Planning a statistical investigation: a week of ready-to-teach maths

Five lessons for Year 9 Statistics. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus 12 question cards, 12 claim cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair: sheet 1 has the question cards (Day 1), sheet 2 the claim cards (Day 5). Cut them before Day 1.
4. Each student needs a calculator and a ruler; on Day 3 each pair needs a stopwatch or a phone timer.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

A statistical investigation runs a cycle: pose a question that data can answer, collect the data fairly, analyse it with a centre, a spread and a display, and conclude and report. A good report says how strong its evidence is: a difference is convincing when it is large compared with the spread, the samples are big enough and fairly chosen, and the limitations are stated. Two things that rise together are associated, but only a fair experiment can show that one causes the other.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these answers

Quartiles are the medians of the lower and upper halves, leaving the middle value out when there is an odd number of values, as the unit page does. Five-number summaries are listed in order: minimum, lower quartile, median, upper quartile, maximum. Answers to the planning and writing tasks are examples: any plan or report that meets the same test is right.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

The week follows the investigation cycle once: Day 1 poses the question, Day 2 plans the collection, Day 3 collects and analyses, Day 4 reports and weighs the evidence, and Day 5 tests claims made by others. It brings together the whole Year 9 strand: surveys and samples, five-number summaries and box plots, and choosing displays.

Materials for the week (one trip)

• From the classroom: calculators, rulers, a stopwatch or a phone timer per pair (Day 3), scissors, envelopes, this pack printed.
• Cut-out sheet 1, one per pair: 12 question cards to sort (Day 1).
• Cut-out sheet 2, one per pair: 12 claim cards to sort (Day 5).
• If you have them: a spreadsheet that draws box plots, to check the Day 3 summaries. No other equipment to buy.

Words for the word wall

statistical question, population, variable, primary data, secondary data, survey, leading question, random sample, bias, five-number summary, interquartile range, parallel box plots, conclusion, limitation, strength of evidence, association, cause, third variable, experiment

Dear families

This week in maths, Year 9 is running a statistical investigation from start to finish: asking a question, collecting fair data, analysing it and judging how strong the evidence is.

Try this at home

• Try the 30-second test together: close your eyes and say “now” when you think 30 seconds have passed. Who is closest?
• Look for a claim in an advertisement or the news that says one thing causes another. What evidence does it give?
• Ask: what question about our home could we answer by collecting data for a week?

My investigation week

Fill one row a day. Write a question you could investigate, how you would collect the data, and whether it would be primary or secondary.

Asking a statistical question

Every investigation runs a cycle: pose a question, collect data, analyse it, then conclude and report, which often raises a new question. A statistical question expects the answers to vary and needs data from many people or things; a question with one fixed answer does not. A good question names its population and its variable, so that it can actually be answered.

We are learning to

• tell a statistical question from one with a single fixed answer,
• sharpen a vague question by naming the population and the variable,
• plan how a question will be answered with data.

Success criteria

• I can sort the twelve question cards and explain what the vague ones are missing.
• I can plan an investigation: population, variable, type of data, collection, sample and display.

You need

The worksheet; the 12 question cards (cut-out sheet 1) per pair.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the sort; start the second with Part B.

Watch for these ideas

• Thinking any question with a number in its answer is statistical.
• Questions that do not say who is asked (the population) or what is measured (the variable).
• Choosing the graph before the question is clear.

Answers

• Part A: the card key is with the mini-check answers.
• Part B: for example, B: “What mark out of 10 do Year 9 students give the canteen’s lunches?” G: “How many hours a day do Year 9 students spend on screens?” I: “Do Year 9 students sleep less now than Year 9 students did in a published survey from 2010?”
• Part C: for example, A: How long do Year 9 students sleep on a school night?; Year 9 students at our school; hours of sleep on a school night: numerical, continuous; a survey we run (primary data); 30 names drawn at random from the Year 9 list; a histogram or a box plot.
• Part D: for example, “Do students who play a sport sleep longer than students who do not?”: two groups, compared on hours of sleep.

Asking a statistical question

A. Sort the question cards

Write each card’s letter in its pile.

B. Sharpen the vague questions

Rewrite each card from your “too vague” pile as a statistical question that names its population and its variable.

C. Plan an investigation

Choose one card from your statistical pile and plan how to answer it.

D. Stretch: a question about two groups

Write a statistical question that compares two groups at our school. Name the two groups and the variable you would compare.

Collecting data

Data you collect yourself, by a survey, an experiment or a measurement, is primary data: you control what is asked. Data someone else collected is secondary: quick to get, but gathered for another purpose. A survey question must be fair: not leading, one question at a time, clear, with answer boxes that hold every answer exactly once. A fair sample gives every member of the population the same chance of being chosen.

We are learning to

• sort sources of data into primary and secondary,
• find and fix unfair survey questions,
• choose a fair sample, and explain why the others are not.

Success criteria

• I can say what is wrong with a survey question and rewrite it fairly.
• I can explain why a random sample beats the people who are easiest to ask.

