Open Math›Year 11›Mathematics Standard›Data analysis›Displaying data

Displaying data

Classify variables as categorical or numerical, construct frequency tables and histograms, describe the shape of a distribution, and identify misleading features in statistical graphs.

Worked examples

Classifying a variable and constructing a frequency table

Straightforward

Problem

A librarian records the number of books borrowed by each of 12 students during one week:

0, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 3

.

(a) Classify the variable "number of books borrowed" as categorical, numerical (discrete) or numerical (continuous).
(b) Construct a frequency table for the data.

Decide whether the variable is categorical or numerical.

The number of books borrowed is found by counting. It takes number values that can be ordered and compared in size, so it is a numerical variable.

Decide whether the numerical variable is discrete or continuous.

You can only borrow a whole number of books — values such as 1.5 are not possible. The variable can only take isolated whole-number values, so it is numerical (discrete).

List the distinct values that appear in the dataset.

Reading through the data, the values that appear are

0, 1, 2

and

3

. List these in order as the rows of the frequency table.

Count the frequency of each value.

Tally each value in the list

0, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 3

:

| Number of books | Frequency |
|---|---|
|

0

4

|
|

1

3

|
|

2

3

|
|

3

2

|
| **Total** | **12** |

Check that the frequencies sum to the total number of students.

4 + 3 + 3 + 2 = 12

✓

Answer

The variable is numerical (discrete). Frequency table:

0 \to 4

1 \to 3

2 \to 3

3 \to 2

(total: 12 students).

Describing distribution shape and calculating relative frequency

Moderate

Problem

The frequency table below shows the distances (in km) that 25 Year 11 students travel to school each day.

| Distance (km) | Frequency |
|---|---|
|

0

–

<5

10

|
|

5

–

<10

8

|
|

10

–

<15

4

|
|

15

–

<20

2

|
|

20

–

<25

1

|

(a) Describe the shape of the histogram drawn from this table.
(b) Find the relative frequency of the class

0

–

<5

km, expressed as a simplified fraction.

Read the frequencies to identify where the data is concentrated.

The frequencies are

10, 8, 4, 2, 1

— they decrease steadily from the first class to the last. Most students travel a short distance; very few travel

15

km or more.

Identify the tail and describe the shape.

The data is concentrated at the left (lower values) with a long tail extending to the right (higher values). This is a positively skewed distribution.

Find the total frequency.

Total

= 10 + 8 + 4 + 2 + 1 = 25

students.

Calculate the relative frequency of the

0

–

<5

km class.

Relative frequency

= \dfrac{\text{class frequency}}{\text{total frequency}} = \dfrac{10}{25} = \dfrac{2}{5}

Answer

The distribution is positively skewed. The relative frequency of the

0

–

<5

km class is

\dfrac{2}{5}

Identifying a misleading feature in a graph

Challenging

Problem

A local council presents a bar chart showing the number of potholes repaired each quarter. The vertical axis begins at

180

rather than

0

, and shows four bars: Q1

= 183

, Q2

= 186

, Q3

= 185

, Q4

= 189

. The council spokesperson claims: "Our repair rate tripled from Q1 to Q4."

Identify the misleading feature of the chart and explain whether the spokesperson's claim is justified.

Identify the misleading feature of the chart.

The vertical axis begins at

180

rather than at

0

. This is called a truncated axis — it cuts off the lower portion of the scale.

Determine the visual impression created by the truncated axis.

With the axis starting at

180

, the visible bar heights above the baseline are:
- Q1:

183 - 180 = 3

units
- Q4:

189 - 180 = 9

units

The Q4 bar appears

\dfrac{9}{3} = 3

times taller than the Q1 bar in the chart.

Calculate the actual change in the number of potholes repaired.

Actual increase

= 189 - 183 = 6

potholes.

Percentage increase

= \dfrac{6}{183} \times 100 \approx 3.3\%

Compare the visual impression with the actual data and evaluate the claim.

