2.11 Looking ahead

We opened this unit by asking why statistics has settled on the sample mean, and the answer is now in hand. It is centred on the truth, its variability shrinks as the sample grows, and larger samples therefore tend to produce estimates closer to the population mean.

But every argument in this unit relied on something we never have in practice.

We drew a thousand samples from the same population. We watched the sampling distribution emerge. We could see exactly where our estimate sat among all the other estimates that might have been obtained.

Real research is nothing like that.

A survey is conducted once, often at considerable expense, and produces exactly one sample. There is no second sample to compare against, no visible sampling distribution, and no way to know how far the estimate lies from the population value.

Yet researchers routinely write things like

The average household income is between ₹23,500 and ₹24,500.

How can anyone say that after observing a single sample?

The next unit answers that question. It shows how the sampling distribution, although invisible, can still be used to measure the precision of an estimate and to state a range of population values consistent with it.