In statistics and probability theory, the law of large numbers is a theorem that describes the result of repeating the same experiment a large number of times. The large numbers theorem states that if the same experiment or study is repeated independently a large number of times, the average of the results of the trials must be close to the expected value. The result becomes closer to the expected value as the number of trials is increased.
The law of large numbers is an important concept in statistics because it states that even random events with a large number of trials may return stable long-term results. Note that the theorem deals only with a large number of trials while the average of the results of the experiment repeated a small number of times might be substantially different from the expected value. However, each additional trial increases the precision of the average result.
Example of Law of Large Numbers
The simplest example of the law of large numbers is rolling the dice. The dice involves six different events with equal probabilities. The expected value of the dice events is:
If we roll the dice only three times, the average of the obtained results may be far from the expected value. Let’s say you rolled the dice three times and the outcomes were 6, 6, 3. The average of the results is 5. According to the law of the large numbers, if we roll the dice a large number of times, the average result will be closer to the expected value of 3.5.
Law of Large Numbers in Finance
In finance, the law of large numbers features a different meaning from the one in statistics. In the business and finance context, the concept is related to the growth rates of businesses.
The law of large numbers states that as a company grows, it becomes more difficult to sustain its previous growth rates. Thus, the company’s growth rate declines as it continues to expand. The law of large numbers may consider different financial metrics such as market capitalization, revenue, and net income.
Let’s consider the following example. Company ABC’s market capitalization is $1 million while Company XYZ’s market capitalization is $100 million. Company ABC experiences a significant growth of 50% per year. For ABC, the growth rate is easily attainable since its market capitalization only grows by $500,000.
For Company XYZ, that growth rate is almost impossible because it implies that its market capitalization should grow by $50 million per year. Note that the growth of Company ABC will decline over time as it continues to expand.