What is the Law of Large Numbers?

April 1, 20237 min read
What is the Law of Large Numbers?
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Key Takeaways

  • Investors can use the law of large numbers to make traditional as well as alternative investment decisions.
  • The theory is at the center of value versus growth stocks, as once a company gains market success, it may find it very difficult to keep up its strategy of rapid growth.
  • In employing the law of large numbers, an investor can reap profits even if the investor is incorrect most of the time about the trajectory of assets.

While often employed in statistics and probability, the law of large numbers can also be used to make investment decisions, be they traditional or alternative. But what is the law of large numbers? Keep reading for that and more.

What is the Law of Large Numbers?

The concept essentially describes how an experiment’s results will move closer to the expected value if it is repeated a substantial number of times and an average is calculated. The idea is that as the sample size goes up, the average of observed results will become increasingly representative of the actual value.

Say someone seeks to establish the average value of a dataset of 200 potential values. They are more apt to get a true average by selecting 40 data points rather than depending on only four. That is because there is more likelihood of the four data points being non-representative of the average, and less of a probability that all 40 data points are outliers.

It is important to note that the theory does not always mean a sample will, every time, mirror actual population characteristics, particularly for smaller samples. Further, as suggested by the Gambler’s Fallacy, if the sample deviates from the population’s true average, there is no guarantee that successive samples will shift the average in the direction of the population mean.

Where is the Law Used?

In addition to its use by statisticians, the law is employed by multiple sectors including artificial intelligence, machine learning, and insurance. As well, companies can utilize the data to calculate risk as well as profit and loss.

On the financial front, the theory maintains that a fast-growing entity cannot keep up that rate of growth forever. The largest of the blue chips, with their astronomical market values, are often broached as examples of this phenomenon. As a company grows, according to the theory, it will become increasingly harder to maintain percentage goals. Why? Because the amount of money needed to do so becomes too excessive and unfeasible.

The same principle is applicable to metrics such as net profit or market capitalization. Thus, investors often make decisions hinged on challenges often experienced by companies that have significantly high market capitalization. This is particularly true when it comes to stock appreciation. This puts the law at the center of value versus growth stocks.

Say the market capitalization of Company ABC is $1 million; $100 million for Company XYZ. Company ABC experiences substantial growth – 50% annually. Because its market capitalization increases only by $500,000, its growth rate is easily attainable. However, such a growth rate is nearly impossible for Company XYZ because it implies that its market capitalization should grow by $50 million annually. Note that, at length, Company ABC’s growth will decline as it continues to expand.

Who Created the Law of Large Numbers?

The law of large numbers goes back hundreds of years and is rooted in the work of Jacob Bernoulli, a 17th-century mathematician, in his book “Ars Conjectandi.” Bernoulli proved an iteration of the law — known as the Bernoulli theorem — which posited that the average of the outcomes of a large number of independent experiments will usually be close to expected values.

More than a century after Bernoulli, Russian mathematician Pafnuty Chebyshev proved a more general version of the law of large numbers.

Law of Large Numbers vs. Central Limit Theorem

Related to the law of large numbers in statistical analysis is the central limit theorem, which states that as the sample size grows, the sample mean will be distributed evenly. A bell-shaped curve is often used here; the curve’s peak signifies the mean while sample data distributions are to the curve’s left and right.

Overall, the law of large numbers relates to a curve’s peak – its mean – while central limit generally relates to a curve’s distribution.

How Can Investors Use the Law of Large Numbers to Their Advantage?

In both traditional and alternative investments such as art and real estate, investors can use the law of large numbers under the premise that trading assets that have a favorable risk-reward ratio in relatively larger numbers will be more effective than trading in smaller numbers. This means that an investor can reap profits even if the investor is incorrect about the trajectory of assets most of the time.

Also, if an investment strategy hinges on targeting enterprises with significant growth potential, and eschewing companies that do not possess such prospects, the law of large numbers would have investors seeking companies that have smaller market capitalizations.

What is an Example of the Law of Large Numbers?

In fiscal year 2020, the electric vehicle purveyor Tesla reported auto sales of $24.6 billion. The following year, the company announced $44.1 billion in sales, a hike of about 79%. As it began experiencing economies of scale, it began to experience rapid success.

However, as the law of large numbers indicates, as Tesla continues to expand, it will become more difficult for it to keep up such productivity levels due to unreasonable underlying dollar values. For example, should its growth rate continue, Tesla’s theoretical revenue of $253.1 billion in 2024 would exceed Apple’s six-month total net sales ending in March of 2022.

Investors can ostensibly use the law of large numbers here to decide how much of a position to take in the company in the future.

What are the Benefits and Limitations of the Law of Large Numbers in Investing?

As with most any investment approach – none of them are without risk — there are benefits and limitations associated with using the law of large numbers.

A concept shortcoming is that one can expect to execute hundreds of trades before the law’s effectiveness can be gauged. If the theory is being employed as a predictor of growth, such a strategy can require too much time for a short-term investor.

Note, too, that not all traders use the law. For one thing, investors have varying levels of technical knowledge and skill, and choose certain assets for a myriad of reasons. Not every investment goal can be achieved by gains prospectively rendered more probable by applying the law of large numbers. For instance, the law may not provide much value to an investor who is choosing stocks based on a wish to bolster disadvantaged communities or work against climate change.

Having said that, the concept’s chief benefit is that it can be used by investors to help them make investment decisions. If the law holds true, an investor can be more often wrong than not about the direction in which certain securities are headed – and still manage to turn a good profit.

That also applies to alternative investments – those that are not directly tied to public markets. Investors are increasingly turning to securities such as private credit, structured notes, legal finance, and education for consistent secondary returns that are not as subject to stock market volatility.

These private-market assets are also used by investors to diversify their holdings, which mitigates overall portfolio risk and is key to successful investing. After all, since 2015, traditional portfolios of stocks and bonds have had annualized returns of 6.5%. Compare that to net annualized returns of 9.7% offered by leading alternative investment platform Yieldstreet.

Discover Yieldstreet’s Historical Performance

See our annualized returns and overall performance.

Alternative Investments and Portfolio Diversification

Traditional portfolio asset allocation envisages a 60% public stock and 40% fixed income allocation. However, a more balanced 60/20/20 or 50/30/20 split, incorporating alternative assets, may make a portfolio less sensitive to public market short-term swings.

Real estate, private equity, venture capital, digital assets, precious metals and collectibles are among the asset classes deemed “alternative investments.” Broadly speaking, such investments tend to be less connected to public equity, and thus offer potential for diversification. Of course, like traditional investments, it is important to remember that alternatives also entail a degree of risk.

In some cases, this risk can be greater than that of traditional investments. This is why these asset classes were traditionally accessible only to an exclusive base of wealthy individuals and institutional investors buying in at very high minimums — often between $500,000 and $1 million. These people were considered to be more capable of weathering losses of that magnitude, should the investments underperform. However, that meant the potentially exceptional gains these investments presented were also limited to these groups.

To democratize these opportunities, Yieldstreet has opened a number of carefully curated alternative investment strategies to all investors. While the risk is still there, the company offers help in capitalizing on areas such as real estate, legal finance, art finance and structured notes — as well as a wide range of other unique alternative investments. Learn more about the ways Yieldstreet can help diversify and grow portfolios.


The law of large numbers generally permits predictions about random variables’ long-term behavior based on the outcomes of a large quantity of experiments. Some investors in traditional as well as alternative assets successfully use the theory to help them decide where to place their capital, while others employ it to determine the frequency with which they should execute trades.