Sampling methods and misleading graphs: 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 a page of 100 rectangles to sample, 12 method 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 is the page of rectangles (Day 1, not cut), sheet 2 the method cards (Day 3). Cut the cards before Day 3.
4. Each student needs a ruler and a calculator; Days 4 and 5 use grid paper.
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

How a sample is chosen shapes what a survey finds. A random sample gives every member the same chance; a stratified sample keeps the population’s groups in proportion; a systematic sample takes every k-th member. A convenience or self-selected sample favours some members and can push the results one way. Once data is collected, the way it is drawn can push a reader too: an axis that does not start at zero, a picture that grows in two directions, or a carefully chosen time window can make the same numbers tell different stories.

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. Results from the class’s own samples vary; the answers give the population values to compare them with. A graph is honest when its axis starts at zero for bars, its scale is even, and its window shows the whole story.

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

Day 1 sees why random sampling matters; Day 2 meets stratified, systematic and convenience samples; Day 3 names the method behind a survey and its likely bias; Day 4 turns to how data is shown and finds three graph tricks; and Day 5 makes one data set support two views. It builds on reading survey reports earlier in Year 9, and it prepares choosing displays and planning investigations later in the year.

Materials for the week (one trip)

• From the classroom: rulers, calculators, grid paper, scissors, envelopes, this pack printed.
• Cut-out sheet 1, one per pair: the page of 100 rectangles (Day 1). Keep it whole.
• Cut-out sheet 2, one per pair: 12 method cards (Day 3).
• If you have them: graphs cut from news sites or advertisements, for Days 4 and 5. No other equipment to buy.

Words for the word wall

population, sample, random, judgement sample, stratified, systematic, convenience, self-selected, bias, representative, axis, truncated, scale, pictogram, window, point of view

Dear families

This week in maths, Year 9 is looking at how the way a survey picks its people can change its results, and how the way a graph is drawn can push a reader towards a point of view.

Try this at home

• Look for a graph in an advertisement or the news. Does its axis start at zero? Would the story change if it did?
• Ask: a radio station asks listeners to phone in their opinion. Who is likely to call? Will the result speak for everyone?
• Ask: a picture of a money bag is drawn twice as tall and twice as wide to show savings doubling. How much bigger does it look? (Four times.)

My graph week

Fill one row a day. Write a graph or survey you saw, where it was, and one thing that could mislead.

Random or judged?

Choosing a sample “by eye”, picking members that look typical, feels sensible but is not: the eye is drawn to what stands out. A random sample gives every member the same chance, so over many samples its estimates gather around the truth. Today the class samples a page of 100 rectangles both ways and finds which way can be trusted.

We are learning to

• choose a random sample with random digits,
• compare estimates from judged samples and random samples,
• explain how a sampling method can push results one way.

Success criteria

• I can choose 5 rectangles at random and find their mean area.
• I can explain why the judged samples tend to give a bigger mean.

You need

The worksheet; the page of rectangles (cut-out sheet 1) per pair; a number line on the board for the class’s estimates.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the judged samples; start the second with the random ones.

Watch for these ideas

• Skipping a random number because its rectangle “looks wrong”: that brings judgement back in.
• Counting the grid squares carelessly on the large rectangles.
• Thinking one random sample must be close to the truth: it is the method that is fair, not every result.

Answers

• Parts A and B: results vary.
• Part C: the true mean area is 5.69 squares. The judged means usually sit above it, because the large rectangles catch the eye; the random means scatter on both sides of it.
• Part D: no. Skip it and read on, so the sample has 5 different rectangles, each still chosen with the same chance.

Random or judged?

A. A sample by eye

Look at the page of rectangles for 10 seconds. Pick 5 that look typical. Count each area in squares.

B. A random sample

Start at your row of the digit block and read pairs of digits: 00 to 99 are the rectangle numbers. Skip a pair you have already used.

C. Which method can we trust?

The true mean area is revealed by your teacher. Where do the class’s judged means sit? The random means? Why?

D. Stretch: a repeat

Your digits give the same rectangle number twice. Should you count it twice? Explain.

Other ways to sample

A stratified sample splits the population into groups and samples each in proportion, so the sample keeps the population’s mix. A systematic sample takes every k-th member of a list. A convenience sample takes whoever is easiest to reach, and if those people differ from the rest, the result is pushed one way. Today the population is a school of 600 whose juniors and seniors answer very differently.

