Lesson 4 ~40 min Data Analysis C · Path +85 XP

Reporting with Data Visualisations

A visualisation is an argument. Which display you choose, and where you start the axis, decides what a reader concludes before they have read a single word of your analysis.

Today's hook: Two bar charts, the same two numbers, $66$ and $78$. On one the second bar is four times the height of the first. On the other it is barely taller. Neither chart contains a false number. Only one of them tells the truth.

0/5QUESTS

Think First

warm-up

You have collected the favourite sport of 40 students and the height of the same 40 students. Sketch, roughly, the display you would use for each. Then say what goes wrong if you swap them: what would a histogram of favourite sport look like, and what would a pie chart of heights tell you?

Record your answer in your workbook.

The Big Idea

+5 XP to read

The display must fit the variable, and the report must fit the question. Choosing a display is a statistical decision: it determines what comparison the reader can make easily, what they can make with effort, and what they cannot make at all.

Categorical → column or pie. Numerical → histogram, dot plot or box plot. Two numerical → scatter plot.

The second half is the axis. A bar chart compares lengths, so its value axis must start at zero or the lengths lie about the ratio. A line graph compares slopes rather than lengths, so a truncated axis there is often legitimate. The rule is not "always start at zero"; it is "start at zero whenever the reader will compare sizes".

Match the variable

A histogram needs a continuous variable; a pie chart needs parts of one whole.

Bars start at zero

Bars are compared by length, so truncating the axis distorts the ratio.

Label everything

Axis names, units, sample size and source. An unlabelled chart cannot be checked.

What You'll Master

objectives

Know

Which display suits a categorical, discrete, continuous or paired-numerical variable
The elements every chart must carry: title, axis labels with units, scale, sample size and source
Why a bar chart's value axis must begin at zero

Understand

Why choosing a display is a statistical decision rather than a decorative one
Why a truncated axis is deceptive on a bar chart but often reasonable on a line graph

Can Do

Choose and justify an appropriate display for a given variable and question
Identify and repair a misleading chart
Structure a report with several visualisations that together answer the aim

Words You Need

vocabulary

HistogramA display of a continuous variable, grouped into intervals, with no gaps between the columns.

Column graphA display of a categorical or discrete variable, with gaps between the columns.

Scatter plotA display of two numerical variables measured on the same subjects, used to show a relationship.

Box plotA display of a numerical variable's spread through its five-number summary, useful for comparing groups.

Truncated axisAn axis that does not begin at zero, which exaggerates differences on a bar chart.

InsightA statement the display makes visible, which is the reason for including it.

Matching the Display to the Variable

+5 XP to read

The variable type from Lesson 1 decides the display, and getting it wrong produces a chart that cannot be read.

Variable	Display	Shows
Categorical	column graph, or pie if parts of one whole	which categories are common
Numerical discrete	column graph or dot plot	the distribution of counts
Numerical continuous	histogram or box plot	shape, centre and spread
Two numerical	scatter plot	whether they move together
One numerical, split by a category	parallel box plots	how groups compare

Two rules follow from the table rather than being extra. A histogram has no gaps, because the intervals are adjacent parts of a continuous scale; a column graph has gaps, because the categories are separate. And a pie chart is only valid when the categories are mutually exclusive parts of one whole: "sports played" fails that test the moment one student plays two.

The Axis Decides the Story

+5 XP to read

The chart above shows two values, $66$ and $78$, drawn twice. On the left the axis begins at $60$ and the second bar looks about four times the first. On the right it begins at $0$ and the bars are close in height. Neither chart contains a false number.

The difference matters because a bar chart is read by comparing lengths. If the axis starts at $60$, the drawn lengths represent $6$ and $18$, a ratio of $1:3$, while the real values are in the ratio $66:78$, about $1:1.18$. The chart makes a claim the data does not support.

A line graph is different. It is read by comparing slopes, and a truncated axis does not distort a slope, only its visual steepness. So starting a temperature line graph at $15°$ rather than $0°$ is often the clearer choice.

