## Definition

In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

## Law Of Large Numbers

## What is the ‘Law Of Large Numbers’

In probability and statistics, the law of large numbers states that as a sample size grows, its mean gets closer to the average of the whole population. In a financial context, the law of large numbers indicates that a large entity that is growing rapidly cannot maintain that growth pace forever. The biggest of the blue chips, with market values in the hundreds of billions, are frequently cited as examples of this phenomenon.

## Explaining ‘Law Of Large Numbers’

In statistical analysis, the law of large numbers can be applied to a variety of subjects. It may not be feasible to poll every individual within a given population to collect the required amount of data, but every additional data point gathered has the potential to increase the likelihood that the outcome is a true measure of the mean.

## The Law of Large Numbers and Statistical Analysis

If a person wanted to determine the average value of a data set of 100 possible values, he is more likely to reach an accurate average by choosing 20 data points instead of relying on just two. For example, if the data set included all integers from 1 to 100, and sample taker only drew two values, such as 95 and 40, he may determine the average to be approximately 67.5. If he continued to take random samplings up to 20 variables, the average should shift towards the true average as he considers more data points.

## Law of Large Numbers and Business Growth

In July 2015, the revenue generated by Wal-Mart Stores Inc. was recorded as $485.5 billion while Amazon.com Inc. brought in $95.8 billion during the same period. If Wal-Mart wanted to increase revenue by 50%, approximately $242.8 billion in revenue would be required. In contrast, Amazon would only need to increase revenue by $47.9 billion to reach a 50% increase. Based on the law of large numbers, the 50% increase would be deemed more difficult for Wal-Mart to accomplish than Amazon.

## Further Reading

- Monitoring the monitor: an incentive structure for a financial intermediary – www.sciencedirect.com [PDF]
- Strong laws of large numbers for sub-linear expectation without independence – www.tandfonline.com [PDF]
- The Law of Large Numbers and Research of Credit Models of Micro and Small Enterprises [J] – en.cnki.com.cn [PDF]
- Power laws in economics and finance – www.annualreviews.org [PDF]
- Law of large numbers and large deviations for dependent risks – www.tandfonline.com [PDF]
- International financial integration and economic growth – www.sciencedirect.com [PDF]
- Power laws in economics and finance: some ideas from physics – www.tandfonline.com [PDF]