What Is the Law of Large Numbers?

What Is the Law of Large Numbers?

Let me explain the law of large numbers directly to you. In statistics, it states that as your sample size grows, the results from that sample get closer to the true average of the entire population you're studying. This makes your sample more representative overall.

Now, in the financial world, I see it used differently. It means that a large entity growing rapidly can't keep up that pace forever. Think of those massive blue-chip companies with market caps in the hundreds of billions—they're prime examples of this.

Key Takeaways

You should know that the law of large numbers says an observed average from a growing sample will approach the true population average. In stock analysis, it ties into how scale affects growth rates. For companies, it's linked to diseconomies of scale, meaning a firm with fast sales growth won't hold that rate indefinitely.

Understanding the Law of Large Numbers

The law of large numbers has distinct definitions in statistics and finance, so let's break it down.

In statistical analysis, you can't poll every single person in a population, right? Researchers use samples instead. The law says the bigger your sample, the better it mirrors the whole population. Each extra data point boosts the chance that your outcome truly reflects the mean.

But remember, this doesn't guarantee every sample will perfectly match the population, and it doesn't support the gambler's fallacy—that repeating a random event makes a payoff more likely.

In business, I apply it to growth rates over time. It suggests maintaining a high percentage growth gets tougher as the base dollar amount swells, even if the percentage stays constant. This applies to businesses or products alike.

Law of Large Numbers Statistical Analysis Example

Consider this: if you're finding the average of a data set with 100 possible values, picking 20 data points gives you a more accurate average than just two. Those two could be outliers, but with 20, it's less likely they're all unrepresentative.

Say the set is integers from 1 to 100, and you pick 95 and 40—your average is about 67.5. Add more random picks up to 20, and it shifts closer to the true average.

Law of Large Numbers and Central Limit Theorem

This law connects to the central limit theorem, which says that as sample size grows, the sample mean distributes evenly, often shown as a bell curve with the mean at the peak.

The law of large numbers focuses on how the sample mean nears the population mean with larger samples—it's about the center of that curve. The central limit theorem covers the overall distribution.

Law of Large Numbers and Business Growth

People sometimes use 'law of large numbers' casually to note that exponential growth doesn't scale well, often due to diminishing marginal returns or diseconomies of scale.

This applies to metrics like market cap or net profit. Huge companies struggle to keep past stock appreciation rates. It's relevant when choosing between growth and value stocks.

Law of Large Numbers Business Example

Take Tesla: in 2020, automotive sales were $24.6 billion, jumping to $44.1 billion in 2021—a 79% increase. As an emerging market player, they hit economies of scale fast.

But the law says maintaining that won't last as the company grows, because the dollar values get unreasonable with steady percentage growth.

Tesla's Actual Revenue Numbers

In reality, Tesla's revenues were: 2021: $53.82 billion (70.67% increase), 2022: $81.46 billion (51.35%), 2023: $96.73 billion (18.8%), 2024: $97.69 billion (0.95%), 2025: $95.72 billion (1.03%). For the quarter ending March 31, 2025, it was $19.33 billion, down 9.23% year-over-year.

Law of Large Numbers and Insurance

In insurance, this law helps calculate and refine risk projections. If a company sets car insurance premiums with a small data set, they can't gauge risks well.

With more data, patterns emerge—like young male drivers being accident-prone. This larger sample represents incidents better, leading to accurate premiums and refined risk criteria.

Why Is the Law of Large Numbers Important?

In statistics, it validates your sample size—small data assumptions might not hold for the real population.

In business, it's key for targets. Doubling revenue one year means the same dollars earned over two years; 50% growth next year might seem less, but percentages mislead as dollars escalate.

How Can Companies Overcome the Challenge of the Law of Large Numbers?

Companies tackle this by acquiring smaller firms for scalable growth, boosting efficiency through size in manufacturing and distribution, or focusing on dollar goals over percentages.

What Is the Law of Small Numbers?

It's the idea that people underestimate variability in small samples, often overestimating based on inadequate data.

The Bottom Line

For researchers, this law guides sample size to represent populations. For stock analysts, it's a caution: huge, fast-growing companies may not sustain that pace forever.