Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Unit 2, lesson 3) What is 10% of 240?
Answer
24. 240 ÷ 10 = 24. - (Unit 1, lesson 3) List all the factors of 12.
Answer
1, 2, 3, 4, 6 and 12. - (Lesson 1 of this unit) Is the number of pets a student has discrete or continuous?
Answer
Discrete. Pets are counted, so there is no value between 2 pets and 3 pets.
Words used in this lesson
- primary data
- data you collect yourself, first-hand. For example: a class survey.
- secondary data
- data someone else collected. For example: a table in a report.
- survey
- a set of questions asked of a group of people. For example: how students travel to school.
- sample
- a smaller group chosen to stand for a whole group. For example: every 20th name on the register.
- leading question
- a question that pushes the answer one way. For example: “Don't you agree it is too noisy?”.
- biased
- leaning one way because of who was asked or how. For example: only asking at the bus stop.
- random sample
- every member has the same chance of being chosen. For example: names drawn from a hat.
- gap and overlap
- a value that fits no box, or a value that fits two. For example: 2 hours fits “0 to 2” and “2 to 4”.
Before the chart, the numbers
Rahul wants to know how students travel to school. He asks six friends at the bus stop and says, “Most people take the bus.” Predict first: would asking the first six students at the bus stop give the right share for the whole school? Keep that in mind while we look at what makes data trustworthy.
Data you collect yourself, by a survey, an experiment or by counting, is primary data. Data someone else collected, such as a table in a report or a website, is secondary data. Both can be good or poor. What matters is whether the questions and the people asked were fair.
Boxes that cover everyone once
Often a survey asks for a number and offers boxes to tick. Each box is a range. Choose a set, then choose a number of hours and see which boxes fit it.
Set A: 4 hours fits 2 boxes, “2 to 4” and “4 to 6”, so a student cannot give one clear answer.
Show me the steps
- Set A, 4 hours: fits “2 to 4” and “4 to 6”. Two boxes. The hours 2, 4 and 6 all do this.
- Set B, 7 hours: fits no box. The hours 6, 7, 8 and 9 have nowhere to go.
- Set C, 0 hours: fits no box, so a student who does no sport cannot answer.
- Set D, any whole number of hours: exactly one box every time. For hours to the nearest hour this set has no gaps and no overlaps.
A good question is clear, asks one thing and does not push for an answer. “Don't you agree the library is far too noisy?” is a leading question. “How noisy is the library? Not noisy, a little noisy, very noisy” is fair.
Answer boxes should cover everyone exactly once: no gaps, so nobody has nowhere to go, and no overlaps, so nobody has two answers.
If the answer is measured, such as 3.5 hours, whole-number boxes leave gaps. Lesson 3 shows the fix: 0 to under 2, 2 to under 4.
Who you ask is part of the data
Back to the bus stop. The figure draws 30 students, each walking, by bus, by car or cycling. Choose who is asked and compare their share with the true share for all 30.
Every third student: 4 of 10 take the bus, 40%. All 30 students: 13 of 30, about 43%.
Show me the steps
- First 6 at the bus stop, looking at Bus: 5 of 6 is about 83%, but all 30 give 13 of 30, about 43%.
- First 6 at the bus stop, looking at Walk: 0 of 6 = 0%, but all 30 give 10 of 30, about 33%.
- Every third student, looking at Bus: 4 of 10 = 40%, close to the true share for all 30, about 43%.
- All 30 students, looking at Bus: 13 of 30, about 43%, which is the whole group.
A sample is a smaller group chosen to stand for the whole group. It is fair when it is picked in a way that does not favour one place or kind of person, such as a draw from a hat or every 3rd name from a list in no special order. A sample that leans one way is biased: taken all from one place, such as the bus stop, it can give a share that is far from the truth. A random sample gives every member the same chance of being chosen.
To use a sample, find its percentage and take that percentage of the whole group.
A common mistake: “I asked real people and wrote down what they said, so the data must be right.” Primary data is first-hand, not automatically accurate. A leading question, boxes with gaps, or asking one corner of the group can all push the answers one way.
Worked example: from a sample to the whole group
A school has 350 students. A survey asks every 5th name on the register, 70 students, and 21 of them cycle to school. Estimate how many of all 350 cycle.
- Share in the sample: 21 ÷ 70 = 0.3, which is 30%.
- Take the same share of everyone: 30% of 350 = 0.3 × 350 = 105.
- Check: 350 ÷ 5 = 70, so each person asked stands for 5 students, and 21 × 5 = 105. About 105 students.
Your go. The first step is given. A chess club has 240 members. A survey asks every 6th name, 40 members, and 10 of them prefer chess to draughts. Estimate how many of all 240 members prefer chess.
- Share in the sample: 10 ÷ 40 = 0.25, which is 25%.
- Now take that share of 240 yourself, then check.
Check the answer
25% of 240 = 60. Check: each person asked stands for 240 ÷ 40 = 6 members, and 10 × 6 = 60. About 60 members.
Check before you go on
1. Which set of options for “How many pets do you have?” covers every student exactly once?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Ines measures the heights of 25 students in her class herself. She then compares them with a national table of heights for the same age group. Which of these is secondary data?
3. A school has 480 students. A survey asks every 8th student on the register. How many students are asked?
Level 2 · Building
4. Which question is the fairest way to ask about school lunches?
5. A form offers three boxes for hours of sleep: “under 6”, “6 to 7” and “8 to 9”. Whole hours from 0 to 12 are possible. How many whole numbers of hours fit no box?
6. A form asks: “What is your favourite sport? Tick one: football, netball, swimming.” What is the main problem?
Level 3 · Stretch
7. A survey asks every 12th student on a register of 420 students, so 35 are asked. Of those, 14 walk to school. Estimate how many of all 420 students walk.
8. Zara says, “We asked our own class, so our data is primary, which means it must be correct.” Which reply is best?
Something to think about
A school has 600 students and the survey team can only ask 30. Is a bigger sample always better, or can a sample of 30 chosen well beat a sample of 100 chosen badly? What would you check first?
One way to think about it
Size helps, but how the sample is picked matters more. 30 students spread through the whole register can give a fair picture, while 100 students all picked from the football team would be wrong however many there are. Check first who could have been asked.
Exit ticket
9. A school has 600 students. Which is the fairest way to choose 30 students to ask about a new timetable?