4. Displaying Data

 

 

 

6

Why Display Data?

A table organizes information, but sometimes we want to see the information more:

visually.

Suppose a survey asks students to choose their favourite fruit:

The table gives us the exact numbers.

A graph makes the comparison easier to see at a:

glance.

Graphical displays allow us to identify:

• largest and smallest categories
• similarities and differences
• patterns
• clusters
• gaps
• unusual values

A well-designed graph turns numerical information into a:

visual message.

From Data to a Display

Creating a graph is not simply drawing bars or plotting points.

You first need to decide:

What kind of data do I have?

What do I want the reader to notice?

Which display communicates the information most clearly?

For this topic, we will focus on three useful displays:

Bar graphs

Pictographs

Line plots

Each represents data in a different way.

Bar Graphs

A bar graph uses rectangular bars to represent numerical values or frequencies.

The length or height of each bar represents the:

value of that category.

Bar graphs are especially useful for comparing:

separate categories.

 

 

 

6

Example: Favourite School Subjects

Suppose 30 students choose their favourite subject.

A bar graph allows us to compare these categories visually.

The tallest bar represents:

Science.

The shortest bar represents:

Music.

The difference between them is:

9 − 3 = 6 students.

Parts of a Good Bar Graph

A good bar graph should normally include:

A Clear Title

The title tells the reader:

what the graph represents.

For example:

Favourite School Subjects

Category Labels

The categories should be clearly identified.

For example:

Science, Mathematics, English, Art, Music

Numerical Axis

The other axis shows the:

frequency or measured value.

Scale

The scale tells us what each interval represents.

Units

If the data involve measurements, the units should be:

clearly stated.

Bars Should Be Separated

In a standard bar graph, the bars normally have:

gaps between them.

Why?

Because each bar represents a separate:

category.

For example:

cats, dogs, fish, birds

are distinct categories.

The gaps help show that the categories are:

separate.

Vertical and Horizontal Bar Graphs

Bar graphs can be drawn:

vertically

or:

horizontally.

Both represent the same type of information.

A horizontal bar graph can be particularly useful when category names are:

long.

For example:

• Walking to school
• Public transportation
• Family vehicle
• School bus
• Bicycle

The choice should make the graph:

easy to read.

Constructing a Bar Graph

Suppose a class records the number of students participating in different activities:

Step 1: Choose the categories

Place:

Football, Basketball, Swimming, and Tennis

along one axis.

Step 2: Determine the largest value

The largest value is:

12.

Step 3: Choose an appropriate scale

A scale from:

0 to 12

using intervals of:

2

would work well.

Step 4: Label the axes

For example:

Activity

and:

Number of Students

Step 5: Add a title

Participation in School Sports

Step 6: Draw the bars

The bar heights should be:

12, 8, 6, and 4.

Choosing a Scale

Choosing the correct scale is an important part of constructing:

any graph.

Suppose your values are:

20, 35, 45, 60, 75.

Using intervals of:

1

would create an unnecessarily large graph.

A scale increasing by:

10

would be more practical.

For example:

0, 10, 20, 30, 40, 50, 60, 70, 80

The goal is to choose a scale that is:

simple, consistent, and appropriate for the data.

Equal Intervals

Graph scales must normally use:

equal intervals.

This scale is correct:

0, 5, 10, 15, 20, 25

Each interval increases by:

5.

This scale is misleading:

0, 5, 10, 20, 25, 50

because the spacing is equal visually but the numerical intervals are:

not equal.

Worked Example 1: Reading a Bar Graph

A school records the number of books borrowed from different sections of its library.

From this information we can determine:

Most popular section: Fiction

Least popular section: Biography

Difference:

32 − 12 = 20 books

Total books borrowed:

24 + 15 + 32 + 12 + 17 = 100 books

A graph can make these comparisons much easier to:

see quickly.

Comparing Categories

One major purpose of bar graphs is to compare:

categories.