You need

The worksheet; a calculator for Part D.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Thinking secondary data is always worse: it can be larger and carefully checked.
• Writing questions that suggest the answer the writer hopes for.
• Choosing the people who are easiest to ask and calling it a sample.

Answers

• Part A: primary 1, 4, 5; secondary 2, 3, 6.
• Part B: 1. leading: it pushes towards yes; for example, How do you rate the canteen’s prices? (too low, about right, too high) 2. two questions in one; for example, Ask about maths and science separately. 3. vague: “a lot” means different things; for example, How many hours did you exercise last week? 4. the boxes overlap at 7 and 8, and leave out under 6 and over 9; for example, Use boxes that do not overlap and cover every answer: under 6, 6 to under 7, 7 to under 8, 8 to under 9, 9 or more.
• Part C: C, because every student has the same chance of being chosen. A: one class may not be like the rest of the year; B: early arrivers may sleep differently from the rest; D: people choose themselves, and keen ones answer most.
• Part D: take the bus: 90 ÷ 180 × 30 = 15; walk: 54 ÷ 180 × 30 = 9; are driven: 36 ÷ 180 × 30 = 6.

Collecting data

A. Primary or secondary?

B. Fix the survey questions

C. Choose a sample

Question: how long do Year 9 students at our school sleep on a school night? There are 180 Year 9 students. Which sample is fairest? Why are the other three not?

D. Stretch: a sample in proportion

Of the 180 students, 90 take the bus, 54 walk, 36 are driven. Choose a sample of 30 with each group in the same proportion. How many from each group?

Collect and analyse

Today the class runs a quick experiment and analyses the results. The question: do students estimate 30 seconds better when they count in their heads? With eyes closed, a student says “now” when 30 seconds seem to have passed, while a partner times them. Analysis summarises each group with a centre and a spread, and parallel box plots on one scale show both at once.

We are learning to

• collect data in a planned way that is the same for everyone,
• summarise two groups with five-number summaries and box plots,
• answer a question by comparing centres and spreads.

Success criteria

• I can find the five-number summary and the IQR of each group.
• I can answer the question with evidence from both the centre and the spread.

You need

The worksheet; a stopwatch or a phone timer per pair; a calculator each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Comparing the medians only: here the medians are close and the spreads are not.
• Changing how the test is run part of the way through.
• Rounding some times to whole seconds and others to tenths.

Answers

• Part A: the class’s own results.
• Part B: counting: 27.8, 29.25, 30, 30.7, 32.1, IQR 1.45; not counting: 22.9, 25.65, 28.8, 34.25, 38.2, IQR 8.6 (seconds).
• Part C: both medians are close to 30 (30 and 28.8), but counting gives far more consistent estimates: an IQR of 1.45 against 8.6. Counting helps: the estimates bunch close to 30 seconds.
• Part D: the median distance from 30 seconds is 0.75 counting and 4.35 without.

Collect and analyse

A. Our test

Same rules for everyone: eyes closed, the partner says “go” and starts the timer, you say “now” at 30 seconds. Times to 0.1 of a second.

B. Another class’s results

Counting: 28.4, 29.6, 30.2, 31.0, 29.1, 30.5, 27.8, 30.9, 29.8, 32.1, 30.3, 29.4.
Not counting: 24.6, 33.8, 27.2, 36.5, 22.9, 31.4, 26.0, 38.2, 29.5, 25.3, 34.7, 28.1.

Draw the two box plots on one scale.

C. Answer the question

Do students estimate 30 seconds better when they count? Use the centres and the spreads as your evidence.

D. Stretch: how far from 30?

Find how far each estimate is from 30 seconds, then the median distance for each group. What does it add to Part C?

How strong is the evidence?

A conclusion is only as strong as its evidence. Evidence that one group scores higher is strong when the difference is large compared with the spread inside the groups and the samples are big and fairly chosen; it is weak when the groups overlap a lot or hold only a few. A full report gives the question, method, results, conclusion and limitations, so that a reader can judge the evidence.

We are learning to

• judge the strength of evidence from the gap, the spread and the sample size,
• write a full report, limitations included,
• suggest how an investigation could give stronger evidence.

Success criteria

• I can say whether evidence is strong or weak and give the reason.
• I can write all five parts of a report.

You need

The worksheet; the Day 3 worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the second picture.

Watch for these ideas

• Calling evidence strong from the gap alone, without the spread or the sample size.
• Leaving out the limitations, or listing ones that do not matter.
• Claiming the conclusion for everyone, everywhere.

Answers

• Part A: School 1 strong: a gap of 5.5 between medians 17 and 11.5, more than either IQR (2.5, 3); 12 in each group. School 2 weak: a gap of 1 between medians 13.5 and 12.5, against IQRs of 6: the groups overlap almost completely. School 3 weak: a gap of 6 between medians 17 and 11, but only 3 in each group: one or two other students could change it.
• Part B: for example, question: is a 30-second estimate better when counting? Method: 12 counted, 12 did not, one timed try each. Results: medians 30 and 28.8, IQRs 1.45 and 8.6. Conclusion: counting makes estimates more consistent. Limitations: one class, 12 per group, one try each.
• Part C: for example, test students chosen at random from several schools; let each try both ways in a random order.
• Part D: for example, a gap of 1 mark against IQRs of 6 is too small to show that readers score higher.

