Reading statistics in the media: a week of ready-to-teach maths

Five lessons for Year 10 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 twelve report cards to sort, twelve bias cards to match, 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. Cut the 12 report cards before Day 2 and the 12 bias cards before Day 3, and keep each set in an envelope.
4. Bring a few real news items with a statistic in them for Day 5, if you like; every task also works from the sheets.
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 report makes claims and draws conclusions from data. A careful reader asks how the data was collected, who was in the sample, what was measured, how it is shown, and whether the conclusion follows. Bias can enter at every step: a sample that is easy rather than fair, a question that leads, a graph with a cut axis, an average that hides a skew, a risk quoted only as a ratio, or a correlation sold as a cause. Collecting and reporting data also carries ethical duties: consent, privacy and honesty.

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

Every report, headline and data set in this pack is invented for teaching; none is a real study or outlet. A claim is what a report says is true; a statistic is a number from the data; an inference is a conclusion drawn beyond the data.

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 separates what a report says from what its data shows; Days 2 to 4 look at three places bias hides, in the display, the sample and the summary; and Day 5 turns to cause, to ethics and to writing a critique. It builds on Year 9, where students judged samples and compared data sets, and it prepares the statistical reasoning of senior mathematics and of everyday citizenship.

Materials for the week (one trip)

• From the classroom: rulers, calculators, scissors, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 bias cards (Day 3).
• Cut-out sheet 2, one per pair: 12 report cards (Day 2).
• A few news items with a statistic in them, if you have time to collect them (Day 5). No other equipment to buy.

Words for the word wall

claim, statistic, inference, conclusion, sample, population, bias, representative, random sample, leading question, truncated axis, outlier, skewed, relative risk, absolute risk, percentage point, correlation, causation, lurking variable, consent, privacy

Dear families

This week in maths, Year 10 is learning to read statistics in the news with care: how the numbers were collected, how they are shown, and whether the conclusions follow, including the ethics of collecting data about people.

Try this at home

• Ask: when you see a number in the news, who was asked, how many, and how were they chosen?
• Ask: a headline says something “doubles your risk”. What would you want to know first? (The risk before and after.)
• Ask: is the average income in a street a good guide to what a typical family earns? (Not if a few earn far more.)
• Ask: two things rise together. Does one cause the other? (Look for a third thing behind both.)

My statistics in the news

Fill one row a day. Write a statistic you saw or heard, where it came from, and one question you would ask about it.

Claims and evidence

A report mixes claims, statistics and inferences. Students take an invented report apart sentence by sentence, write the questions that would test each part, and decide which conclusions its data can support.

We are learning to

• tell a claim, a statistic and an inference apart,
• write questions that test a claim against its evidence,
• judge whether a conclusion follows from the data given.

Success criteria

• I can label each sentence of a report as a claim, a statistic or an inference.
• I can write three questions that would test the report’s main claim.

You need

The worksheet; highlighters in two colours if you have them.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Taking a statistic as proof of a cause.
• Accepting volunteers who answered an online poll as all teenagers.
• Checking only the numbers: the wording and the conclusion need checking too.

Answers

• Part A: 1 claim, 2 statistic, 3 inference, 4 statistic, 5 inference.
• Part B: for example, who answered and how were they chosen; how was sleep measured; is there earlier data to compare; do the phone users in this poll sleep less than the others.
• Part C: 1 supported, for these readers only; 2 not supported: no comparison of users and non-users, and a link would not show a cause; 3 not supported: there is no earlier data.
• Part D: a random sample across many schools, sleep diaries or trackers over two weeks, phone use recorded for each student, a comparison with earlier years, and consent from students and families.

Claims and evidence

A. Take the report apart

Label each sentence: claim, statistic or inference. The first is done for you.

B. Questions to ask

C. Does the conclusion follow?

D. Stretch: a fairer study

Graphs that mislead

The same numbers can be drawn honestly or to mislead. Students redraw a cafe’s sales chart with an axis from zero, name other graph tricks, and sort twelve reports into fair, misleading and can’t tell yet.

We are learning to

• spot a cut axis, a stretched picture or a chosen time window,
• redraw a misleading graph honestly,
• decide whether a report is fair, misleading, or cannot be judged yet.

