Mathematics Year 7: Population and sample: who do you ask, and how do you choose them?
Mathematics, Year 7 (Portugal): Tell the whole group you want to know about from the part you actually study, and plan a draw that makes the sample represent it.
The whole group and the part you study
The population is the whole set of elements your question is about, and you must say exactly which: “the 240 Year 7 students of the school”. The sample is the part of it from which you really collect data, for example 40 of those 240. If you collect data from the whole population, that is a census; if only from a sample, you then talk about the population with care. The population depends on the question, not on who is easy to ask.
When a sample is needed, and why it is uncertain
You use a sample when studying everyone is impossible (testing how long a battery lasts destroys it; a factory cannot open every box) or too costly (a whole country). But a sample brings uncertainty: of 40 students drawn, 14 use the bus (14/40 = 35%); another draw of 40 could give 11 (11/40 = 27.5%). Neither is the exact value for the 240. The conclusion is about the population, but it is less certain than a census.
Planning a draw of 40 from 240
The 240 students are in 8 classes of 30. Step 1: list everyone and number them from 1 to 240. Step 2: the sample has 40 students, and 240 ÷ 8 = 30 per class, so to keep every class present take 40 ÷ 8 = 5 from each. Step 3: in each class, draw 5 numbers at random (paper slips or a random-number tool). Check: 8 × 5 = 40, and 5/30 = 1/6 = 40/240, the same fraction of every class.
A bigger sample is not always a better one
To find out how many of the 240 like staying indoors at break, Sílvia asks the 60 students she finds in the corridor; Antónia draws 40 numbers from the list. Sílvia's sample is bigger (60/240 = 25%, against 40/240 = 1/6), but those students are in the corridor because they like being indoors, so the sample leans one way: it is biased. Antónia's smaller draw gives everyone the same chance, so it represents the 240 better.
Polls, quality control and censuses
Opinion polls ask a sample instead of every citizen, while a census tries to count everyone. Factories test a few items from each batch, because testing all would destroy them or take too long. Before believing a number, ask: who is the population, how was the sample chosen, and how many answered? A poll answered only by those who chose to reply describes only those who replied.
Test what you learned
A school has 240 Year 7 students. To find out how they travel to school, a teacher asks 40 of them, drawn at random. What is the population?
- a) The 40 students who are asked
- b) The number 240
- c) All the students in the school, of every year
- d) The 240 Year 7 students
Correct answer: d) The 240 Year 7 students — Right: the population is the whole group the question is about. The 40 drawn are a sample of it.
The 240 students are in 8 classes of 30. A sample of 40 should include the same number of students from each class. How many per class?
- a) 5
- b) 8
- c) 6
- d) 30
Correct answer: a) 5 — Right: 40 ÷ 8 = 5 per class, and 5 out of 30 is 1/6, the same fraction as 40 out of 240.
To find how many hours a week the 240 Year 7 students read, Bruno asks the 60 students he finds in the library at lunch. Carla draws 40 numbers from the list of 240. Whose sample is better?
- a) Bruno's, because 60 is more than 40
- b) Both are equally good
- c) Carla's, because every student had the same chance of being chosen
- d) Neither: only asking all 240 counts
Correct answer: c) Carla's, because every student had the same chance of being chosen — Right: a random draw from the full list represents the 240 better than a bigger group chosen by where students happen to be.
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