Repeated random observations fluctuate, but their average becomes increasingly stable as more observations are collected. The law of large numbers formalizes this behavior.

Let

$$X_1,X_2,\ldots$$

be independent observations with common expected value $\mu$. Their sample mean after $n$ observations is

$$\bar X_n=\frac{X_1+\cdots+X_n}{n}.$$

Under the usual conditions,

$$\bar X_n\to\mu$$

as $n$ becomes large, in the probabilistic sense specified by the theorem.

What this means

If a fair coin is encoded as $1$ for heads and $0$ for tails, then $E[X]=1/2$. The proportion of heads in a long sequence of independent flips is the sample mean, so it tends to stabilize near

$$\frac12.$$

The same principle explains why repeated averages often settle near an expected value.

What it does not mean

The law does not say that short sequences must look representative. Ten fair coin flips can easily contain seven or eight heads.

It also does not say that the average moves steadily closer to the expectation at every step. Random fluctuations continue; they simply become smaller relative to the growing number of observations.

The law of large numbers explains long-run stabilization of averages. It does not describe the precise shape of their remaining fluctuations; that is the role of the central limit theorem.