大数法则是什么？

What is the law of large numbers?

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The law of large numbers applies to probability and statistics. It states that as the sample size grows, the mean of the sample will get closer to the mean of the entire population because the larger sample is more representative of the entire population.

In a financial context, the law of large numbers suggests that large, rapidly growing entities cannot sustain this rate of growth forever. Blue-chip stocks with market capitalizations in the hundreds of billions are often cited as prime examples of this phenomenon.

Main points

• The law of large numbers states that the observed mean of a large sample will be close to the true population mean, and this proximity will deepen as the sample size increases.
• The law of large numbers does not guarantee that a particular sample will reflect true population characteristics, especially when the sample is small.
• It also does not guarantee that a sample that does not reflect the characteristics of the true population will subsequently be balanced by another sample.
• The law of large numbers states that a larger sample will be more representative of the population average. The central tendency theorem states that a larger sample will represent the distribution of the population.
• The term "Law of Large Numbers" is sometimes used with different meanings in business to express the relationship between size and growth rate.

Understand the law of large numbers

The law of large numbers can be understood from two aspects.

In statistical analysis, the law of large numbers can be applied to a variety of topics. It may not be practical to survey every individual in a given population to collect the required amount of data, but each additional data point potentially improves the probability that the result is a true measure of the mean.

The law of large numbers does not mean that a given sample or series of consecutive samples always reflects the true population characteristics, especially when the sample or series of samples deviates from the mean of the true population. Nor does it guarantee that successive samples will push the observed mean toward the population mean, as the gambler's fallacy suggests.

The term "Law of Large Numbers" is sometimes used in business in connection with growth rates. It is expressed as a percentage, implying that as a business expands, it becomes increasingly difficult to maintain a growth rate. Even if the percentage of growth remains the same, the dollar amount behind it keeps increasing.

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

Statistical analysis examples of the law of large numbers

If you want to determine the mean of a data set consisting of 100 possible values, you are more likely to get an accurate mean by choosing 20 data points rather than relying on just two data points. There is a high probability that two data points are outliers, or non-representative, while there is a lower probability that all 20 data points are non-representative.

If the data set contained all integers from 1 to 100, and you only sampled two values, 95 and 40, your average would probably be determined to be approximately 67.5. If you continue to randomly sample up to 20 variables, the average should gradually move closer to the true average as more data points are considered.

Law of Large Numbers and Central Limit Theorem

The law of large numbers and the central limit theorem are related in statistical analysis. The central limit theorem states that the sample mean will become evenly distributed as the sample size increases. This usually appears as a bell-shaped curve, with the peak of the curve being the mean and an even distribution of sample data on the left and right sides of the curve.

The law of large numbers also states that as the sample size increases, the data becomes more accurate, and the law of large numbers moves closer to the center of the bell curve. It states that the mean of a sample will become closer to the mean of the population as the sample size increases. Thus, the law of large numbers relates to the peak or mean of a curve, while the central limit theorem relates to the distribution of a curve.

The law of large numbers and business growth

The term "law of large numbers" is sometimes used colloquially in business and finance to refer to exponential growth rates, a phenomenon that is often difficult to sustain as it scales. This has nothing to do with the law of large numbers, but may be a result of the law of diminishing marginal returns or diseconomies of scale.

The same principle applies to other metrics such as market capitalization or net profit. Investment decisions can be guided by the difficulties associated with stock appreciation experienced by companies with extremely high market capitalizations. This concept is quite central between growth and value stocks, as companies may use this to maintain a business strategy of rapid growth after market success.

Business examples of the law of large numbers

Tesla reported vehicle sales of 44.125 billion the following year, an increase of about 79%. "[1] Electric vehicles are an emerging market, and Tesla is beginning to experience economies of scale and achieve rapid success.

The law of large numbers suggests that as Tesla continues to grow, it will become more difficult to maintain this level of production. Clearly, Tesla cannot maintain its current growth trajectory assuming a few years of steady growth because the dollar value behind it becomes unreasonable.

Theoretical Tesla Car Revenue

The Law of Large Numbers and Insurance

The law of large numbers is equally evident in the insurance industry. It is used to calculate and refine expected risk.

Imagine a scenario where an insurance company evaluates how much to charge a customer for car insurance. If the data set is small, companies will not be able to adequately determine an appropriate risk profile.

As the amount of data collected increases, the company will experience the law of large numbers. It may soon be discovered that young male drivers are most likely to be involved in accidents. This larger sample is more representative of the occurrence of driving incidents, and insurance companies can make more accurate insurance premium judgments.

The law of large numbers also allows insurance companies to refine the criteria for evaluating insurance premiums by analyzing what characteristics lead to higher risk.

Why is the law of large numbers important?

The law of large numbers is important in statistical analysis because it provides validity for sample size. When you perform analysis with small amounts of data, the assumptions you make may not translate appropriately to the actual population, so it is important to ensure that you capture enough data points to appropriately represent the entire data set.

The law of large numbers is also important when setting company goals. A company may double its revenue in one year. If it only grew revenue by 50% over the next year, it would have made the same amount of money over the past two years. Percentages can be misleading because large dollar values are exploding.

How can companies overcome the challenges of the law of large numbers?

Companies often overcome the challenges of the law of large numbers by acquiring smaller growth companies that can achieve scalable growth. They are also trying to become more efficient and take advantage of their scale in manufacturing, ordering or distribution. Companies can focus more on dollar targets rather than percentage targets.

What is the law of decimals?

The law of small numbers is when people underestimate the variability in small samples. When the sample size is too small, they often overestimate the value of the population based on the incorrect sample size.

What does the law of small numbers mean in psychology?

In psychology, the law of large numbers manifests itself in the fact that a greater number of trials usually leads to more accurate expectations. As the number of trials increases, the predictions become closer to the correct medical assessment.

Summary

When analyzing a data set, be sure to understand the law of large numbers to determine whether your sample size is representative of the entire population. But when analyzing companies, pay attention to sample size. The law of large numbers states that as a company grows in size, it becomes more difficult to maintain a growing percentage change because the amount of change behind it increases.

References

[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."

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