Population and Sample

Suppose you want to know the average height of every adult on Earth. There are roughly eight billion of them. You are not going to measure them all — you would still be measuring long after everyone you started with had grown old. Or suppose a factory wants to know how long its light bulbs last. The only way to be certain is to switch every bulb on and wait for it to die… which leaves a warehouse of dead bulbs and nothing to sell.

So statistics is built on a bargain. We give up the impossible dream of measuring everything, and instead measure a small sample. From that handful we infer what the whole is like. We trade certainty for feasibility — and, done well, the trade is astonishingly good: a few thousand carefully chosen people can tell you what a nation of millions thinks, to within a percent or two.

The two words, and the two kinds of number

The population is the whole collection we want to know about — every voter, every light bulb on the line, every tree in the forest. The sample is the part we actually get our hands on and measure. Almost all of statistics lives in the gap between them: we can see only the sample, yet the questions we care about are about the population. The whole craft is using the part to reason about the whole.

Once you have those two, two matching kinds of number fall out. A number that describes the population is a parameter. It is fixed but unknown — a single true value sitting out there that we never get to read directly. The population mean and standard deviation get Greek letters:

\mu = \text{population mean}, \qquad \sigma = \text{population standard deviation}.

A number we compute from the sample is a statistic. It is knowable — we have the data — but it varies: take a different sample and you get a different value. The sample mean and standard deviation get Roman letters:

\bar{x} = \text{sample mean}, \qquad s = \text{sample standard deviation}.

The link between them is the entire point: we use the statistic \bar{x} as our best estimate of the parameter \mu. The parameter is the target; the statistic is the arrow. (How to build a good arrow — a point estimate — and how far it typically misses is a story picked up in point estimates.)

A memory hook: parameter goes with population; statistic goes with sample. Greek letters for the unknown truth, Roman letters for what your data actually hands you.

Worked example 1 — name the four things

A biologist wants the average wingspan of the monarch butterflies in a large reserve. She nets 40 of them, measures each, and finds an average wingspan of 9.8 cm. Pin down the four ideas:

Population — every monarch butterfly in the reserve.
Sample — the 40 butterflies she actually netted and measured.
Parameter — the true mean wingspan \mu of all monarchs in the reserve. She will never know it exactly.
Statistic — the sample mean \bar{x} = 9.8 cm, computed from her 40. This is her estimate of \mu.

Notice the pattern: the population and its parameter are the things she cares about; the sample and its statistic are the things she can actually get.

Worked example 2 — why won't the statistic just equal the parameter?

Even if the biologist did everything perfectly — a truly random net, flawless measuring — her \bar{x} would still almost never land exactly on \mu. Why?

Because a sample is only some of the butterflies. This particular 40 happened to include a couple of unusually large males; the next random 40 might catch a few small ones instead. Each sample is a slightly different slice of the whole, so each produces a slightly different mean. That built-in wobble — the fact that \bar{x} changes from sample to sample purely by the luck of the draw — is called sampling variability. It is not a mistake and it is not bias; it is the unavoidable price of looking at a part instead of the whole. (The shape of that wobble — how \bar{x} is distributed across all possible samples — is exactly what the sampling distribution of the mean describes.)

One population, many samples

Below is a fixed population of points with its true mean \mu marked — a value you would normally never know. Slide to pick a different sample (the highlighted points). Each sample has its own mean \bar{x}, and you can watch that line jitter around \mu as the sample changes. The statistic chases the parameter, but it never lands exactly on it — that jitter is sampling variability, made visible.

The population is everything; a sample is the part we measure.
A parameter (\mu, \sigma) describes the population — fixed but unknown.
A statistic (\bar{x}, s) is computed from the sample — known, but it varies sample to sample.
We use the statistic to estimate the parameter: \bar{x} estimates \mu.

Worked example 3 — does a bigger sample help?

Yes — with an important string attached. Imagine estimating the average height of students in a school of 1,000 by measuring a random handful:

A random sample of 5 could easily miss by several centimetres — one tall basketball player throws the whole average off.
A random sample of 50 will usually land much closer; the tall and short students start to balance out.
A random sample of 500 would nail it to within a whisker.

As long as the sample is random, more data means less sampling variability — the estimate gets more precise. Roughly, the typical error shrinks with the square root of the sample size, so to halve your error you need about four times as much data. Bigger is better… but only when the sample is drawn fairly. That last clause is where people come spectacularly unstuck — see the warning below.

A bigger sample does not fix a biased one. Size buys you precision, not correctness — and only if the sample is representative in the first place. If your sampling method is skewed (say you only survey people who own telephones, or who happen to walk past your stall), then collecting a million of them just pins down the wrong answer more confidently. A poll of a million from a lopsided source is worse than a small but truly random sample of a few thousand, because it wraps a wrong answer in a reassuring cloak of "but look how many people we asked!"

The lesson: how you sample matters more than how many. A large sample from a bad method is a large mistake. (Where those skews sneak in, and how randomness defends against them, is the whole subject of sampling and bias.)

In 1936 the respected magazine Literary Digest ran the biggest poll anyone had ever seen. They mailed out ballots and got back a staggering 2.4 million replies — and confidently predicted that Alf Landon would beat Franklin Roosevelt in a landslide.

Roosevelt won 46 states out of 48. The Digest was humiliated and folded soon after. What went wrong with 2.4 million people? Their address lists came from telephone directories, club memberships and car registrations — and in the depths of the Great Depression, the people with phones and cars were disproportionately wealthy, and disproportionately voting Landon. The sample was enormous and rotten.

Meanwhile a young pollster named George Gallup called the result correctly using a sample of only a few thousand — chosen to mirror the electorate. It remains the most famous demonstration in the history of statistics that a small representative sample crushes a giant biased one. How you sample beats how many, every time.

Sometimes! When a country counts every resident, that is a full population measurement — a census — with no sampling error at all. But censuses are so slow and costly that most nations run one only once a decade, and even then rely on samples to fill the gaps and check the count. For almost every everyday question — quality control, opinion polls, medical trials, tasting the soup — a good sample is not a compromise you settle for; it is the smart, deliberate choice.