Reading surveys: 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 60 population cards, 12 report cards, a mini-check and every answer with its working.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Print the two cut-out sheets, one copy per pair: sheet 1 has the 60 population cards (Day 2), sheet 2 the report cards (Day 3). Cut them before Day 2 and keep each set in an envelope.
- Each student needs a calculator.
- 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 survey asks a sample, not the whole population, so its mean or median is an estimate of the population’s. How good the estimate is depends on how the data was obtained: how many were asked, who they were, how they were chosen, and whether the method could favour some answers. A large random sample gives a trustworthy estimate; a small or self-selected one may not. When a few values are extreme, the median describes a typical value better than the mean.
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
Means are given to two decimal places at most. A median of an even number of values is the mean of the middle two. Results from the class’s own samples vary; the answers give the population values to compare them with.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Students learn and do | On screen |
|---|---|---|---|
| 1 | Mean and median | Find the mean and median of small data sets, even and odd, and build a data set to match a given mean and median | Mean: balance the values; Median: the middle of sorted data; Even number: average two middles |
| 2 | Samples and populations | Draw random samples of 10 from 60 student cards, estimate the population mean and median, and compare the class’s estimates with the truth | A sample stands for a population |
| 3 | How was the data obtained? | Question survey reports with a checklist and sort 12 reports into trustworthy, biased and not enough information | How was the data obtained? |
| 4 | Mean or median? | See how a few extreme values pull a mean in house prices and pay, and estimate a population total from a sample mean | An outlier pulls the mean |
| 5 | Writing a report | Write a survey report that says how the data was obtained, improve a weak one, explain why two reports disagree, and play the quiz | Self-check quiz as a class game |
How the week builds
Day 1 finds the mean and median; Day 2 uses samples to estimate them for a population; Day 3 judges how a report’s data was obtained; Day 4 decides between mean and median when values are extreme; and Day 5 writes and critiques reports. It builds on Year 8 work on sampling and on the mean and median, and it prepares comparing distributions and planning investigations later in Year 9.
Materials for the week (one trip)
- From the classroom: calculators, scissors, envelopes, this pack printed.
- Cut-out sheet 1, one per pair: 60 population cards, each a student’s trip to school in minutes (Day 2).
- Cut-out sheet 2, one per pair: 12 report cards (Day 3).
- If you have them: news articles or web pages that report a survey, for Day 3 and Day 5. No other equipment to buy.
Words for the word wall
population, sample, survey, census, random sample, sample size, bias, self-selected, estimate, mean, median, outlier, skewed, report, representative
Dear families
This week in maths, Year 9 is reading survey reports and asking how the data was obtained before trusting the numbers: how many people were asked, who they were and how they were chosen.
Try this at home
- Find a news story that quotes a survey. Does it say how many people were asked and how they were chosen?
- Ask: an online poll on a fishing website says most people love fishing. Why might that not be true of everyone?
- Ask: house prices are often reported as a median, not a mean. Why? (A few very expensive houses would pull the mean up.)
My survey week
Fill one row a day. Write a survey claim you saw or heard, where it came from, and one question you would ask about it.
| Day | The claim | Where from | My question |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Mean and median
The mean shares the total out equally: add the values and divide by how many there are. It is the balance point of the data. The median is the middle value once the data is in order; with an even number of values it is the mean of the middle two. The two often differ, and the gap between them tells us something about the data.
We are learning to
- find the mean and the median of a data set,
- find the median of an even number of values,
- build a data set to match a given mean and median.
Success criteria
- I can find the mean and median of any small data set, sorting it first.
- I can explain why the mean and the median of a data set can differ.
You need
The worksheet; a calculator each; a place on the board for the class data (Part C).
Lesson flow (about 60 minutes)
| 12 min | Balance and middle Show the board (see the box). For 3, 7, 7, 2, 9: the sum is 28, so the mean is 28 ÷ 5 = 5.6; in order, 2, 3, 7, 7, 9, the median is 7. Ask: “Why must the data be sorted before finding the median, but not before finding the mean?” |
| 25 min | Four data sets Pairs do Parts A and B. Ask: “In Part B, which number did you choose first, and why?” |
| 23 min | Our class Part C: each student gives one number, the class finds the mean and median; early finishers do Part D. |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Taking the middle of the unsorted list as the median.
- For an even number of values, taking one of the middle two instead of their mean.
- Dividing by the wrong count, such as the largest value.
Answers
- Part A: goals in six netball quarters: mean 7, median 7; minutes on the bus, seven days: mean 24, median 22; hours of sleep, five nights: mean 7.6, median 8; points in eight basketball games: mean 59.25, median 60.
- Part B: many answers; for example 2, 4, 5, 9, 10: the middle is 5 and the total 30 = 5 × 6.
- Part C: results vary.