We are learning to

• work out a stratified sample in proportion to the groups,
• set up a systematic sample from a list,
• predict how a convenience sample would push a survey’s result.

Success criteria

• I can find how many juniors and seniors belong in a stratified sample of any size.
• I can explain why surveying only the senior study centre would mislead.

You need

The worksheet; 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

• Taking equal numbers from each group instead of proportional numbers.
• Thinking a bigger convenience sample fixes its bias.
• Starting a systematic sample at the top of the list every time, instead of at a random start.

Answers

• Part A: 50: 30 juniors and 20 seniors; 80: 48 juniors and 32 seniors; 120: 72 juniors and 48 seniors (juniors are 60% of the school).
• Part B: 600 ÷ 50 = 12: choose a random start from 1 to 12, then every 12th student on the roll.
• Part C: the whole school’s mean is (360 × 0.8 + 240 × 6.5) ÷ 600 = 3.08 hours; the study centre would report about 6.5 hours, more than twice as much, because it misses every junior.
• Part D: a stratified sample gives about 3.08 hours; a sample of juniors only gives about 0.8. The sample’s mix decides the answer.

Other ways to sample

A school has 360 juniors (Years 7 to 9) and 240 seniors (Years 10 to 12). A survey asks how many hours of paid work each student does a week. Juniors average 0.8 hours; seniors average 6.5 hours.

A. Stratified samples

B. A systematic sample

Describe how to choose a systematic sample of 50 students from the school roll of 600.

C. The study centre

A student surveys the people in the senior study centre at lunch. What mean would that survey report? What is the mean for the whole school? Explain the difference.

D. Stretch: the mix decides

Find the mean a stratified sample would give, and the mean a sample of juniors only would give. What does this show?

Name the method

Every survey report rests on a sampling method, even when the report does not name it. Naming the method is the first step to judging the result: random, systematic and stratified samples give every member a fair chance, while convenience and self-selected samples let some people in more easily than others, and the result leans towards them.

We are learning to

• name the sampling method in a survey description,
• predict which way a biased method would push a result,
• suggest a fairer method for the same question.

Success criteria

• I can sort the twelve method cards and give a reason for each.
• I can say which way a convenience sample would push a result, and why.

You need

The worksheet; the 12 method cards (cut-out sheet 2) 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

• Calling any sample with a rule “systematic”, even the first 30 through a gate.
• Thinking a self-selected sample is fair because anyone could answer.
• Mixing up stratified (in proportion to groups) with sampling equal numbers from each group.

Answers

• Part A: the card key is with the mini-check answers.
• Part B: the canteen line over-represents students who buy lunch; the website poll over-represents those who care most about the topic; friends in one class share a year level and often opinions.
• Part C: for example, a stratified sample by year level, chosen at random within each year from the roll.
• Part D: the gate rule takes whoever arrives first, so early arrivers, such as bus travellers, fill the sample; every 10th customer all day spreads across the whole day.

Name the method

A. Sort the method cards

Write each card’s letter in its pile, with a few words of reason.

B. Which way would it lean?

Three of the biased cards were used for a survey about school lunches: the canteen line, the website poll and friends in one class. For each, say whose views it would over-represent.

C. A fairer method

Design a fair way to survey the whole school about lunches. Name the method and how the students are chosen.

D. Stretch: two rules

“The first 30 through the gate” and “every 10th customer through the door all day” both follow a rule. Explain why only one is fair.

Graphs that mislead

The same numbers can be drawn to look dramatic or dull. Three tricks are common: a bar axis that starts above zero, so small differences look huge; a picture scaled in two directions, so doubling a value quadruples its area; and a time window chosen to show only the part of the story the author wants. Each is fixed by redrawing honestly.

We are learning to

• spot a truncated axis, a pictogram scaled two ways and a chosen window,
• work out how much a trick exaggerates,
• redraw a misleading graph honestly.

Success criteria

• I can explain how much bigger shop B looks than it really is.
• I can redraw a graph so that it shows the whole story fairly.

You need

The worksheet; grid paper and a ruler each.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Reading bar heights instead of the axis numbers.
• Thinking a broken or truncated axis is always dishonest: it can be fine for a line graph if it is clearly marked.
• Judging a trend from a few chosen months.

