Math 9 · Probability and statistics
Who you ask, and what it predicts
A survey result is only as good as the question asked and the people who answered it. This unit is about the choices that make data trustworthy, and about how probability — in weather forecasts, insurance, sport and medicine — turns past data into a prediction.
- 1. The words, first
- 2. Issues in collecting data
- 3. Populations and samples
- 4. Probability in society
- 5. What costs marks
The words, first
The idea: Population and sample are the pair everything else hangs on: who you want to know about, and who you actually asked.
| Word | What it means |
|---|---|
| Population | The whole group you want to know about: every student in the school, every voter in Alberta. |
| Sample | The part of the population you actually ask or measure. Asking everyone is a census. |
| Bias | Anything that pushes results away from the truth in one direction: a leading question, a lopsided sample, bad timing. |
| Representative sample | A sample whose makeup matches the population's, so its results can stand in for everyone's. |
| Random | Every member has the same chance of being chosen. |
| Probability | How likely an event is, from 0 (impossible) to 1 (certain), as a fraction, decimal or percent. |
| Theoretical probability | Worked out from equally likely outcomes. On a fair die, rolling a 5 has probability 1/6. |
| Experimental probability | Worked out from what actually happened: successes ÷ trials. |
| Assumption | Something taken as true without being checked. For a prediction, usually that the future will behave like the past. |
Issues in collecting data
The idea: Seven things can spoil data before any maths is done. Most of them are about the question and the moment, not the numbers.
| Issue | Example |
|---|---|
| Bias | Asking whether the town needs a skatepark, only at the skatepark |
| Use of language | “Don't you agree the ugly old gym should be replaced?” |
| Ethics | Collecting answers “for a class project” and then passing them to an advertiser |
| Cost | Testing how long every light bulb in a factory lasts — none would be left to sell |
| Time and timing | Asking about favourite seasons during a January cold snap |
| Privacy | A named form asking about your family's income |
| Cultural sensitivity | Asking the whole school which gift they got for Christmas |
Worked example: fixing a question. “Don't you agree that the school should stop wasting money on new uniforms?” has two problems. “Don't you agree” invites a yes, and “wasting” has already decided the answer. A neutral version: “Should the school buy new uniforms this year? Yes / No / No opinion.” It names the issue, takes no side, and lets people who do not care say so.
Populations and samples
The idea: A sample is only useful if it looks like the population. Choosing at random is how you get there without having to guess what matters.
A school of 600 students wants to know how students get to school. Asking 60 of them is a sample. Six ways to choose the 60:
| Method | How the 60 are chosen | Fair? |
|---|---|---|
| Simple random | A computer picks 60 names from all 600 | Yes |
| Systematic | Every 10th name on the list, from a random start | Yes, if the list has no hidden pattern |
| Stratified | A random 10% of each grade: 18 of 180 Grade 7s, 20 of 200 Grade 8s, 22 of 220 Grade 9s | Yes, with every grade fairly represented |
| Cluster | Two homerooms of 30 picked at random, everyone in them asked | Usually, if homerooms are alike |
| Convenience | The first 60 through the front doors | No: bus riders may use another door |
| Voluntary response | A poll on the school website | No: mostly people with strong feelings reply |
Worked example: from sample to population. In a fair sample of 60, 18 students ride the bus. That is 18 ÷ 60 = 0.30, so the prediction for the whole school is 0.30 × 600 = 180 bus riders. The prediction assumes the sample is representative. Had the 60 been the first through the front doors, it would undercount bus riders, and every number built on it would be wrong in the same direction.
Probability in society
The idea: A probability is a prediction built from data and assumptions. Knowing the assumption tells you how far to trust it.
Theoretical and experimental. On a fair die, P(5 or 6) = 2/6 = 1/3. Roll it 60 times and get a 5 or a 6 on 23 of them, and the experimental probability is 23/60 ≈ 0.38. The two differ because 60 rolls is not many; with more rolls, the experimental value tends to settle closer to 1/3.
Worked example. A basketball player has made 34 of her last 40 free throws. Predict how many of her next 20 she will make.
- Experimental probability: 34 ÷ 40 = 0.85, or 85%.
- Prediction: 0.85 × 20 = 17 free throws.
- The assumptions: she keeps shooting as she has been — no injury, no extra pressure. Change that and the prediction weakens.
Where you meet it.
- Weather. A 40% chance of rain means that on days forecast like this one, measurable rain falls at a given spot about 4 times in 10 — not rain for 40% of the day.
- Insurance. Prices are set from claims data. New drivers have more collisions on average, so they usually pay more: a prediction about a group, applied to each person in it.
- Sport. A goalie's save percentage of .915 is an experimental probability: 915 saves for every 1000 shots.
- Medicine. A side effect in 2 of every 100 patients in trials predicts your chance only as far as you are like the people tested.
A probability is not the whole decision. A 30% chance of rain might mean no umbrella for the walk to school, but a rented tent for a wedding. What is at stake decides.
What costs marks
The idea: Most of these are naming the wrong method, or not naming the problem at all.
- Calling a convenience sample random. Easy to reach is not the same as random.
- Mixing up voluntary response and convenience. Voluntary means people chose themselves; convenience means you chose whoever was handy.
- Saying a question is biased without saying why. Name the words that lead.
- Treating an experimental probability as exact. It is an estimate, and it improves with more trials.
- Forgetting the assumption when asked how reliable a prediction is.