Statistical enquiry & sampling
Statistical enquiry & inference: pose statistical questions; conjecture from data; identify a data set as a sample of a population.
A statistical enquiry starts with a good question and answers it with data — but usually we can only look at some of the things we care about, then reason about the rest.
What it means
A statistical question is one you expect to answer with data that varies — where different measurements give different answers. “How tall is Sam?” is not statistical (there is one answer). “How tall are the children in Year 5?” is statistical, because the heights vary and you need many measurements to describe them.
The whole group you want to know about is the population — every Year 5 child, say. Often it is too big, too costly, or simply impossible to measure everyone. So you measure a sample: a smaller part of the population that you actually collect data from. From the sample you make a conjecture — a reasoned guess about the whole population.
The key idea is that a sample stands in for the population. If the sample is chosen fairly, so that every part of the population has a chance of being picked, then what is true in the sample is probably close to what is true overall. If the sample is chosen unfairly — only your tall friends, only the fast readers — it is biased, and your conjecture will be off.
So a statistical enquiry is a loop: pose a question → collect a sample → look at the data → conjecture about the population → ask whether the sample was fair.
Worked examples
1. Question or not?
| Question | Statistical? | Why |
|---|---|---|
| ”What is my shoe size?” | No | one fixed answer |
| ”What shoe sizes are in my class?” | Yes | sizes vary |
| ”Do children prefer apples or grapes?” | Yes | preferences vary |
2. Sample to population. A jar holds 500 marbles. You scoop out 20 without looking: 8 are red. In the sample, 8 / 20 = 40% are red. Conjecture: about 40% of all 500, so roughly 0.40 × 500 = 200 red marbles. You did not count all 500 — you inferred from a sample.
3. Spotting bias. To learn the class’s favourite sport, you ask only the football team. That sample is biased — it over-represents football fans, so the conjecture “everyone loves football” is unsafe. A fairer sample: draw ten names from a hat.
The generative-art connection
Sampling is something you can literally watch happen. In Seeing Theory’s probability visual, dots keep dropping from a population and the sample’s shape slowly grows to match the true one — the picture is the idea that more samples give a steadier estimate.
You can make your own with the dot-multiplier tool: think of the full starburst of dots as the population, then “sample” by lighting up only a handful and reading their colours or positions. A small handful looks patchy and random; take more, and the pattern of the whole starts to show through the sample. The art makes visible the central move of statistics — a part standing in for the whole, and the estimate sharpening as the sample grows.
Common misconceptions
- “A bigger sample is always right.” Bigger helps, but a large biased sample (only football players, however many) is still wrong. Fairness matters as much as size.
- “The sample must exactly match the population.” It won’t — samples wobble. A conjecture is about the population, expected to be close, not identical.
- “Any question that uses numbers is statistical.” Only if the answer varies. A single fixed fact is not a statistical question.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
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Sort these into "statistical question" or "not statistical", and say why: (a) `How old is our teacher?` (b) `How old are the people who live on my street?` (c) `How many minutes did I sleep last night?` (d) `How many hours do children in my school sleep each night?`
Answer
Not statistical: (a) and (c) — each has one fixed answer. Statistical: (b) and (d) — the answers vary from person to person, so you need many measurements to describe them. A question is statistical when you expect the data to vary.
Art hook Build a Canvas 2-D 'variability meter': draw a horizontal number line and drop a dot for each answer. A non-statistical question stacks all dots on one spot; a statistical one scatters them into a spread. Clicking a question toggles between a single tall column and a wide sprinkle of dots.
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A library has `2000` books. Without looking, a librarian pulls `50` books off the shelves and finds `15` of them are picture books. Estimate how many of all `2000` books are picture books, and explain your reasoning.
Answer
In the sample, `15 / 50 = 0.30 = 30%` are picture books. Conjecture: about 30% of all books, so `0.30 × 2000 = 600` picture books. The sample stands in for the whole collection.
Art hook Draw a 40×50 grid of tiny book-spine rectangles (2000 cells). Randomly recolour 30% as picture books, then let the user 'sample' by clicking any 50 cells; a counter shows the sample percentage wobbling near 30% as they pick different patches.
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You want to find out the favourite school lunch of everyone in your school. You only ask the children waiting in the pizza queue. Explain why this sample is unfair (biased), and describe a fairer way to choose who to ask.
Answer
The pizza queue over-represents children who already like pizza, so the sample is biased and the conjecture will lean toward pizza unfairly. A fairer sample gives everyone a chance to be picked — e.g. draw names randomly from the whole school register, or ask a few children from every class.
Art hook Canvas scene: a crowd of dots coloured by lunch preference. A biased 'net' only catches dots near the pizza corner; a fair sampler drops a random highlight ring anywhere. Show the two resulting bar charts side by side so the bias is visible.
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Nadia measured the arm-span of `12` children in her class and made this claim: "My sample of 12 tells me the exact average arm-span of every 11-year-old in the world." What is wrong with her claim? Rewrite it as a sensible conjecture.
Answer
Her class of 12 is a small sample and cannot give the *exact* value for the whole population — samples wobble and 12 children from one class may not represent every 11-year-old. A sensible conjecture: "Based on my sample, the average arm-span of 11-year-olds is probably *close to* my class average," and a larger, fairer sample would make the estimate steadier.
Art hook Animate a running-average line: add sampled arm-spans one at a time and plot the mean as a jittery line that jumps a lot at first, then settles as more dots are added — visually showing why 12 is shaky and more samples steady the estimate.
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In a survey of `40` children (a fair sample of a `600`-child school), `24` said they walk to school. Write a conjecture about the whole school, estimate the number of walkers, and state one reason your estimate might not be exactly right.
