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Statistics · Sampling

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Populations & samples

What a population and a sample are and why we sample at all; why a representative sample matters more than a big one; and the two calculations that earn the marks — scaling a sample up to estimate a population total (including capture–recapture and its assumptions) and sharing a stratified sample out in proportion.

Statistics · Sampling

Populations & samples

What a population and a sample are and why we sample at all; why a representative sample matters more than a big one; and the two calculations that earn the marks — scaling a sample up to estimate a population total (including capture–recapture and its assumptions) and sharing a stratified sample out in proportion.

Why it works

The population is everything you want to know about — every battery in the day's production, every fish in the lake, every member of the club. A sample is a subset of it that you actually look at. Counting the whole population is a census, and it is the accurate option, so the first question is: why would anyone ever settle for a sample?

Three reasons, and exam questions turn on all three. Cost — asking 40 000 people costs vastly more than asking 400. Time — a survey that takes a year answers a question you needed settled in a month. And destructive testing — to find out how long a battery lasts you have to run it flat. A census of battery lifetimes destroys the entire day's production; there would be nothing left to sell. In that last case a sample isn't the cheap option, it is the only option.

What you buy with a sample, and what you give up. A sample is only useful if what is true of the sample is roughly true of the population — if it is representative. A representative sample is the population in miniature: the same kinds of people or items, in roughly the same proportions.

The trap, made concrete. A council wants to know how often people in a town exercise, and asks 4040 people at a leisure centre. 3434 of them — 85%85\% — exercise at least three times a week. That is obviously too high: everyone standing in a leisure centre chose to go to a leisure centre. So the council repeats it with 20002000 people, at the same leisure centre, and gets… about 85%85\% again. The true figure for the town might be 20%20\%. Making the sample 5050 times bigger did not move the answer one step closer to the truth; it just made a wrong answer look more convincing.

That is the whole point about size. A bigger sample is generally more reliable, because random ups and downs average out — one unusual person shifts a sample of 20002000 far less than a sample of 4040. But that only shrinks the random error. If the way you chose people leaves part of the population out altogether, no amount of extra data brings them back. Size improves precision; only the choosing can give you accuracy.

The workhorse: scaling a sample up. Nearly every calculation in this topic is one idea — *the proportion in the sample is your best estimate of the proportion in the population*. A charity with 50005000 members surveys 200200 of them, and 3434 say they volunteer monthly. That is 34200=0.17\frac{34}{200} = 0.17, so estimate

0.17×5000=850 members.0.17 \times 5000 = 850 \text{ members.}

Check it the other way round: 8505000=0.17=34200\frac{850}{5000} = 0.17 = \frac{34}{200}. ✓ That check — putting your answer back and confirming it reproduces the sample's proportion — catches almost every slip in this topic.

Capture–recapture is that same idea in disguise. You cannot count the fish in a lake, so: catch 6060, mark them, put them back. Later catch 5050 and find 1212 marked. The marked fish have mixed back through the lake, so the proportion marked in your net should match the proportion marked in the lake:

marked in 2nd samplesize of 2nd sample=marked totalpopulation⟹1250=60N.\frac{\text{marked in 2nd sample}}{\text{size of 2nd sample}} = \frac{\text{marked total}}{\text{population}} \quad\Longrightarrow\quad \frac{12}{50} = \frac{60}{N}.

So N=60×5012=250N = \frac{60 \times 50}{12} = 250 fish. Check: 60250=0.24\frac{60}{250} = 0.24 and 1250=0.24\frac{12}{50} = 0.24. ✓ Put the 1212 in the wrong place and you get 60×1250=14.4\frac{60 \times 12}{50} = 14.4 — a lake holding fewer fish than you have already pulled out of it, which the check would have caught instantly.

Because that argument leans on the marked fish behaving like the rest, you must be able to state an assumption — it is a real mark:
  • the population has not changed between the two catches (no births, deaths,
arrivals or departures);
  • the marked individuals mixed back in evenly, and the second sample was taken
from the whole population;
  • the marks did not come off, and being marked did not change an individual's
chance of being caught or surviving.

Stratified sampling: sharing the sample out in proportion. If a population splits into groups (strata) — year groups, sites, age bands — a representative sample should contain each group in the same proportion as the population:

number from a stratum=size of stratumsize of population×sample size.\text{number from a stratum} = \frac{\text{size of stratum}}{\text{size of population}} \times \text{sample size}.

A school of 900900 students has 320320 in Year 9, 300300 in Year 10 and 280280 in Year 11, and wants a stratified sample of 4545: 320900×45=16\frac{320}{900}\times 45 = 16, 300900×45=15\frac{300}{900}\times 45 = 15, 280900×45=14\frac{280}{900}\times 45 = 14. Always re-add: 16+15+14=4516 + 15 + 14 = 45. ✓

Usually the numbers don't land whole. A youth theatre with 240240 members in four age bands of 63,57,66,5463, 57, 66, 54 wants a sample of 2525: 6.5625, 5.9375, 6.875, 5.6256.5625,\ 5.9375,\ 6.875,\ 5.625 — which do add to 2525 exactly, as they must. Round each to the nearest whole number and you get 7,6,7,67, 6, 7, 6, a "sample of 2525" containing 2626 people. So round, then re-add, then adjust one of them: send down whichever only just made it up — 6.56256.5625 was the closest to the halfway mark — giving 6,6,7,6=256, 6, 7, 6 = 25. ✓ Nearest-whole-number rounding is a starting point in this topic, never the finish.