- Part D: the mean rises by 30 ÷ 7 ≈ 4.29 to about 28.29; the median stays 22, because the largest value only moves further from the middle.
Mean and median
A. Four data sets
Find the mean (to two decimal places at most) and the median of each set.
| Data | Values | Mean | Median |
|---|---|---|---|
| Goals in six netball quarters | 8, 5, 11, 6, 9, 3 | ||
| Minutes on the bus, seven days | 22, 18, 25, 19, 40, 21, 23 | ||
| Hours of sleep, five nights | 8, 7, 9, 8, 6 | ||
| Points in eight basketball games | 52, 61, 48, 70, 55, 66, 59, 63 |
B. Build a data set
Write 5 whole numbers with a mean of 6 and a median of 5. Show that yours work.
C. Our class
Each student gives the number of minutes their trip to school took today. Record the class data, then find the mean and median.
| How many values | Mean | Median |
D. Stretch: change one value
On the bus, the slowest trip took 40 minutes. Suppose it had taken 70 minutes instead. What happens to the mean? To the median? Explain.
Samples and populations
A population is every member of the group we want to know about; a sample is the part we actually measure. The sample’s mean and median are estimates of the population’s. Different random samples give different estimates, but they gather around the true values, and larger samples gather more tightly. Today the class knows the whole population, so it can see how good its estimates are.
We are learning to
- draw a random sample from a population,
- use a sample’s mean and median to estimate the population’s,
- see how estimates from different samples vary.
Success criteria
- I can draw a random sample of 10 cards and find its mean and median.
- I can explain why the class’s estimates are spread out, and where they gather.
You need
The worksheet; the 60 population cards (cut-out sheet 1) in an envelope per pair; calculators; a number line on the board for the class’s sample means.
Lesson flow (about 60 minutes)
| 12 min | One sample Show the board (see the box). Then a demo sample of 10 cards, 01, 05, 15, 25, 29, 32, 35, 42, 46, 59: 31, 4, 31, 32, 27, 34, 8, 8, 15, 5 minutes, mean 19.5 and median 21. Ask: “Is the demo sample wrong because its median is not the population’s?” |
| 28 min | Sample and plot Pairs shake the envelope, draw 10 cards without looking, and do Part A; then three more samples. Every pair adds its sample means to the class number line (Part B). Ask: “Where do the class’s sample means bunch up?” |
| 20 min | The truth Reveal the population mean and median; pairs do Part C; early finishers do Part D. |
Two shorter sessions instead? Split after the first sample; start the second with the other three.
Watch for these ideas
- Choosing cards by looking, which is not random.
- Expecting a sample to give the population’s values exactly.
- Thinking a sample is wrong because it differs from another pair’s.
Answers
- Part A: results vary.
- Part B: the class’s sample means gather around the population mean, with some well above and some well below.
- Part C: the population mean is 19.8 minutes and the median 16 minutes; most sample estimates fall within a few minutes of them.
- Part D: samples of 20 vary less: their means bunch more tightly around 19.8, so one sample of 20 is a better estimate than one of 10.
Samples and populations
A. Four samples of 10
Draw 10 cards without looking. Record the values, the mean and the median. Put the cards back and shake before the next sample.
| Sample | The 10 values | Mean | Median |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 |
B. The class’s estimates
Copy the class’s sample means onto the dot plot (minutes).
C. The truth
| The population mean and median (from your teacher) | |
| How far was your closest sample mean from the population mean? | |
| Where do the class’s sample means gather? |
D. Stretch: bigger samples
Predict what the class dot plot would look like if every sample had 20 cards instead of 10. Try two samples of 20 to test it.
How was the data obtained?
A survey’s mean or median is only as good as the way its data was obtained. Four questions test a report: how big was the sample, who was surveyed, how were they chosen, and could the method favour some answers? A report that answers all four well gives a trustworthy estimate; one with a lopsided sample is biased; one that answers none cannot be judged at all.
We are learning to
- question how a survey report’s data was obtained,
- recognise samples that are too small, self-selected or drawn from the wrong group,
- judge whether a report’s estimate can be trusted.
Success criteria
- I can ask the four questions of any survey report.
- I can sort the twelve reports and give a reason for each.
You need
The worksheet; the 12 report cards (cut-out sheet 2) per pair; any real survey reports you have collected.
Lesson flow (about 60 minutes)
| 12 min | Four questions Show the board (see the box). Ask: “The biased report answers every question. Why is it still not trustworthy?” |
| 28 min | Sort the reports Pairs sort the 12 cards into trustworthy, biased, not enough information (Part A; the key is with the mini-check answers), writing one reason for each. Ask: “Which card was hardest to place? What would you need to know to be sure?” |
| 20 min | Fix a report Parts B and C; early finishers do Part D. |
Two shorter sessions instead? Split after the sort; start the second with Part B.