Suppose:

Team A = 18 points

Team B = 21 points

Team C = 12 points

Team D = 24 points

We can ask:

Which team scored the most?

Team D

Which team scored the least?

Team C

How many more points did Team D score than Team C?

24 − 12 = 12

Graphs allow us to combine:

visual interpretation and numerical reasoning.

Pictographs

A pictograph uses pictures or symbols to represent:

data.

 

 

 

6

For example:

Favourite Pets

But the pictures do not necessarily represent:

one item each.

That is why every pictograph needs a:

key.

The Pictograph Key

Suppose the key says:

🐾 = 2 students

Then:

🐾 🐾 🐾 🐾

represents:

4 × 2 = 8 students.

The symbol itself is not the value.

Its meaning comes from the:

key.

This is one of the most important rules when reading pictographs.

Worked Example 2: Reading a Pictograph

Suppose a pictograph shows books read by four students.

Key: 📘 = 3 books

Anna: 📘 📘 📘

Ben: 📘 📘

Carlos: 📘 📘 📘 📘

Dana: 📘

Therefore:

Anna:

3 × 3 = 9 books

Ben:

2 × 3 = 6 books

Carlos:

4 × 3 = 12 books

Dana:

1 × 3 = 3 books

Total:

9 + 6 + 12 + 3 = 30 books.

Partial Symbols

Sometimes a pictograph uses part of a symbol.

Suppose:

★ = 4 students

Then:

½ ★ = 2 students.

If a category contains:

★★½

the value is:

4 + 4 + 2 = 10 students.

Partial symbols allow pictographs to represent values that are not exact multiples of the:

full symbol.

Choosing a Good Pictograph Key

Suppose the values are:

10, 20, 30, 40.

A sensible key might be:

● = 10 items.

But if the values are:

100, 200, 300, 400,

using:

● = 1 item

would require hundreds of symbols.

Instead, we might use:

● = 100 items.

A good key makes the pictograph:

simple and readable.

Constructing a Pictograph

Suppose a shop sells:

Choose:

🍦 = 5 ice creams

Then:

Monday:

🍦 🍦 🍦 🍦

Tuesday:

🍦 🍦 🍦 🍦 🍦 🍦

Wednesday:

🍦 🍦 🍦

Thursday:

🍦 🍦 🍦 🍦 🍦

Always include the key:

🍦 = 5 ice creams.

Without the key, the pictograph is:

incomplete.

Advantages of Pictographs

Pictographs can be:

• visually appealing
• easy to understand
• useful for simple comparisons
• memorable
• effective for small datasets

They are especially useful when communicating information to a:

general audience.

Limitations of Pictographs

Pictographs become less useful when:

• there are many categories
• values are very large
• precise comparisons are required
• many partial symbols are needed
• the key is complicated

For more detailed numerical information, a:

bar graph

may be more effective.

Line Plots

A line plot displays numerical data along a:

number line.

Each observation is usually represented by a mark such as:

X

above its value.

Line plots are sometimes called:

dot plots

when dots are used instead of X marks.

 

 

 

7

Line Plots Are Not Line Graphs

These terms are easy to confuse.

A line plot displays individual data values along a:

number line.

A line graph usually shows how a quantity changes across an ordered variable, often:

time.

They are different graphical displays.

For this topic, we are focusing on:

line plots.

Example of a Line Plot

Suppose ten students record the number of books they read:

1, 2, 2, 3, 3, 3, 4, 4, 5, 3

The frequencies are:

The line plot would contain:

            X
            X
      X     X     X
X     X     X     X     X
--------------------------------
1     2     3     4     5
       Number of Books

Each X represents:

one observation.

Reading a Line Plot

From the previous plot:

The most common value is:

3 books.

The smallest value is:

1 book.

The largest value is:

5 books.

The number of students who read 4 books is:

2.

The total number of observations is found by counting all the X marks:

10.