How strong is the evidence?

A. Three schools

Each school asked: do students who read for fun score higher on a vocabulary quiz (out of 20)? Is each school’s evidence strong or weak? Why?

B. Report Day 3

C. Make it stronger

Give two changes to the Day 3 method that would make its evidence stronger.

D. Stretch: answer a headline

A newsletter uses School 2’s results under the headline “Readers score higher!”. Write a reply about the strength of the evidence.

Association or cause?

Two variables can rise and fall together without one causing the other: a hidden third variable can drive both. Only an experiment that puts people into groups at random can show a cause. Judging a claim means asking who was measured, how many, how they were chosen, and whether an association is being reported as a cause.

We are learning to

• judge a claim by its sample and by how the data was collected,
• find a hidden third variable behind an association,
• explain why a random experiment can show a cause and a survey cannot.

Success criteria

• I can sort the claim cards and say what each weak claim is missing.
• I can name a third variable that could explain an association.

You need

The worksheet; the 12 claim cards (cut-out sheet 2) per pair; the unit’s self-check quiz.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the sort; start the second with Part B and the quiz.

Watch for these ideas

• Thinking a strong association proves a cause.
• Thinking a big sample fixes a biased one: a phone-in poll can have thousands of callers.
• Thinking any claim with a number in it is well supported.

Answers

• Part A: the card key is with the mini-check answers.
• Part B: for example, C: richer countries buy more chocolate and spend more on research; H: families who read a lot both buy books and read more together; J: bigger towns have more of both.
• Part C: for example, card A: ask a random sample of 50 students from every year level, not five friends.
• Part D: for example, split 40 students at random into two groups of 20; both study the same words for the same time, one with music; both sit the same quiz; compare the medians and IQRs.

Association or cause?

A. Sort the claim cards

Write each card’s letter in its pile.

B. Find the hidden cause

For each card in your “association” pile, name a third variable that could drive both things.

C. Fix a weak claim

Choose one card from your “weak sample” pile. Rewrite how its data should be collected so that its conclusion would be stronger.

D. Stretch: design an experiment

Design an experiment to test whether listening to music while studying changes quiz scores. How do you make it fair?

Question cards

Day 1, one set per pair: 12 cards. Sort them: a statistical question; one fixed answer; too vague to investigate.

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

Claim cards

Day 5, one set per pair: 12 cards. Each reports a conclusion and where it came from. Sort them: well supported; weak sample: too small or biased; association reported as a cause.

Teacher note: the key, with the hidden cause behind each association, is with the mini-check answers, so this sheet can go to pairs whole.

What we know: Planning a statistical investigation

Work on your own. Show your working.

1. Which is a statistical question: “How many students are in Room 12?” or “How long do Year 9 students take to get to school?” Why? ______________________
2. You weigh your own school bag every day for a week. Is this primary or secondary data? ______________________
3. What is wrong with the survey question “You agree that homework is useful, don’t you?” ______________________
4. To find how far Year 9 students live from school, which sample is fairer: 30 names drawn at random from the Year 9 list, or the 30 students on the first bus? Why? ______________________
5. Find the median and the IQR of 5, 7, 8, 8, 10, 12. ______________________
6. Two groups’ medians differ by 1 second, and both IQRs are about 9 seconds. Is this strong evidence of a difference? Why? ______________________
7. A survey of 8 students from one class makes a claim about all Year 9 students in Australia. Give two reasons the evidence is weak. ______________________
8. Students who sleep more get higher marks. Does this show that more sleep raises marks? Explain. ______________________

Answers and marking guide

Answers with working

1. The second: the answers vary from student to student, so data from many students is needed.
2. Primary: you collected it yourself.
3. It is leading: it pushes towards “yes”.
4. The random names: every student has the same chance; the students on one bus all live along one route.
5. Median 8; Q1 = 7, Q3 = 10, so the IQR is 3.
6. No: a gap of 1 second is small against a spread of about 9 seconds; the groups overlap a lot.
7. The sample is very small, and one class at one school may not be like all Year 9 students.
8. No: it is an association. A third variable, such as a regular routine, could drive both; only an experiment could show a cause.

A quick three-level guide

Eight questions, four ideas. A student at standard plans a fair investigation, compares groups by centre and spread, and judges the evidence.

Card keys

Question cards (Day 1): a statistical question: A, D, F, K, L; one fixed answer: C, E, H, J; too vague to investigate: B, G, I.

Claim cards (Day 5): well supported: B, E, I, K; weak sample: too small or biased: A, D, F, G, L; association reported as a cause: C, H, J. Hidden causes: C, richer countries buy more chocolate and spend more on research; H, families who read a lot both buy books and read more together; J, bigger towns have more of both.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: asking questions, collecting, analysing, the strength of evidence, and association or cause.