2.6 Why does the sample mean become more reliable?

Two properties, established separately. What do they amount to together?

We can now answer the question this unit opened with.

The sample mean is used almost universally because its sampling distribution sits in the right place and tightens as the sample grows. It is centred on \(\mu\) at every sample size, and the width of that distribution, \(\sigma/\sqrt{n}\), falls towards zero as \(n\) increases. A distribution centred on the correct value and becoming ever narrower must concentrate on that value. Large samples are therefore increasingly likely to produce averages close to \(\mu\).

That last sentence is the Law of Large Numbers, which Section 1.5 introduced as a description of observed behaviour. It is no longer a description. Averages settle down because the procedure that produces them is unbiased and increasingly precise, and there is nothing else to the result.

Both halves are load-bearing, and it is worth seeing why neither would do alone.

An estimator that was centred but whose spread never shrank would remain permanently unreliable — the “use the first observation” rule from the previous section, which stays honest and stays useless however much data you give it.

An estimator whose spread shrank around a value that was not \(\mu\) would be worse still. It would become reliably wrong, converging with increasing confidence on the incorrect answer, and every additional observation would harden the mistake rather than expose it. Survey B is exactly this: had the urban-only surveyor collected ten thousand households instead of a hundred, their estimates would have clustered even more tightly around a number that was still ₹6,800 too high.

The sample mean earns its place through two properties together:

\[E(\bar{X}) = \mu \qquad\qquad \mathrm{Var}(\bar{X}) = \frac{\sigma^2}{n}\]

Centred on the truth, and increasingly precise. Neither is sufficient on its own, and both depend on assumptions about how the sample was collected.