Law of Averages

Date
Sept. 24th 2024

By
Protothesis

Discipline
Statistics

Related Concepts

Parity Principle
Law of Means
Red Queen Effect
Power Laws

Imagine you're flipping a coin. Each flip is independent: heads or tails, both are equally likely. Now, say you start flipping repeatedly. In the first few flips, you might get a string of heads, or maybe tails dominates early on. But over time, as you flip more and more, you'll notice something: the results start evening out. The ratio of heads to tails begins to stabilize, inching closer and closer to an equal 50/50 split.

This is an intuitive glimpse of the Law of Averages: over a large number of trials, the outcomes will tend to reflect the expected probabilities. But let’s dive deeper into the why behind it.

Analogies

Law of Averages

At its core, the Law of Averages is about balance across many trials. Think of it like filling an empty bucket with water, one cup at a time.

Sometimes, you pour a little too much and it splashes over the sides. Sometimes, too little and the bucket stays dry at the top.

But after many pours, the bucket reaches a balanced, even state — just like the outcomes of coin flips or dice rolls over many trials.

Diagram

Law of Averages

Imagine a graph: on the y-axis, you have the ratio of heads to total flips. On the x-axis, the number of flips increases from left to right. After a few flips, the line on the graph jumps up and down wildly — you may have four heads and only one tail at first.

But as you flip more, the peaks and valleys start to flatten. The line converges towards a smooth 50%, reflecting how, in the long run, the outcomes average out.

This “flattening” is the Law of Averages in visual form: over many trials, the actual outcomes approach the expected probability.

Examples

Law of Averages

Let’s take this into a real-world context. Consider a basketball player with a career free-throw average of 80%. In any given game, they might shoot way above or below that average. But across a whole season, their performance tends to align with their expected 80% success rate.

Similarly, casinos rely on the Law of Averages. While individual gamblers may win big on occasion, over thousands of hands, spins, or rolls, the house edge ensures the casino turns a profit — this is statistical inevitability playing out over large numbers.

Principles

Law of Averages

The Law of Averages stems from a broader statistical principle known as the Law of Large Numbers. In essence, the more trials you have, the closer your actual outcomes will come to the expected probabilities. If you flip a coin just 10 times, you might not get an even split of heads and tails. But if you flip it 1,000 or 10,000 times, the ratio will gravitate towards 50/50.

This principle applies to any probabilistic event where there’s an expected outcome — whether it’s coin flips, dice rolls, or the free throws of our basketball player.

Technical Definition

Law of Averages

Finally, let’s introduce the formal mathematical language that captures the Law of Averages. The Law of Large Numbers states that as the number of experiments increases, the sample mean (the observed average outcome) approaches the population mean (the theoretical expected value).

In the coin flip example, the population mean is 50% heads, and the sample mean is the ratio of heads after a certain number of flips.

The Law of Averages is simply our intuitive way of expressing this convergence over time. More flips (or trials) = more consistency with expected outcomes.