The chart makes Q4 look three times as high as Q1, but the actual increase is only about

3.3\%

— far from a tripling. The spokesperson's claim is not justified. If the axis started at

0

, all four bars would be nearly identical in height, accurately showing that the number of repairs changed very little across the year.

Answer

The misleading feature is the truncated vertical axis, which starts at

180

instead of

0

. The Q4 bar appears three times taller than Q1 in the chart (9 vs. 3 units above the baseline), creating the impression of a tripling. The actual increase is

6

potholes — approximately

3.3\%

. The spokesperson's claim is not justified.

Practise

Q1·Straightforward

A survey asks students how many text messages they send each day. Which type of variable is "number of text messages sent per day"?

Numerical (discrete)Numerical (continuous)Categorical (nominal)Categorical (ordinal)

Explanation

The number of text messages must be a whole number — you cannot send 2.7 messages. It is numerical because it is found by counting, and discrete because only whole-number values are possible.

Q2·Straightforward

An athletics coach records the time taken by each student to run 100 metres. Which type of variable is "time to run 100 m"?

Numerical (continuous)Numerical (discrete)Categorical (nominal)Categorical (ordinal)

Explanation

Time can take any value within a range — for example, 13.47 s or 13.472 s. It is numerical because it is measured, and continuous because any value is theoretically possible within the range, not just whole numbers.

Q3·Straightforward

Eight students reported the number of books they read last month:

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

. How many students read exactly

2

books?

Explanation

Reading through the list:

\mathbf{2}, 5, \mathbf{2}, 3, 5, \mathbf{2}, 3, 5

. The value 2 appears three times, so its frequency is

3

Q4·Straightforward

Ten quiz scores were recorded:

8, 10, 8, 6, 10, 8, 6, 10, 8, 6

. How many scores of

8

are there?

Explanation

Reading through the list:

\mathbf{8}, 10, \mathbf{8}, 6, 10, \mathbf{8}, 6, 10, \mathbf{8}, 6

. The value 8 appears at positions 1, 3, 6 and 9, giving a frequency of

4

Q5·Moderate

A frequency table shows the number of hours per week that 30 students spend on social media:

| Hours per week | Frequency |
|---|---|
|

0

–

<5

3

|
|

5

–

<10

6

|
|

10

–

<15

12

|
|

15

–

<20

6

|
|

20

–

<25

3

|

Which description best matches the shape of the histogram drawn from this table?

Approximately symmetricPositively skewedNegatively skewedUniform

Explanation

The data peaks in the middle class (

10

–

<15

hours, frequency

12

). The frequencies on either side are mirror images:

3, 6

below the peak and

6, 3

above it. The distribution is approximately symmetric — neither tail is longer than the other.

Q6·Moderate

A researcher records how long customers wait in a queue at a bank. Most customers wait between 1 and 4 minutes, but a small number of customers wait much longer — up to 25 minutes. Which description best matches the shape of the histogram of waiting times?

Positively skewedNegatively skewedApproximately symmetricBimodal

Explanation

Most values are clustered near the lower end (1–4 minutes) with a few very large values extending to 25 minutes. This long tail stretches to the right (towards higher values), so the distribution is positively skewed.

Q7·Moderate

A frequency table records Year 11 test scores:

| Score | Frequency |
|---|---|
|

40

–

<50

3

|
|

50

–

<60

8

|
|

60

–

<70

11

|
|

70

–

<80

6

|
|

80

–

<90

2

|

How many students sat the test in total?

Explanation

Total

= 3 + 8 + 11 + 6 + 2 = 30

students.

Q8·Moderate

In a class of

24

students, a frequency table shows that

8

students scored in the range

50

–

<60

. Express the relative frequency of this class as a simplified fraction.

Explanation

Relative frequency

= \dfrac{8}{24}

. Dividing numerator and denominator by

8

gives

\dfrac{1}{3}

Q9·Challenging

A bar chart displays the average monthly electricity bills for four households. The vertical axis begins at

\$200

rather than

\$0

and shows values

\$204

\$218

\$212

and

\$224

. Which statement best identifies the misleading feature and explains its effect?