Success criteria

• I can redraw the sales chart from zero and say how its message changes.
• I can explain why “can’t tell yet” is sometimes the right answer.

You need

Rulers; the report cards (cut-out sheet 2), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Judging a graph without reading its axis numbers.
• Bar heights compared on a cut axis: the bars no longer show the amounts.
• Calling a report misleading when it only lacks information: that is “can’t tell yet”.

Answers

• Part A: April’s bar looks 2.5 times as tall as January’s (25 against 10 above 480), but sales grew only about 3%. From 0 the bars look almost equal.
• Part B: for example, a picture graph that grows in height and width (the area shows four times, not two); a chosen time window; uneven gaps on an axis; a 3D pie that swells the front slice.
• Part C: the card key is with the mini-check answers.
• Part D: yes, when small changes matter and the axis is clearly marked with a break, such as body temperature over a day; never for comparing bar heights.

Graphs that mislead

A. A cafe’s sales

An invented cafe advertises its “soaring” coffee sales (cups a month) with the chart on the left.

B. Other tricks

C. Sort the cards: fair, misleading or can’t tell yet?

D. Stretch: never from zero?

Samples and bias

A sample can only speak for the population it fairly represents. Students judge three ways of choosing 50 students from a school of 800, rewrite a leading question, and match six surveys to the bias each risks.

We are learning to

• judge whether a sampling method gives a representative sample,
• recognise leading questions and other sources of bias,
• name the bias a survey risks and suggest a fix.

Success criteria

• I can explain why 50 names drawn at random from the roll are fairer than the first 50 in the canteen queue.
• I can rewrite a leading question so that it is neutral.

You need

The bias cards (cut-out sheet 1), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Thinking a big sample cures bias: thousands of volunteers can still be biased.
• Calling a sample of friends random.
• Missing the wording: a fair sample can still be asked a leading question.

Answers

• Part A: 1 biased: only canteen users, a convenience sample; 2 fair: every student has the same chance; 3 biased: volunteers with strong views reply.
• Part B: a neutral version, such as “How do you rate the canteen’s prices: too high, about right, or too low?”; other wording biases include loaded words, double questions and missing options.
• Part C: the card key is with the mini-check answers.
• Part D: the 500 chosen at random: the 10 000 online replies come from people who chose to answer, so size does not remove the bias.

Samples and bias

A. Three ways to choose

A school of 800 students wants to know what students think of the canteen. Is each way of choosing 50 students fair or biased? The first is done for you.

B. The wording

C. Match the biases

Match each survey card to the bias card it risks. Write the letters.

D. Stretch: bigger or fairer?

Averages, percents and risk

True numbers can still mislead when the summary is chosen to impress. Students compare the mean and median pay of a small business, turn a “doubled risk” into actual numbers, and tell a percent change from a change in percentage points.

We are learning to

• choose the average that describes skewed data fairly,
• turn a relative risk into an absolute risk,
• tell a percent change from a change in percentage points.

Success criteria

• I can say whether the mean or the median gives a fairer picture of the pay, and why.
• I can explain what “doubles the risk” means in actual numbers.

You need

Calculators; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Quoting the mean of skewed data as typical.
• Hearing “doubles” as large: double a tiny risk is still tiny.
• Mixing percent and percentage points.

Answers

• Part A: mean $11 940 ÷ 9 ≈ $1326.67; median $910 (the middle of 9). Only one worker earns more than the mean, so the median is the fairer picture.
• Part B: from 1 to 2 in 50 000: doubled, but only 1 more case in every 50 000. The headline is true but alarming without the actual numbers.
• Part C: 4% to 5% is up 1 percentage point, and up 25% of the old rate. Both are true; “up 25%” sounds larger.
• Part D: without $4 600, the mean of the other 8 is $917.50, close to the median: one value was dragging the mean up.

