The Law of Large Numbers, in probability and statistics, asserts that as a sample size grows, its mean will approach the average of the entire population. This occurs because a larger sample tends to be more representative of the population. In a financial context, this principle reveals that a large, rapidly growing entity cannot sustain that growth pace indefinitely.
Key Concepts
- The law asserts that the observed sample average from a large sample will be close to the true population average and become even closer as the sample size increases.
- It doesn’t guarantee that any given sample, particularly small ones, will reflect true population characteristics or that a non-representative sample will be balanced by a subsequent sample.
- The law implies a bigger sample will represent a population mean, while the central limit theorem indicates a bigger sample will resemble a population’s distribution.
- In business, this term is occasionally used to describe the challenge of maintaining growth rates as a company expands because the underlying dollar amounts become too large and unmanageable.
Delving into the Law of Large Numbers
The Law of Large Numbers can be applied to multiple contexts. In statistical analysis, although polling every individual within a population is impractical, each additional data point increases the likelihood of accurately measuring the mean.
It does not imply that a given sample will always represent the true population, particularly small samples. Additionally, it doesn’t ensure that successive samples will move the observed average toward the population mean (as opposed to what the Gambler’s Fallacy suggests). In business, the term applies when percentage growth rates become difficult to maintain as companies scale up.
Statistical Significance and the Law of Large Numbers
Consider a scenario to establish the average value of a data set of 100 possible values. Sampling two values might yield non-representative outliers. Increasing the sample size to 20 data points is more likely to yield a result approximating the true average. For instance, in a data set ranging from 1 to 100, taking just 2 values (e.g., 95 and 40) may produce an average around 67.5. Increasing the sampling to 20 points would result in an average aligning more accurately with the true population average.
Relationship with Central Limit Theorem
The central limit theorem states that as the sample size grows, the sample mean becomes normally distributed. The Law of Large Numbers is closely related, indicating data refinement as the sample size grows. The law directly relates to the peak (mean) of the distribution curve found in the central limit theorem.
Application in Business Growth
In business, particularly in finance, the Law of Large Numbers illustrates that exponential growth rates often do not scale. This can result from the principle of diminishing marginal returns or diseconomies of scale. High market capitalization companies might struggle with rapid growth due to the immense underlying dollar values.
Business Example: Tesla
In 2020, Tesla’s automotive sales were $24.604 billion. In 2021, it skyrocketed to $44.125 billion, marking a 79% increase. As Tesla continues to expand, maintaining such high growth rates will be challenging.
| Year | Revenue | Notes |
|---|---|---|
| 2021 | $44.1 billion | Actuals |
| 2022 | $79.0 billion | |
| 2023 | $141.4 billion | |
| 2024 | $253.1 billion | Comparable to Apple’s half-year sales for March 2022. |
| 2025 | $453.0 billion | |
| 2026 | $810.9 billion | Multiple times Ford’s 2021 full-year revenue of $136.3 billion. |
| 2027 | $1.451 trillion | Roughly equivalent to combined 2021 sales of the top 20 automakers, which totaled $1.7 trillion. |
Insurance and Risk Assessment
The Law of Large Numbers is essential in the insurance industry for accurate risk assessment. With a smaller data sample, risk profiles are inadequately determined. As more data is collected, patterns, such as young, male drivers being higher risk, become evident, allowing finer adjustments in insurance premiums.
The Importance of the Law of Large Numbers
In statistical analysis, the law validates your sample size. Small samples can lead to incorrect assumptions not representative of the population. Capturing a large number of data points ensures a more accurate reflection of the entire dataset.
In business, understanding this law is vital for setting realistic targets. Doubling revenue in one year and then achieving 50% growth the following year results in substantial differences in real-world dollars. Be cautious with percentage goals and interpret them within larger financial contexts.
Overcoming Business Challenges of the Law of Large Numbers
Companies can combat this challenge by acquiring smaller, high-growth firms, enhancing operational efficiency, and focusing on absolute dollar goals versus percent growth. These strategies help sustain momentum even as scale increases.
Law of Small Numbers
The Law of Small Numbers posits that people underestimate variability within small samples, often leading to overreliance on incorrect data representations.
Law of Large Numbers in Psychology
In psychology, larger trial numbers contribute to more accurately anticipated outcomes, reflecting true medical assessments as the number of observations grows.
Conclusion
When analyzing data, understand the Law of Large Numbers to ensure your sample size is representative. In business analyses, recognize that as a company grows, surpassing previous growth rates becomes harder due to increasing dollar amounts.
Related Terms: Central Limit Theorem, Law of Small Numbers, Gambler’s Fallacy, mean, Insurance, Market Capitalization.
References
- U.S. Securities and Exchange Commission. “Form 10-K, Tesla, Inc.”
- Apple. “Form 10-Q, Q2 2022”.
- Ford. “Strategic Progress of Ford+ Growth Plan, Solid Financials in ‘21 Position Company for Connected EV Leadership in 2022, Beyond”.
- Factory Warranty Tools. “Top Automakers by Revenue”.
