analysis, the law of large numbers can be applied to a variety of subjects. It may not be possible to survey every individual in a given population to collect the amount of data required, but each additional data point collected has the potential to increase the likelihood that the outcome is a true measure. of the average.

In business, the term “law of large numbers” is sometimes used in relation to growth rates, expressed as a percentage. He suggests that as a business grows, the percentage growth rate becomes increasingly difficult to maintain.

The law of large numbers does not mean that a given sample or a group of successive samples will always reflect the true characteristics of the population, especially for small samples. It also means that if a given sample or series of samples deviates from the true population mean, the law of large numbers does not guarantee that successive samples will shift the observed mean to the population mean (like the suggests Gambler’s Fallacy).

The law of large numbers should not be confused with the law of means, which states that the distribution of results in a sample (large or small) reflects the distribution of results of the population.

The law of large numbers and statistical analysis

If a person wants to determine the average value of a dataset of 100 possible values, they are more likely to reach a precise average by choosing 20 data points instead of relying on just two. For example, if the dataset included all the integers from one to 100 and the sampler only pulled two values, such as 95 and 40, he can determine that the average is about 67, 5. If he continued to take random samples of up to 20 variables, the mean should move to the true mean since it considers more data points.

Law of big numbers and business growth

In business and finance, this term is sometimes colloquially used to refer to the observation that exponential growth rates are often not to scale. This is not really related to the law of large numbers, but may be the result of the law of diminishing marginal returns or diseconomies of scale.

For example, in July 2020, revenues generated by Walmart Inc. were recorded at $ 485.5 billion while Amazon.com Inc. reported $ 95.8 billion over the same period. If Walmart wanted to increase revenues by 50%, approximately $ 242.8 billion in revenue would be required. On the other hand, Amazon would only have to increase its revenues by $ 47.9 billion to reach an increase of 50%. Based on the law of large numbers, the 50% increase would be considered more difficult for Walmart than Amazon.

The same principles can be applied to other parameters, such as market capitalization or net profit. Therefore, investment decisions can be guided based on the associated difficulties that companies with very large market capitalizations may encounter in terms of stock appreciation.

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