Watch for these ideas
- Trusting a report because its sample is large, when the people were not chosen at random.
- Calling every online poll trustworthy because many people answered.
- Treating “studies show” as information about how the data was obtained.
Answers
- Part A: the card key is with the mini-check answers.
- Part B: for example: “A random sample of 300 Year 9 students from five schools, chosen from the class rolls, were asked how many hours of sport they played last week: median 4 hours.” Any answer that states the size, the population, a random method and the question is correct.
- Part C: the poll is self-selected (people who care choose to answer), drawn from one website’s visitors, and gives no sample size.
- Part D: a census asks every member of the population; it is used when the population is small or the answer must be exact, and a sample when a census would cost too much or take too long.
How was the data obtained?
A. Sort the report cards
Ask the four questions of each report. Write each letter and a short reason in its pile.
| trustworthy | biased | not enough information |
|---|---|---|
| 1 | How big was the sample? |
| 2 | Who was surveyed: the population the claim is about? |
| 3 | How were they chosen: at random, or did they choose themselves? |
| 4 | Could the method favour some answers? |
B. Write a trustworthy report
Rewrite this report so that it answers all four questions: “Teenagers play about 4 hours of sport a week.”
C. A poll on the news
A news website ran a poll: “Should the school day start later? Click yes or no.” It reported that 78% of people want a later start. What is wrong with how the data was obtained?
D. Stretch: sample or census?
A census asks every member of the population. When is a census worth it, and when is a sample better?
Mean or median?
A few extreme values pull the mean towards them but barely move the median. When data is skewed, like house prices or pay, the median describes a typical value better, which is why reports of house prices use the median. A sample mean can also estimate a population total: multiply it by the size of the population.
We are learning to
- see how extreme values affect the mean and the median,
- choose the measure that describes a typical value, and say why,
- use a sample mean to estimate a population total.
Success criteria
- I can explain why house prices are reported as a median.
- I can estimate a total for a whole population from a sample mean.
You need
The worksheet; a calculator each.
Lesson flow (about 60 minutes)
| 10 min | Pulled away Show the board (see the box). Ask: “The outlier is one value of five. Why does it move the mean so far?” |
| 30 min | Houses and pay Pairs do Parts A and B. Ask: “A report says the cafe’s staff earn $1170 a week on average. Is that honest?” |
| 20 min | From a sample to a total Part C; early finishers do Part D. |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Thinking the mean is always the best “average”.
- Deleting an outlier just because it is large: some are real and matter.
- Forgetting to sort before finding the median of a long list.
Answers
- Part A: in order, 520, 545, 560, 575, 590, 610, 620, 635, 650, 680, 700, 720, 760, 1450, 2300; the median is $635 thousand and the mean $794.33 thousand. Only 2 sales are above the mean, so the median describes a typical sale.
- Part B: the mean is $1170 and the median $820; without the owner’s $4000 the mean is $816.25. The median is fairer: the mean is above every staff member’s pay.
- Part C: about 4.5 × 1200 = 5400 hours a week; the estimate is fair because the sample was random, but another sample would give a slightly different number.
- Part D: the mean, because a total is the mean times the count; the median would not give the total.
Mean or median?
A. House sales
A real estate report lists 15 sales in a suburb last month, in thousands of dollars: 610, 545, 720, 590, 2300, 635, 520, 680, 575, 650, 1450, 560, 700, 620, 760.
| The median price | |
| The mean price | |
| Which describes a typical sale, and why? |
B. Pay at a cafe
The weekly pay of everyone at a cafe, in dollars: 780, 820, 760, 900, 850, 800, 740, 880, 4000. The largest is the owner’s. Find the mean and the median. Which would you report as typical pay, and why?
C. From a sample to a total
A school has 1200 students. A random sample of 50 students did a mean of 4.5 hours of homework last week. Estimate the total hours of homework done by all 1200 students. How sure can you be?
D. Stretch: which measure for a total?
To estimate a total, should you use the sample mean or the sample median? Explain.
Writing a report
A good survey report lets a reader check it: it names the question and the population, says how many were sampled and how they were chosen, gives the estimates, and says how far to trust them. Two reports on the same question can disagree for good reasons, and the way their data was obtained usually explains which to believe.
We are learning to
- write a report that says how the data was obtained,
- improve a weak report,
- explain why two reports on one question disagree, and which to trust.
Success criteria
- I can write a report on my Day 2 sample that a reader could check.
- I can say which of two reports to trust, and why.
You need
The worksheet; each pair’s Day 2 worksheet; the unit’s self-check quiz.