Constructing a Line Plot

Consider:

2, 3, 4, 3, 5, 4, 3, 2, 4, 3

Step 1: Find the smallest value

2

Step 2: Find the largest value

5

Step 3: Draw an appropriate number line

2 — 3 — 4 — 5

Step 4: Add one X for each observation

The frequencies are:

2 → 2

3 → 4

4 → 3

5 → 1

Step 5: Add a clear title and label

The reader should know:

what the numbers represent.

Why Line Plots Are Useful

Unlike a bar graph showing broad categories, a line plot allows us to see the:

distribution of individual numerical values.

We can quickly identify:

• common values
• clusters
• gaps
• extremes
• range
• unusual observations

This makes line plots particularly useful for:

small numerical datasets.

Clusters

A cluster is a group of data values concentrated in a particular region.

Suppose most observations fall between:

6 and 8.

We might say:

The data cluster between 6 and 8.

Clusters help us describe where observations are:

concentrated.

Gaps

A gap is a region of the scale containing no observations.

For example, if values occur at:

2, 3, 4, 7, 8

there is a gap between:

4 and 7.

Gaps may reveal interesting features of a:

dataset.

Outliers

An outlier is a value that is unusually far from most of the other observations.

Suppose the data are:

10, 11, 11, 12, 12, 13, 12, 11, 35

The value:

35

is far from the rest of the data.

It may be an:

outlier.

An outlier is not automatically an error.

It may represent a:

real but unusual observation.

Range

The range measures the spread from the smallest to the largest value.

Range = maximum − minimum

For:

3, 4, 5, 5, 6, 8, 9

the range is:

9 − 3 = 6.

A line plot makes the minimum and maximum easy to:

identify visually.

Bar Graph vs Pictograph vs Line Plot

Choosing the correct graph depends on:

the type of data and the purpose of the display.

Which Graph Should I Choose?

Suppose you want to compare the favourite sports of students.

Use a:

bar graph or pictograph.

Suppose you want to display the heights of 20 plants.

A:

line plot

may be useful if the measurements use a manageable set of values.

Suppose you want to communicate a simple survey to young students.

A:

pictograph

might be especially effective.

The graph should fit:

the data and the audience.

Labels Matter

Consider a graph with bars of heights:

5, 10, 15, 20.

Without labels, we do not know whether the graph represents:

• kilograms
• students
• dollars
• centimetres
• animals
• test scores

Numbers without context communicate:

very little.

A good graph should answer:

What is being measured?

What do the categories represent?

What units are being used?

Titles Matter

Compare:

Graph 1

with:

Average Daily Temperature in June

The second title immediately tells the reader:

what the graph represents.

Titles should be:

brief, specific, and informative.

Scales Matter

Suppose a graph displays:

12, 14, 16, 18, 20.

A scale of:

0, 100, 200, 300

would make the bars appear almost identical and waste most of the graph.

A smaller scale would show the values more:

clearly.

The scale should match the:

range of the data.

Zero and Truncated Axes

Bar graphs often begin at:

zero.

This is important because bar length represents:

magnitude.

Suppose:

Company A = 98

Company B = 100

If the vertical axis begins at 97, the difference may look:

very large.

In reality, the numerical difference is only:

2.

Always inspect the graph's:

scale.

Misleading Graphs

A graph can contain correct numbers and still create a misleading:

visual impression.

Common problems include:

• unequal scale intervals
• missing units
• truncated axes
• exaggerated pictures
• unclear keys
• distorted shapes
• missing labels

Good graphical communication requires:

accuracy as well as appearance.

 

 

 

6

Misleading Pictographs

Suppose:

🚗 = 10 cars.

Town A has 20 cars, so it receives:

🚗 🚗

Town B has 40 cars, so it should receive:

🚗 🚗 🚗 🚗

Instead of adding symbols, someone might make Town B's car picture:

twice as tall and twice as wide.