The rule stated properly

Start the axis at zero whenever the reader will compare sizes. If a truncated axis is genuinely needed, say so on the chart.

What Every Chart Must Carry

+5 XP to read

A chart a reader cannot check is a decoration. Five elements make it checkable.

A title that states the finding, not the topic. "Year 10 sleep and mathematics marks" names the topic; "Students sleeping over eight hours scored higher on average" states what the chart shows, and the reader can then verify it.

Axis labels with units. "Sleep" is not a label; "Mean nightly sleep (hours)" is.

A visible scale, including where the axis starts.

The sample size. "n = 40" changes how much weight the reader gives the chart.

The source and date. Your own collection counts as a source, and saying when it was collected lets a reader judge whether it is still current.

The first of these is the one students most often skip, and it is the one that turns a chart from something to look at into something that makes a claim.

Building a Report from Several Displays

+5 XP to read

A report is not a gallery. Each visualisation should answer a different part of the aim, and the text should say what each one shows and what it does not.

For the sleep-and-marks inquiry from Lesson 1, three displays do different jobs:

A histogram of sleep hours establishes what the sample looks like: where most students sit, whether the distribution is symmetric, whether there are unusual values. It answers "who did we measure?"

A scatter plot of sleep against mark addresses the aim directly: do the two move together, and how strongly?

Parallel box plots splitting students into two sleep groups makes the comparison in the hypothesis concrete, showing not just the difference in centre but whether the two groups overlap heavily.

Three displays, three jobs. Adding a pie chart of favourite subject would add a picture and no insight, and a reader is entitled to ask what it is doing there.

Common Pitfalls

+5 XP to read

Truncating the vertical axis on a bar chart to make a difference look bigger.

Fix: bars are compared by length, so the axis must start at zero. If you genuinely need a truncated axis, label it clearly and explain why.

Using a pie chart for categories that are not parts of one whole, so the slices sum to more than 100%.

Fix: if a subject can be in two categories at once, use a column graph of counts instead.

Including a display because the report looks better with more pictures.

Fix: for each chart, write the one sentence it makes visible. If you cannot write that sentence, remove the chart.

Watch Me Solve It · 3 examples

Watch Me Solve It · Choosing displays for an aim

+15 XP per step

PROBLEM

An inquiry compares the time taken to travel to school for students who walk and students who take a bus. Choose two displays and justify each.

1

Classify the variables

Travel time is numerical continuous; method of travel is categorical with two values. The aim compares one numerical variable across two groups.
2

Choose the primary display

Parallel box plots. They show both centres and both spreads on one scale, so the reader sees not only which group is faster but whether the groups overlap, which a pair of means would hide.
3

Choose a supporting display

A histogram of all travel times, to establish what the sample looks like before the comparison: whether times are clustered, and whether any unusual values are driving the result.
4

Say what is NOT chosen and why

A pie chart would be wrong twice over: travel time is not categorical, and the two groups are not parts of one whole being divided.

AnswerParallel box plots for the comparison, with a histogram to describe the sample

Watch Me Solve It · Repairing a misleading chart

+15 XP per step

PROBLEM

A chart headed "Massive rise in canteen prices" shows two bars, last year and this year, with a vertical axis running from $\$4.00$ to $\$4.60$. The bars are $\$4.20$ and $\$4.50$. Evaluate and repair it.

1

Compute the real change

$\frac{4.50 - 4.20}{4.20} \times 100 \approx 7.1\%$

A rise of about seven per cent, which is real but not massive.
2

Compute the drawn change

$\frac{4.50 - 4.00}{4.20 - 4.00} = \frac{0.50}{0.20} = 2.5$

On an axis starting at $\$4.00$ the second bar is two and a half times the length of the first, suggesting a rise of 150%.
3

Name the fault

The axis is truncated on a display read by comparing lengths, so the drawn ratio misrepresents the real ratio. The numbers are accurate and the picture is not.
4

Repair it

Start the axis at $\$0$, retitle it with the finding rather than a judgement, for example "Mean canteen item price rose 7% from $\$4.20$ to $\$4.50$", and label the axis "Mean price per item (dollars)" with the sample size.