Lesson flow (about 60 minutes)
| 10 min | What makes it checkable? Read one strong and one weak real report aloud, or the two in Part C. Ask: “What would you need to know to check each one yourself?” |
| 35 min | Write and judge Pairs write Part A from their Day 2 samples, then do Parts B and C; early finishers do Part D. Ask: “Your report and your neighbour’s give different estimates. Can both be good reports?” |
| 15 min | Self-check quiz as a class game Run the unit’s self-check quiz on the board. Teams vote on each answer, then the reason is read aloud. |
Two shorter sessions instead? Split after Part A; start the second with Part B and the quiz.
Watch for these ideas
- Reports that give a number but not how the data was obtained.
- Claiming an estimate is exact.
- Believing the report with the more striking number.
Answers
- Part A: reports vary; a strong one says: the population is the 60 students on the cards, a random sample of 10 was drawn from an envelope, the estimates are its mean and median, and another sample would differ (the population’s were 19.8 and 16 minutes).
- Part B: add the sample size, who was asked, how they were chosen and when; give the median as well as the mean if the data is skewed.
- Part C: trust Report B: 400 students chosen at random from the roll, against 20 friends of the writer, a small sample that may not be like the rest of the school.
- Part D: a larger random sample, or several samples whose estimates agree; a report from a source that has nothing to gain from the answer.
Writing a report
A. Report on your Day 2 sample
Fill the frame from your first Day 2 sample.
| The question | |
| The population | |
| The sample: how many, and how they were chosen | |
| The estimates: mean and median | |
| How far to trust them, and why |
B. Improve a weak report
“Our survey found the average Year 9 student spends 3 hours a day on a phone.” List what is missing, then rewrite it so a reader could check it.
C. Two reports disagree
Report A: “I asked 20 of my friends how long their trip to school takes: median 25 minutes.” Report B: “We asked 400 students chosen at random from the school roll: median 15 minutes.” Which do you trust, and why might they disagree?
D. Stretch: more convincing
What would make a survey’s estimate more convincing still? Give two ways.
Sixty population cards
Day 2, one set per pair, in an envelope. The whole population: 60 Year 9 students at one school, and each one’s trip to school this morning, in minutes.
Survey report cards
Day 3, one set per pair: 12 cards. Ask the four questions of each report, then sort: trustworthy, biased, or not enough information.
Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.
What we know: Reading surveys
Work on your own. Show your working.
- Find the mean of 4, 8, 6, 10, 7. ______________________
- Find the median of 3, 9, 4, 12, 8, 6. ______________________
- A random sample of 40 of a school’s 600 students has a mean height of 162 cm. Estimate the mean height of all 600 students. ______________________
- A fishing club’s website polled its visitors: “80% say fishing is the best hobby.” What is wrong with how the data was obtained? ______________________
- Which gives a better estimate of a population median: a random sample of 25 or of 250? Why? ______________________
- A report on incomes includes a few very large ones. Should it give the mean or the median as a typical income? ______________________
- A report says only “a survey of 12 people”. Which of the four questions does it answer, and which does it leave open? ______________________
- Find the mean and median of 2, 3, 3, 4, 48. Which describes the centre better? ______________________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers with working
- 35 ÷ 5 = 7.
- In order 3, 4, 6, 8, 9, 12: (6 + 8) ÷ 2 = 7.
- About 162 cm: a random sample’s mean estimates the population mean.
- Its visitors chose themselves and already like fishing, so the sample is biased; it gives no sample size.
- A random sample of 250: larger random samples vary less, so its median is likely closer.
- The median: the few very large incomes pull the mean up.
- It answers how big the sample was; it leaves open who was surveyed, how they were chosen, and whether the method could favour some answers.
- Mean 60 ÷ 5 = 12, median 3; the median, because one value (48) pulls the mean.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Mean and median | Takes the middle of unsorted data, or divides by the wrong count | Finds the mean and median of any small data set | Builds data sets to a given mean and median |
| Samples and estimates | Expects a sample to give the population value exactly | Uses sample statistics to estimate population ones and expects them to vary | Explains why larger random samples vary less |
| How data was obtained | Trusts a number without asking how it was found | Asks the four questions and spots biased and thin reports | Rewrites a weak report so it can be checked |
| Choosing a measure | Uses the mean whatever the data | Chooses the median for skewed data and the mean for totals | Explains the choice with the data’s shape |
Eight questions, four ideas. A student at standard finds the measures, asks how the data was obtained, and chooses a measure.
Card key
Report cards (Day 3): trustworthy: A (random, from the whole roll, a fair size); D (large and random across the country); F (random households and most replied); K (random matches from the whole season); L (random sales across the whole year). biased: C (self-selected visitors of a gaming site); E (library users read more than most students); H (only bus users at peak hour); J (gym members exercise more than most adults). not enough information: B (no size, no population, no method); G (which studies, who, how many?); I (no size, no method, no source).
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Mean, median | Samples | Reports | Which measure | Writing |
|---|---|---|---|---|---|
The five columns are the five days: mean and median, samples, reports, which measure, and writing a report.