That makes its visual area approximately:

four times as large.

The picture can therefore exaggerate the difference.

Pictographs should normally use symbols of:

consistent size.

Interpreting Graphs

When you see a graph, do not immediately look at the tallest bar or largest symbol.

First ask:

What is the title?

What do the axes represent?

What is the scale?

What are the units?

Is there a key?

Then examine the:

data.

This prevents many interpretation errors.

Comparing Categories

Suppose a bar graph shows:

We can make comparisons such as:

Birds were observed most frequently.

Beetles were observed least frequently.

There were:

18 − 9 = 9

more bird observations than beetle observations.

There were:

15 − 12 = 3

more bee observations than butterfly observations.

Difference vs Total

Graphs often require more than simply reading one value.

Suppose:

Red = 12

Blue = 8

The difference is:

12 − 8 = 4.

The combined total is:

12 + 8 = 20.

Be careful to determine whether the question asks for:

difference or total.

Worked Example 3: Science Data

Students record the number of insects found in four habitats.

A bar graph would be appropriate because the habitats are:

separate categories.

The largest number was found:

under logs.

The smallest number was found in:

soil.

Difference:

25 − 12 = 13 insects.

Graphs in Science

Graphs are essential tools in:

science.

Scientists use graphical displays to communicate:

• experimental results
• population counts
• environmental observations
• measurements
• comparisons
• distributions

A graph allows a reader to recognize patterns much faster than reading a long:

list of numbers.

Graphs in Everyday Life

Graphical displays appear in:

• news reports
• sports statistics
• weather forecasts
• business reports
• health information
• advertisements
• school reports
• transportation information
• social media

Understanding graphs is therefore an important part of:

everyday numerical literacy.

 

 

 

5

Communicating Information Visually

A successful graph should allow someone unfamiliar with the dataset to understand:

what happened.

The reader should not need the creator standing beside the graph to:

explain it.

A strong visual display is:

self-explanatory.

The CLEAR Graph Check

Before finishing a graph, check that it is:

C — Correct

Are all values plotted accurately?

L — Labelled

Are the axes and categories labelled?

E — Equal

Are scale intervals equal?

A — Appropriate

Is this graph suitable for the data?

R — Readable

Can someone understand it easily?

This provides a useful checklist for creating:

effective graphs.

Worked Example 4: Choosing the Display

Situation A

A class wants to compare how many students prefer:

pizza, noodles, burgers, or salad.

Best choices:

bar graph or pictograph.

Why?

The data consist of:

categories.

Situation B

A teacher records these quiz scores:

6, 7, 8, 8, 8, 9, 9, 10, 7, 8, 9, 6

A useful choice is:

line plot.

Why?

The data consist of repeated:

numerical values.

Situation C

A children's club wants an attractive display showing the number of animals seen during a nature walk.

A useful choice could be:

pictograph.

Why?

The symbols provide an:

easy visual comparison.

Worked Example 5: From Raw Data to Line Plot

Suppose students measure the lengths of 15 leaves to the nearest centimetre:

5, 6, 7, 6, 8, 7, 7, 9, 6, 7, 8, 5, 7, 6, 8

First organize the frequencies:

The line plot would contain:

            X
      X     X
      X     X     X
X     X     X     X
X     X     X     X     X
--------------------------------
5     6     7     8     9
        Leaf Length (cm)

The most common length is:

7 cm.

The range is:

9 − 5 = 4 cm.

Most leaves are clustered between:

6 cm and 8 cm.

From Table to Graph

A useful data workflow is:

Collect data

↓

Organize data

↓

Create a table

↓

Choose an appropriate graph

↓

Select a scale

↓

Add labels and units

↓

Plot accurately

↓

Interpret the display

↓

Communicate the pattern

The graph is therefore part of a larger process of:

working with data.

Digital Graphs

Graphs can be created using spreadsheet and data-analysis software.

