Law of Large Numbers

Law of Large Numbers

Key takeaways

• In statistics, the law of large numbers (LLN) states that as a sample size grows, the sample average converges on the population average.
• In business and finance, the term is often used to describe the difficulty large organizations have in maintaining high percentage growth rates as their base size expands.
• LLN underpins practices in insurance, forecasting, and any analysis that relies on representative samples.

What the law of large numbers is

The law of large numbers has two related meanings depending on context:

• Statistical meaning: With more independent observations, the sample mean tends to get closer to the true population mean. Larger samples reduce the chance that the average is skewed by outliers and increase confidence that the sample reflects the population.
• Business/financial usage: It describes the practical challenge that maintaining high percentage growth becomes harder as an organization grows, because the same percentage implies ever-larger absolute increases.

LLN in statistical analysis

Researchers typically cannot measure an entire population, so they use samples. The LLN explains why increasing sample size improves the reliability of estimates of population parameters (like the mean). It does not guarantee that any specific small sample will reflect the population, nor does it imply that random events will “even out” in short runs (avoiding the Gambler’s fallacy).

Example: If you randomly draw two values from the integers 1–100, those two values might be far from the true mean (50.5). Drawing 20 or more values makes it far more likely the sample mean will be close to the population mean.

Relation to the central limit theorem

The LLN and the central limit theorem (CLT) are complementary:
– LLN concerns convergence of the sample mean toward the population mean as sample size increases.
– CLT describes how the distribution of the sample mean tends toward a normal (bell-shaped) distribution as sample size increases, regardless of the underlying distribution (under mild conditions).

In short, LLN addresses the mean’s accuracy; CLT describes the sampling distribution’s shape.

LLN in business and finance

When applied to firm growth, the LLN is a practical observation: a company doubling revenue from a small base is far easier than doubling from a multi-billion-dollar base. As absolute size grows, sustaining the same percentage growth becomes more difficult, often due to market saturation, operational complexity, or diminishing returns.

This idea is related to concepts such as diseconomies of scale and the law of diminishing marginal returns. It helps explain why investors temper growth expectations for very large companies.

Example: Tesla (illustrative trend)

Tesla’s revenue growth illustrates the point. After a large percentage increase in one year, subsequent percentage growth rates declined even as revenue rose in absolute terms:
– 2020 → 2021: very large percentage increase
– 2021 → 2022: high double-digit increase
– 2022 → 2023: mid-teens increase
– 2023 → 2024: growth slowed to around zero to low single digits
This pattern shows how maintaining early-stage growth rates becomes harder as a company scales.

Insurance and other applications

Insurance companies rely on the LLN to estimate risk accurately. With many independent policies and loss events, the aggregated experience becomes predictable: larger portfolios allow insurers to estimate frequencies and severities more reliably and set premiums that reflect actual risk.

Other applications include quality control, polling, forecasting, and any domain where representative sampling improves decision-making.

Why LLN matters

• For researchers: it guides sample-size decisions to achieve desired accuracy and confidence.
• For business leaders and investors: it provides a realistic perspective on growth rates, target setting, and valuation assumptions.
• For risk managers: it justifies pooling exposures to reduce uncertainty.

Strategies to address growth challenges implied by LLN

Companies facing slower percentage growth as they scale can pursue:
* Acquisitions of smaller, faster-growing businesses to boost consolidated growth.
* Operational efficiencies and cost improvements to enhance absolute dollar returns even if percentage growth slows.
* Shifting focus from percentage targets to absolute dollar goals or profitability metrics.

The law of small numbers

This is a cognitive bias related to LLN: people often overinterpret results from small samples and underestimate how variable small-sample outcomes can be. Small samples produce unstable estimates and are more likely to mislead if treated as representative.

Conclusion

The law of large numbers is a foundational statistical principle with practical implications across research, insurance, and business strategy. It explains why larger samples produce more reliable estimates and why very large organizations often experience slower percentage growth. Recognizing LLN helps set realistic expectations and informs better decisions about sampling, forecasting, and growth planning.