Law of Large Numbers: What It Is, How It's Used, Examples

Law of Large Numbers

Investopedia / Julie Bang

What Is the Law of Large Numbers?

The law of large numbers, in probability and statistics, states that as a sample size grows, its mean gets closer to the average of the whole population. This is due to the sample being more representative of the population as the sample become larger.

In a financial context, the law of large numbers indicates that a large entity which 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.

Key Takeaways

  • The law of large numbers states that an observed sample average from a large sample will be close to the true population average 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 population characteristics or that a sample that does not reflect the true population will be balanced by a subsequent sample.
  • The law of large numbers indicates a bigger sample will represent a population mean, while the central tendency theorem states a bigger sample will represent a population's distribution.
  • In business, the term "law of large numbers" is sometimes used in a different sense to express the relationship between scale and growth rates. 
  • As a company becomes bigger, it will experience difficulties maintaining percentage targets because the underlying dollars may become too large and unfeasible.

Watch Now: What Is the Law of Large Numbers?

Understanding the Law of Large Numbers

The law of large numbers can refer to two different topics. First, 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 does not mean that a given sample or group of successive samples will always reflect the true population characteristics, especially for small samples. This also means that if a given sample or series of samples deviates from the true population average, the law of large numbers does not guarantee that successive samples will move the observed average toward the population mean (as suggested by the Gambler's Fallacy).

Second, the term "law of large numbers" is sometimes used in business in relation to growth rates, stated as a percentage. It suggests that, as a business expands, the percentage rate of growth becomes increasingly difficult to maintain. This is because the underlying dollar amount is actually increasing even if the growth rate as a percentage is to remain constant.

The Law of Large Numbers is not to be mistaken with the Law of Averages, which states that the distribution of outcomes in a sample (large or small) reflects the distribution of outcomes of the population.

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. This is because there is greater probability of the two data points being outliers or non-representative of the average, while there is lower probability in all 20 data points being non-representative.

For example, if the data set included all integers from one 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 Central Limit Theorem

In statistical analysis, the law of large numbers is related to the central limit theorem. The central limit theorem states that as the sample size increases, the sample mean will be evenly distributed. This is often depicted as a bell-shaped curve where the peak of the curve depicts the mean and even distributions of sample data fall to the left and right of the curve.

In a related manner, the law of large numbers also states that data is refined as the sample grows. However, the law of large numbers more closely relates to the center of the bell curve. The law of large numbers indicates that as a sample size increases, the mean of the sample will more closely resemble the mean of the population. Therefore, the law of large numbers relates to the peak (the mean) of a curve, while the central limit theorem relates to the distribution of a curve.

Law of Large Numbers and Business Growth

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

The same principles can be applied to other metrics, such as market capitalization or net profit. As a result, investing decisions can be guided based on the associated difficulties that companies with very high market capitalization can experience as they relate to stock appreciation. This concept is somewhat central to growth versus value stocks, as a company may find it to maintain its business strategy of rapid growth once it achieves market success.

Law of Large Numbers in Business Example

In fiscal year 2020, Tesla reported automotive sales (not gross sales) of $24.604 billion. The next year, the company reported $44.125 billion, an increase of roughly 79%. As electric vehicles are an emerging market and Tesla is beginning to finally experience economies of scale, the company is started to experience success very quickly.

The law of large numbers indicates that as Tesla continues to grow, it will become harder for the company to maintain this level of productivity. For example, assuming a steady growth rate of the next several years, it becomes quickly apparent that Tesla simply cannot maintain its current growth trajectory due to the underlying dollar values becoming unreasonable.

Theoretical Tesla Automobile Revenue
Year Revenue Notes
2021 $44.1 billion Actuals
2022 $79.0 billion
2023 $141.4 billion
2024 $253.1 billion Would exceed Apple's six-month total net sales ending March 2022.
2025  $453.0 billion
2026  $810.9 billion Would be almost six times as large as Ford's full year 2021 revenue ($136.3 billion).
2027  $1.451 trillion Would almost equal total 2021 car sales for the top 20 automakers combined was $1.7 trillion.
Assuming Consistent Growth Rate in Revenue From 2020 to 2027 using 2020-21 Rate

Law of Large Numbers and Insurance

The law of large numbers is also prominent in the insurance industry to calculate and refine projected risk. Imagine a situation where an insurance company is assessing how much to charge different customers for car insurance. Should the company have a small data set, it will not be able to adequately determine appropriate risk profiles.

As the insurance agency collects more data, it experiences the law of large numbers, they may soon find that young, male drivers are most likely to cause an accident. This larger sample becomes more representative of driving incidents, and the insurance company can arrive at more accurate conclusions about the appropriate insurance premiums to charge.

In addition, the law of large numbers allows insurance companies to deeply refine the criteria in which to assess premiums by analyzing what traits cause higher risk. For example,

Why Is the Law of Large Numbers Important?

In statistical analysis, the law of large numbers is important because it gives validity to your sample size. When working with a small amount of data, the assumptions you make may not appropriately translate to the actual population. Therefore, it is important to make sure enough data points are being captured to adequately represent the entire data set.

In business, the law of large numbers is important when setting targets or goals. A company may double its revenue in a single year. Should the company obtain only 50% growth in revenue the next year, it will have earned the same amount of money each of the last two years. Therefore, it is important to be mindful that percentages can be misleading as large dollar values escalate.

How Can Companies Overcome the Challenge of the Law of Large Numbers?

Companies often strive to overcome the challenge of the law of large numbers by acquiring smaller growth companies that can infuse scalable growth. They also attempt to become more efficient and utilize their size for manufacturing, ordering, or distribution benefits. Last, companies can be more attentive to dollar goals as opposed to percent goals.

What Is the Law of Small Numbers?

The law of small numbers is the theory that people underestimate the variability in small sample sizes. This means that when people study a sample size that is too small, they usually overestimate the population's value based on the incorrect sample size.

What Is the Law of Large Numbers in Psychology?

Similar to other examples above, the law of large numbers in psychology translates to how a larger number of trials often leads to a more accurate expected value. As more trials are performed, the closer the projection is to being a correct medical assessment.

The Bottom Line

When analyzing a data set, ensure you understand the law of large numbers to determine whether or not your sample size is representative of your population. On the other hand, when analyzing a company, be mindful of its size. As a company becomes larger, the law of large numbers states it will become more difficult for a company to maintain a percentage change (growth) due to the underlying large change in dollar amounts.

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  1. U.S. Securities and Exchange Commission. "Form 10-K, Tesla, Inc."

  2. Apple. "Form 10-Q, Q2 2022."

  3. Ford. "Strategic Progress of Ford+ Growth Plan, Solid Financials in '21 Position Company for Connected EV Leadership in 2022, Beyond."

  4. Factory Warranty Tools. "Top Automakers by Revenue."