Census vs Population

In a census you record data on every individual in the population, so what you compute is the parameter and not an estimate of it. There is no sampling distribution around it, no standard error, no margin of error, because nothing was sampled. What makes a study a census is coverage of the population as you defined it, which means the same set of responses can be a census of one group and a sample of a larger one.

A department has 8 teachers with 3, 5, 6, 8, 11, 12, 14, and 21 years of experience. If the population is that department, the mean $\frac{80}{8} = 10$ years is $\mu$ (mu), exact and final: no interval is needed because no other value is possible. Ask instead about all teachers in the district and those 8 become a sample, the same 10 years becomes $\bar{x}$ (x-bar), and it turns into an estimate carrying uncertainty. The arithmetic did not move. The population did.

"A census has no error" holds for exactly one kind of error. It removes sampling error, the sample-to-sample variation that exists because you measured a part instead of the whole. Every other flaw survives: households an enumerator never reaches are undercoverage, people who refuse at the door are nonresponse, and a leading question produces the same response bias at full coverage that it produces in a sample of 300.

There is also nothing left to infer. When the data cover the population of interest, a confidence interval has no unknown parameter to bracket and a significance test has no population claim to weigh. If two fully measured departments average 10 and 11.4 years of experience, that 1.4 year gap is a description of those two departments, and testing it for significance answers a question nobody needed to ask.

A census is rare for practical reasons rather than statistical ones: cost, time, populations that change while you count them, and measurements that destroy what they measure, since testing every battery until it fails leaves none to sell.

The population is fixed by the question, not by the data. Defining one means writing a membership rule precise enough to sort any individual in or out: not "voters" but "the 12,400 people registered to vote in this county on October 1". A number computed from the whole population is a parameter, written $\mu$ (mu) for a mean, $p$ for a proportion, $\sigma$ (sigma) for a standard deviation. A parameter is one fixed number, usually unknown, and it does not move when you take a different sample.

Stay with that county. The parameter is $p$, the proportion of all 12,400 registered voters who plan to vote yes. A random sample of 600 gives $\hat{p} = 0.52$ (p-hat), an estimate of $p$ and not $p$ itself. The population size enters in one place only: the sample is $600/12400 = 0.048$ of the population, under the 10 percent ceiling the standard error formula needs, so $\sqrt{(0.52)(0.48)/600} = 0.020$ can stand as the standard error.

The wrong sentence is short and common: "the population is the 600 voters who were surveyed." Those 600 are the sample. The population is the group the conclusion is about, and it exists whether or not anyone measures it. Two smaller slips travel with it. A population need not be people; it can be 4,000 laptop batteries, or every 20-minute interval in a factory's day. And a population is not everyone, because it stops exactly where the question stops, so this survey says nothing about the state.

One piece of intuition is worth killing off. A bigger population does not need a bigger sample. Precision comes from $n$; apart from that 10 percent check, the population size appears nowhere in $\sqrt{p(1-p)/n}$, so a county of 12,400 and a country of 300 million take about the same sample for the same margin of error.

The population you want and the population your method can reach are different things. The list actually drawn from is the sampling frame, and anyone in the population but off that list has no chance of selection. Random sampling is topic 1.11 in Unit 1.