Answer
In the sample `24 / 40 = 0.60 = 60%` walk. Conjecture: about 60% of the school walks, so roughly `0.60 × 600 = 360` children. It may not be exact because a sample wobbles — a different fair sample of 40 could give a slightly different percentage.
Art hook Colour-wheel pie: draw a ring split 60% 'walk' / 40% 'other' from the sample, beside a 600-dot field where 360 dots pulse to show the estimated walkers. Re-sampling reshuffles which 40 dots are read, nudging the ring each time.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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Which of these can have more than one answer depending on who you ask? (a) `What is the date today?` (b) `What is your favourite colour?`
Answer
(b) — favourite colour varies from person to person. (a) has one fixed answer today. Questions whose answers vary are the statistical ones.
Art hook Make a colour-vote wheel: each person's favourite colour adds a wedge to a growing pie chart, so the class's variety becomes a bright ring of unequal slices.
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True or false: "How many legs does a spider have?" is a statistical question. Explain.
Answer
False. Every spider has 8 legs, so the answer does not vary — it is a single fixed fact, not statistical.
Art hook Draw 8-legged spider dots in a grid; every one identical shows there is no spread, i.e. nothing to sample.
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Turn this non-statistical question into a statistical one by changing just a few words: `How tall am I?`
Answer
For example: `How tall are the children in my class?` Now the answer varies across many people, so it needs data — it is statistical. (Any version where heights vary is correct.)
Art hook Height number line: your single dot sits alone; switching to the class version sprinkles many dots along the line, making the 'spread' appear.
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A gardener has a big tray of `100` seedlings. She checks `10` of them and `2` are yellow. What fraction of the ones she checked are yellow?
Answer
`2 / 10 = 1/5 = 20%` of the checked seedlings are yellow.
Art hook 10×10 grid of seedling dots; the sampled 10 glow, and 2 of them turn yellow — the sampled fraction shown as a lit patch of the whole tray.
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Using the same tray from the last question (`10` checked, `2` yellow), estimate how many of all `100` seedlings are yellow.
Answer
If `20%` of the sample is yellow, conjecture about 20% of all 100, so roughly `0.20 × 100 = 20` yellow seedlings.
Art hook Extend the grid: recolour ~20 of the 100 dots yellow to show the conjecture painted across the whole tray, next to the small sampled patch it came from.
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Name the population and the sample: To learn what fruit `500` festival-goers like best, an organiser asks `30` of them at the entrance.
Answer
Population: all 500 festival-goers. Sample: the 30 people asked at the entrance. The sample is the part actually surveyed; the population is the whole group we want to know about.
Art hook A field of 500 dots with a small ring lassoing 30 of them — a clear picture of 'sample inside population'.
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Spot the mistake: "I want to know the favourite pet of everyone in town, so I asked 20 people leaving the dog park." What is unfair here?
Answer
People at the dog park almost all like dogs, so the sample is biased toward dog-lovers. It does not fairly represent the whole town. A fairer sample would pick people randomly from all over town.
Art hook Show a town of mixed pet-preference dots; the dog-park sampler only grabs dogs, skewing the resulting bar chart — toggle to a random sampler to see it balance out.
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A jar has coloured beads. You scoop `25` beads and get `5` blue. What percentage of your scoop is blue, and what would you conjecture about the whole jar?
Answer
`5 / 25 = 1/5 = 20%` of the scoop is blue. Conjecture: about 20% of all the beads in the jar are blue (assuming the scoop was a fair mix).
Art hook A jar of scattered bead-dots; a scoop circle lifts 25, of which 5 pulse blue, and a mini-bar predicts '20% blue' for the full jar.
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Reasoning: Ben scoops `20` beads and gets `4` blue (20%). He scoops another `20` and gets `6` blue (30%). Why don't the two scoops give the same answer, and which is likelier to be closer to the truth for the whole jar: one scoop of 20, or both scoops combined as 40?
Answer
Samples wobble — different fair scoops naturally give slightly different results. Combining both into a sample of 40 (`10 / 40 = 25%`) is likelier to be closer to the truth, because a bigger fair sample gives a steadier estimate.
Art hook Running-percentage line: plot the blue-% after each bead is drawn; it jumps around early and settles toward the true value as the sample grows — the 'steadying' made visible.
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In a fair sample of `50` books from a `1000`-book library, `20` are non-fiction. Estimate the number of non-fiction books in the whole library, then say why your answer is an estimate and not an exact count.
Answer
`20 / 50 = 0.40 = 40%` non-fiction in the sample, so conjecture `0.40 × 1000 = 400` non-fiction books. It is an estimate because you only looked at 50 of the 1000 books — a different fair sample could give a slightly different percentage, so the real number is probably *near* 400, not certainly exactly 400.
Art hook 1000-dot shelf grid: light the sampled 50, colour 40% of them, then paint ~400 dots across the whole grid as the conjecture — with a faint 'wobble band' around 400 to show it is an estimate.
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Open-ended: You want to find out the most popular playground game in your whole school (say `300` children). Write a statistical question, describe a *fair* way to choose a sample of about 30 children, and say how you would use your results to make a conjecture about all 300.
Answer
Example question: `Which playground game do children in our school play most?` (answers vary → statistical). Fair sample: put all 300 names in a list and randomly pick 30, or pick 5 children from each of 6 classes so every class is represented. Conjecture: whichever game most of the 30 pick, estimate that about the same *percentage* of all 300 prefer it — e.g. if 12 of 30 (40%) say tag, conjecture roughly `0.40 × 300 = 120` children school-wide, remembering the estimate can wobble.
Art hook Interactive survey-sim: 300 dots each secretly assigned a game-colour; a fair random sampler highlights 30, tallies them into a live bar chart, then projects the winning colour's percentage across all 300 dots — re-run to watch the conjecture wobble but stay near the true split.