Digital tools can quickly:

• create bars
• calculate scales
• add labels
• change graph types
• organize large datasets

However, the computer does not automatically know which graph is:

most appropriate.

The person analyzing the data must still make good decisions about:

representation and interpretation.

Check Your Understanding

1. What is the main purpose of a bar graph?

2. Why are there usually gaps between bars?

3. Name four features that should appear on a well-designed bar graph.

4. Why should scale intervals be equal?

5. When might a horizontal bar graph be useful?

6. What is a pictograph?

7. Why must a pictograph include a key?

8. If ★ = 5 students, what does ★★★ represent?

9. If ● = 8 items, what does half a ● represent?

10. Give one advantage and one limitation of pictographs.

11. What is a line plot?

12. Explain the difference between a line plot and a line graph.

13. What does each X usually represent on a line plot?

14. What is a cluster?

15. What is a gap?

16. What is an outlier?

17. Calculate the range of:

4, 5, 5, 7, 8, 10

18. Which display would you choose to compare favourite school subjects? Explain your choice.

19. Which display would you choose for a small dataset of repeated numerical measurements? Explain your choice.

20. Explain two ways that a graph could mislead its audience.

Key Terms

• Graphical display: Visual representation of data.
• Bar graph: Graph using separated bars to compare categories or discrete values.
• Bar: Rectangular shape whose length or height represents a numerical value.
• Category: Group used to classify data.
• Axis: Reference line used to organize values or categories on a graph.
• Scale: Numerical intervals used along an axis.
• Interval: Difference between consecutive values on a scale.
• Label: Text identifying an axis, category, or quantity.
• Title: Description identifying what a graph represents.
• Pictograph: Graph using pictures or symbols to represent quantities.
• Key: Explanation of the numerical value represented by a symbol.
• Partial symbol: Fraction of a pictograph symbol representing part of its full value.
• Line plot: Graph showing individual numerical observations as marks above a number line.
• Dot plot: Line-plot style display using dots to represent observations.
• Line graph: Graph showing changes across an ordered variable, often time.
• Frequency: Number of times a value or category occurs.
• Cluster: Region where many data values are concentrated.
• Gap: Region containing no observations.
• Outlier: Observation unusually distant from most other values.
• Range: Difference between the maximum and minimum values.
• Truncated axis: Axis that begins above zero or omits part of the numerical range.
• Visual representation: Method of communicating information using graphical features.

Key Takeaways

• Graphs transform numerical data into visual information.
• Bar graphs, pictographs, and line plots serve different purposes.
• Bar graphs are particularly useful for comparing categories.
• Bar height or length represents the numerical value of a category.
• Bars representing separate categories normally have gaps between them.
• Bar graphs may be vertical or horizontal.
• Every graph should have a clear and informative title.
• Axes and categories should be clearly labelled.
• Measurement units should be included where appropriate.
• Graph scales should use equal intervals.
• The scale should suit the range and size of the data.
• Pictographs use symbols or pictures to represent quantities.
• Every pictograph requires a key.
• One symbol may represent more than one observation.
• Partial symbols can represent fractions of the value given in the key.
• Pictographs are useful for simple and visually engaging comparisons.
• Line plots display individual numerical observations along a number line.
• Line plots and line graphs are different types of displays.
• Line plots are useful for identifying clusters, gaps, extremes, and repeated values.
• The range is calculated using maximum − minimum.
• Different displays are appropriate for different types of data.
• Bar graphs and pictographs work well for categorical data.
• Line plots work well for relatively small sets of numerical data.
• Graphs can be misleading if scales, symbols, axes, or labels are poorly designed.
• Reading the title, scale, units, labels, and key should come before interpreting the data.
• Good graphical displays are correct, labelled, appropriately scaled, suitable, and readable.
• Effective graphs communicate information accurately without requiring additional explanation.
• Choosing the correct graph is part of analyzing and communicating data, not merely drawing it.