A sample speaks for a population

A statistical study collects data from a sample to answer an investigative question about a larger population. The question comes first, and it names a group: What proportion of the district's seniors work a part-time job? What is the mean lifetime of the batteries this factory shipped in March?

Studies work this way out of necessity. When the population is too large, too scattered, or too expensive to measure item by item - and when the measurement destroys the item, as lifetime-testing a battery does - collecting data from every individual is not an option. A subset is measured instead, and the study's entire job is to say something defensible about the group that was not measured.

Two small words carry the raw material. A single recorded piece of information about one item or individual is a datum; the collection of them is a data set. Everything else in the course is built by summarizing data sets and asking how far the summaries reach.

The population consists of all items or individuals of interest - every unit the investigative question is about. Its size is written $N$. The sample is the subset of the population from which data are actually obtained, and the number of items in it, the sample size, is written $n$.

1. A poll interviews 850 of a city's 61,000 registered voters: the population is all 61,000, the sample is the 850, so $n = 850$.
2. An inspector tests 50 of the 2,000 chargers a plant made today: population 2,000 chargers, sample 50.
3. A biologist nets 40 trout from a lake: the sample is the 40; the population is every trout in the lake, and $N$ is not even known.

The test that never fails: the population is the group the question is about; the sample is the group the data came from. When those are the same group - every unit measured - the study is a census, and a census supports a different, stronger kind of claim than a sample ever can.

A parameter is a numerical summary of the variable of interest for the population. A statistic is the same kind of summary computed for the sample. The pairing is strict: proportion of all registered voters supporting the incumbent, parameter; proportion of the 850 interviewed, statistic. In practice the parameter is unknown - it is the reason the study exists - and the statistic is the number in hand.

The two are almost never equal. Different random samples of the same population produce different statistics, scattered around the fixed parameter. That scatter is not error in the sense of a mistake; it is what sampling does, and later units measure it precisely. A statistic is the basis for making inferences about the parameter. It is not a stand-in that can be announced as the population's value.

Notation enforces the bookkeeping early: a population proportion is written $p$, and a sample proportion is written $\hat{p}$. The hat marks the estimate. Swapping the symbols swaps the claim.

An investigative question for a specific study should have a defined purpose and should not be changed based on the data analysis or results. A question rewritten to chase whatever pattern showed up is no longer being answered by the data; it is being dictated by them, and the apparent finding cannot be trusted at face value.

The question should also be posed so that the required data can be collected and analyzed:

1. A defined population. Households in Maple City can be listed and sampled; people everywhere cannot.
2. A recordable variable. Visited a park in June, yes or no, is measurable; loves parks is not.
3. A summary that would answer it. A proportion, a mean, a comparison - name it before collecting.

Finally, every component and every calculation should be tied back to the real-world setting it came from: the variable, its units, and the population. That identification is what the exam means by in context, and it is graded relentlessly. A proportion with no group attached is not yet an answer.