Averages, percents and risk

A. The average worker

The weekly pay of the 9 staff of an invented small business, the owner included:

$820, $860, $880, $900, $910, $950, $980, $1 040, $4 600

B. “Doubles your risk!”

C. Percent or percentage points?

D. Stretch: one value

Cause, ethics and a critique

A correlation alone never proves a cause, and collecting data about people carries duties. Students find the lurking variable behind three invented correlations, weigh the ethics of a survey plan, write a short critique of a report, and play the self-check quiz.

We are learning to

• name a lurking variable that could explain a correlation,
• identify ethical issues in collecting and reporting data,
• write a balanced critique of a statistical report.

Success criteria

• I can explain why towns with more fire trucks have more fire damage without trucks causing damage.
• I can name two ethical duties of anyone running a survey.

You need

The worksheet; any news items with statistics you collected; the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; start the next with Part C.

Watch for these ideas

• Treating a strong correlation as proof of a cause.
• Thinking ethics only means not lying: consent, privacy and harm matter too.
• A critique that only praises or only attacks: weigh the evidence.

Answers

• Part A: 1 town size; 2 age; 3 hot, sunny weather.
• Part B: for example, names make answers identifiable (collect none); students may feel pressured (make it voluntary, with consent from students and families); publishing by class could expose small groups (report only school-wide totals); stress is sensitive (give support contacts).
• Part C: a critique covers the sample, the measure, the display, whether the conclusion follows, and any ethical issue, with evidence for each.
• Part D: an experiment: volunteers are split at random into two groups, one with phones out of the bedroom and one as usual, for a few weeks, and their sleep is compared.

Cause, ethics and a critique

A. Find the lurking variable

In each pair the two things really do rise together, but neither causes the other. Name a third thing that could drive both. The first is done for you.

B. Is this survey ethical?

C. Write a critique

D. Stretch: test the cause

Surveys and the biases they risk

For Day 3, one set per pair. Six cards describe an invented survey; six name a bias. Match each survey to the bias it risks most.

Teacher note: the key is with the mini-check answers.

Fair, misleading, or can’t tell yet?

For Day 2, one set per pair. Each card describes an invented report or graph. Sort the cards into three piles: fair, misleading, and can’t tell yet (something important is missing).

Teacher note: the key is with the mini-check answers.

What we know: Reading statistics in the media

Work on your own. Every report here is invented.

1. A bar chart shows 96, 97, 99, 98 with its axis starting at 95. What is misleading, and how would you fix it? ______________________
2. An online poll on a sports website asks whether sport should get more funding. Name a bias it risks. ______________________
3. Five salaries, in thousands of dollars, are 40, 42, 45, 48, 250. Find the mean and the median. Which is fairer to quote, and why? ______________________
4. A risk rises from 1 in 20 000 to 2 in 20 000. Describe the change as a relative risk and as an absolute risk. ______________________
5. A rate rises from 10% to 12%. By how many percentage points, and by what percent? ______________________
6. A strong correlation between two variables shows that, circle one:   one causes the other  ·  they move together, but the cause is unknown  ·  the sample was random
7. Name two ethical problems with publishing survey answers with students’ names attached. ______________________
8. Write three questions you would ask about the claim “Nine out of ten users love our app.” ______________________

Answers and marking guide

Answers with working

1. The cut axis makes differences of a few units look large; start the axis at 0, or mark a clear break and do not compare bar heights.
2. Voluntary response (only those who choose to vote), from an audience of sports fans: not representative.
3. Mean 85, median 45; the median, because one large salary pulls the mean above every other value.
4. Relative: the risk doubles. Absolute: 1 more in every 20 000.
5. Up 2 percentage points; up 20% of the old rate.
6. The second: they move together, but the cause is unknown.
7. For example: no consent to be named, loss of privacy, possible embarrassment or harm, and answers people would not give honestly.
8. For example: how many users were asked and how were they chosen; what exactly were they asked; who ran the survey and who paid for it.

A quick three-level guide

Eight questions, four ideas. A student at standard asks where the numbers came from before deciding what they show.

Card keys

Bias cards (Day 3): G with A; D with J; E with L; B with H; K with F; I with C. Report cards (Day 2): fair: N, Q, S, W; misleading: M, P, R, U, X; can’t tell yet: O, T, V.

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: claims and evidence, graphs, samples and bias, averages and risk, and cause and ethics.