Sample

A sample is the set of individuals you actually measure, drawn from the larger population you want to describe. Its size is written $n$. A number computed from a sample is a statistic, such as the sample mean $\bar{x}$ (x-bar) or the sample proportion $\hat{p}$ (p-hat), while the matching fact about the population is a parameter, written $\mu$ (mu) or $p$. The sample is the group of individuals, not the number you compute from them.

Poll 1,000 randomly selected voters and suppose 520 say yes. Then $\hat{p} = 520/1000 = 0.52$, an estimate of the unknown $p$. A different random 1,000 would not land on 0.52 again: the standard error is $\sqrt{\frac{0.52 \times 0.48}{1000}} = 0.0158$, so a repeat poll coming back at 0.50 or 0.54 would be ordinary rather than a sign that something went wrong.

Notice what is absent from that calculation. The population size never enters it, which is where the usual objection fails. "A poll of 1,000 cannot speak for 250 million people" sounds like arithmetic, but precision is governed by $n$, not by the fraction of the population sampled. Those 1,000 voters give the same 0.0158 whether the electorate is 250 million or 250 thousand, provided the sample is a small share of it.

What the formula does assume is that chance chose the sample. None of the above survives a sample that was not random: a self-selected sample of 250,000 has a standard error you can still compute and a bias you cannot, which is the point of does a bigger sample fix bias.

Measuring every member of the population instead is a census, and it returns parameters rather than estimates. Sampling exists because most populations are too large, too expensive, or too scattered for that, and because a careful sample often produces better data than a rushed attempt at everyone.