Answers

• Part A: the visible bars are 2 and 6 tall, so B looks 3 times A; really 56 ÷ 52 ≈ 1.08. Redrawn from 0, the bars are nearly equal.
• Part B: the value doubled (1 to 2), but twice the height and twice the width is 4 times the area. Draw two bags of the same size, or a bar twice as tall.
• Part C: months 10 to 12 rise from 39 to 43, but over the year the price peaked at 53 in month 6 and fell; the window hides the fall.
• Part D: squashing the scale (0 to 1000 for values near 40 to 53) flattens real changes, so a big rise or fall looks like nothing.

Graphs that mislead

A. The truncated axis

B. The pictogram

C. The chosen window

An advertisement shows only the left graph with the headline “Rising fast!”. What does the right graph show that the left one hides?

D. Stretch: the squashed scale

A graph of the share price uses a vertical axis from 0 to 1000. What would it hide?

One data set, two stories

The person who draws a graph chooses its axis, its scale and its window, and those choices can support a point of view. Drawing one data set twice, once for each side, shows how much the choices matter; a reader’s defence is a short checklist and the actual numbers.

We are learning to

• draw one data set to support two opposite headlines,
• judge which drawing is fair, and why,
• check any graph with a short list of questions.

Success criteria

• I can draw the scores to look like a big rise and like no change at all.
• I can write a fair headline that uses the numbers.

You need

The worksheet; grid paper and a ruler each; the unit’s self-check quiz.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Changing the data to make the story, instead of the drawing.
• Calling any graph with an axis above zero a lie: say what it hides.
• Headlines without numbers.

Answers

• Part A: “soaring”: a line graph with the axis from 70 to 75; “flat”: bars or a line with the axis from 0 to 100. Both use the same six scores.
• Part B: the fair graph marks its axis clearly, and the fair headline gives the numbers: “Mean score up 3 points in 5 years, from 71 to 74.”
• Part C: answers vary; every graph should be checked against all five questions.
• Part D: a rise of 3 points in 5 years is about 0.6 a year: real but small.

One data set, two stories

A. Two headlines

A school’s mean test score each year:

On grid paper, draw one graph for “Results soar!” and one for “Six years, no change”. Note the axis you chose for each.

B. The fair version

Which of your graphs is fairer, or what would a fair one look like? Write a fair headline that uses the numbers.

C. Check a real graph

Choose a graph from the news or an advertisement (or one from Day 4). Answer each question.

D. Stretch: how big is the change?

How many points a year did the score rise on average? Is that a big change for a test out of 100?

One hundred rectangles

Day 1, one copy per pair; do not cut. The population: 100 rectangles, numbered 00 to 99. Each area is the number of grid squares inside it.

Sampling method cards

Day 3, one set per pair: 12 cards. Each describes how a survey chose its sample. Sort them: random, systematic or stratified, or convenience or self-selected.

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

What we know: Sampling methods and misleading graphs

Work on your own. Show your working.

1. A sample is chosen by drawing names from a hat that holds every name. What kind of sample is it? ______________________
2. Every 20th name on a roll is chosen. What kind of sample is it? ______________________
3. A school is 70% junior and 30% senior. How many of each belong in a stratified sample of 60? ______________________
4. A poll on a website lets anyone who wants to answer. What is the problem with this sample? ______________________
5. A bar chart compares 64 and 68 with its axis starting at 60. How many times taller does the second bar look, and how many times bigger is it? ______________________
6. A picture is drawn twice as tall and twice as wide to show a value doubling. How many times bigger does it look? ______________________
7. Only Year 12 students are asked about the new uniform for the whole school. What kind of sample is this, and is it biased? ______________________
8. A company’s sales grew from 120 to 126. How could its graph make the growth look large? ______________________

Answers and marking guide

Answers with working

1. A random sample: every name has the same chance.
2. A systematic sample.
3. 42 juniors and 18 seniors.
4. It is self-selected: people with strong views answer, so it is biased.
5. Visible bars 4 and 8: it looks 2 times taller; really 1.06 times.
6. Four times: twice the height times twice the width.
7. A convenience sample of one group; biased, because the other year levels are not asked.
8. Start the vertical axis just below 120, so the 6-unit rise fills the graph.

A quick three-level guide

Eight questions, four ideas. A student at standard names sampling methods, spots bias and finds the trick in a graph.

Card key

Method cards (Day 3): random: B, G, I; systematic or stratified: C, E, H, J; convenience or self-selected: A, D, F, K, L.

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: random sampling, other methods, naming bias, misleading graphs, and two stories.