The vertical axis starts at $\$200$ rather than $\$0$, making the differences between bills appear much larger than they actually areThe chart uses dollars, which makes comparison between households impossibleThe vertical axis should begin at $\$100$ for a fair comparisonUsing average monthly bills rather than total annual bills distorts the data

Explanation

When the axis starts at

\$200

, the visible bar heights above the baseline are very small differences (e.g.,

\$4

\$24

), which are magnified visually. A reader glancing at the chart would think the highest bill (

\$224

) is many times larger than the lowest (

\$204

), when it is actually only about

10\%

more. Starting the axis at

\$0

would show all bars nearly equal in height, reflecting the small actual differences.

Q10·Challenging

A histogram displays income data for 20 employees using unequal class intervals. The class

\$20\,000

–

<\$40\,000

has a frequency of

8

(width

\$20\,000

) and the class

\$40\,000

–

<\$80\,000

has a frequency of

10

(width

\$40\,000

). Bar heights are drawn equal to frequency. A colleague says: "Incomes are more concentrated in the

\$40\,000

–

<\$80\,000

class because its bar is taller." Which response correctly evaluates this claim?

The claim is incorrect — the $\$40\,000$–$<\$80\,000$ class is twice as wide, so its frequency density ($10 \div 40 = 0.25$ per $\$1000$) is lower than that of $\$20\,000$–$<\$40\,000$ ($8 \div 20 = 0.40$ per $\$1000$), meaning incomes are actually more concentrated in the lower classThe claim is correct — a taller bar always indicates a higher concentration of dataThe claim is incorrect — both bars should be drawn with the same height when class widths differThe claim is correct, but only when all class widths are equal

Explanation

Frequency density measures how many data values fall per unit of the class interval. For

\$20\,000

–

<\$40\,000

8 \div 20 = 0.40

per

\$1000

. For

\$40\,000

–

<\$80\,000

10 \div 40 = 0.25

per

\$1000

. The lower class has a higher density — incomes are more concentrated there. Using bar height equal to raw frequency when class widths differ gives a misleading visual impression of concentration.

Q11·Challenging

A 3D pie chart is used in a report to compare the market share of four companies: A

= 25\%

, B

= 25\%

, C

= 25\%

, D

= 25\%

. Due to the 3D perspective, the front slices (C and D) appear larger than the back slices (A and B). Which statement best explains why this is misleading?

The 3D perspective distorts the apparent size of slices — front slices look larger than back slices even when they represent equal values, so a reader incorrectly concludes that C and D have greater market share than A and BA pie chart cannot display more than three categoriesMarket share data must always be shown as a bar chartThe percentages no longer add to $100\%$ when a 3D effect is applied

Explanation

All four companies hold exactly

25\%

market share, so each slice should be identical in size. The 3D tilt makes slices near the front of the chart appear wider and larger than slices at the back, even though they represent the same value. A reader would incorrectly conclude that C and D dominate the market. A flat (2D) pie chart would display all four slices equally and avoid this distortion.

Q12·Challenging

A student surveys 30 classmates about their favourite music genre (Pop, Rock, Jazz, Classical) and plots the results as a line graph, connecting the four data points with straight lines. A teacher says the graph is misleading. Which statement best explains why?

A line graph implies that values change continuously between categories, suggesting a trend that does not exist — genre is a categorical variable with no meaningful order or numerical relationship between categoriesA line graph can only display two data points at a timeThe line should be drawn as a curve, not straight segmentsFrequency data can only be shown as a table, never as a graph

Explanation

Line graphs are appropriate when the horizontal axis represents a continuous or ordered scale (such as time or temperature), so the line between points carries meaning. Genre is a categorical variable — the categories Pop, Rock, Jazz and Classical have no numerical order. Connecting them with a line falsely implies a continuous trend between genres. A bar chart or column graph would correctly display these frequencies without suggesting any relationship between the categories.