AnswerTruncated axis on a bar chart; redraw from zero and retitle with the actual figure

Watch Me Solve It · Turning a chart into a claim

+15 XP per step

PROBLEM

A student's report contains a scatter plot titled "Sleep and marks". Rewrite the title, and state what the chart can and cannot support.

1

Identify what the title does

It names the topic, so the reader must work out the finding themselves and cannot check whether the author read the chart correctly.
2

Rewrite it as the finding

For example: "Students who slept longer tended to score higher, n = 40". The reader can now check the claim against the plot.
3

State what it supports

That in this sample the two variables moved together, and roughly how strongly and in which direction.
4

State what it does not support

That sleep CAUSED the higher marks. A scatter plot shows association only, and a third factor, such as organised study habits, could produce both.

AnswerA title stating the finding with the sample size, plus an explicit note that association is not causation

Brain Trainer · 4 problems

Brain Trainer · Pick the display

4 problems

Four quick problems. Work each one, then reveal the answer.

1 Which display suits the heights of 60 students?

Height is numerical continuous, so it groups into intervals.A histogram
2 Which display suits favourite subject?

Categorical, with separate categories.A column graph
3 Should a bar chart's value axis start at zero?

Bars are compared by length.Yes
4 Which display shows whether two numerical variables move together?

Each subject contributes one point.A scatter plot

Complete in your workbook.

Multiple Choice · 5 questions

MC1

Histogram or column graph

+10 XP

A histogram differs from a column graph in that:

MC2

The truncated axis

+10 XP

Starting a bar chart's value axis at $60$ rather than $0$ is misleading because:

MC3

When truncation is acceptable

+10 XP

A truncated vertical axis is often reasonable on:

MC4

Pie charts

+10 XP

A pie chart is an appropriate display only when:

MC5

What a title should do

+10 XP

The best title for a chart in a report is one that:

Short Answer · 3 questions

Choose and justify

+15 XP

SHORT ANSWER

For each, name the display you would use and justify it in one sentence.
(a) The method of travel to school of 200 students.
(b) The distribution of times taken by 50 students to complete a puzzle.
(c) Whether hours of part-time work is related to hours of sleep, for 60 students.
(d) A comparison of test marks between three classes.

Write your working in your book.

Repair a chart

+15 XP

SHORT ANSWER

A chart is headed "Homework time explodes". It has two bars, Year 9 and Year 10, with a vertical axis from $6$ to $9$ hours. The values are $6.8$ and $8.1$ hours. There are no axis labels and no sample size.
(a) Calculate the actual percentage increase, and the ratio of the drawn bar lengths.
(b) Identify three faults.
(c) Describe the repaired chart in full.

Write your working in your book.

Plan a report

+15 XP

SHORT ANSWER

You have completed the inquiry into travel method and tiredness on arrival from Lesson 1.
(a) Choose three visualisations for your report and state the job each one does.
(b) For each, write the title you would give it.
(c) Explain why a fourth chart showing the favourite subject of respondents should not be included.

Write your working in your book.

Stretch Challenge · The same data, three arguments

+25 XP

CHALLENGE

A council reports that reported bicycle thefts rose from $180$ to $198$ in a year.
(a) Draw or describe a chart that makes this look like a crisis, and state exactly which choices produce that impression.
(b) Draw or describe a chart that makes it look negligible, and state which choices produce that.
(c) Describe the chart you would publish, and explain what additional information the reader needs before the figure means anything at all.

Quick Review

recap

Match

Display to variable type

Bars

Compared by length, so start at zero

Lines

Compared by slope, so truncation is often fine

Every chart

A finding as its title, and a sample size

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