Open Math›Year 9›Statistics and data›Displaying data

Displaying data

Read and interpret stem-and-leaf plots, dot plots and histograms, and select the most appropriate display for different types of data.

Worked examples

Reading a stem-and-leaf plot

Straightforward

Problem

The stem-and-leaf plot shows the ages (in years) of

9

people at a sports club.

\begin{array}{r|l} \text{Stem} & \text{Leaf} \\ \hline 1 & 5 \quad 8 \quad 8 \\ 2 & 0 \quad 3 \quad 7 \\ 3 & 1 \quad 4 \quad 9 \end{array}

Key:

1 \mid 5

means

15

.

Find (a) the mode and (b) the median.

Read the plot to list all values in order. The stem gives the tens digit and each leaf gives the units digit.

15, \ 18, \ 18, \ 20, \ 23, \ 27, \ 31, \ 34, \ 39

Find the mode by identifying the value that appears most often.

18

appears twice; all other values appear once. Mode

= 18

Find the median. With

9

values, the median is the

5

th value in the ordered list.

\underbrace{15, \ 18, \ 18, \ 20, \ \mathbf{23}, \ 27, \ 31, \ 34, \ 39}_{9 \text{ values}}

Median

= 23

Answer

(a) Mode

= 18

; (b) Median

= 23

Reading a histogram

Moderate

Problem

The table below summarises a histogram showing the number of hours

40

students spent studying in one week.

\begin{array}{|c|c|} \hline \text{Hours} & \text{Frequency} \\ \hline [0, 2) & 4 \\ [2, 4) & 8 \\ [4, 6) & 12 \\ [6, 8) & 10 \\ [8, 10) & 6 \\ \hline \end{array}

Find (a) the modal class, (b) the relative frequency of the

[4, 6)

class, and (c) the number of students who studied for fewer than

4

hours.

Find the modal class by identifying the interval with the highest frequency.

Frequencies:

4, \ 8, \ 12, \ 10, \ 6

. The highest is

12

, in the class

[4, 6)

. Modal class

= [4, 6)

Calculate the relative frequency of

[4, 6)

by dividing its frequency by the total.

Total

= 4 + 8 + 12 + 10 + 6 = 40

. Relative frequency

= \dfrac{12}{40} = 0.3

Add the frequencies for all classes below

4

hours.

Classes

[0, 2)

and

[2, 4)

4 + 8 = 12

students.

Answer

(a) Modal class

= [4, 6)

; (b) Relative frequency

= 0.3

; (c)

12

students

Choosing the right statistical display

Challenging

Problem

For each situation below, identify the most appropriate statistical display and give a reason.

(a) A teacher records the individual test scores (out of

50

) for

30

students.

(b) A scientist measures the heights (in cm) of

1500

seedlings, ranging from

5

cm to

85

cm.

(c) A shop records its daily revenue every day for two years.

For (a): identify the data type and quantity, then match to an appropriate display.

The data is numerical with

30

individual values. A stem-and-leaf plot groups values by tens digit, preserves each individual score, and clearly shows the shape of the distribution for a moderate-sized dataset.

For (b): consider the large number of values and wide range.

With

1500

values spanning

5

85

cm, a histogram groups values into class intervals. This makes the distribution readable without the overcrowding that a dot plot or stem-and-leaf would produce.

For (c): consider that the data is recorded over a sequence of time periods.

Daily revenue over two years is a time-series dataset. A line graph connects consecutive data points, making trends, seasonal patterns and unusual days easy to identify.

Answer

(a) Stem-and-leaf plot; (b) Histogram; (c) Line graph

Practise

Q1·Straightforward

The stem-and-leaf plot below shows the scores of

9

students on a maths quiz.

\begin{array}{r|l} \text{Stem} & \text{Leaf} \\ \hline 1 & 3 \quad 5 \quad 7 \\ 2 & 2 \quad 2 \quad 6 \quad 9 \\ 3 & 0 \quad 4 \end{array}

Key:

1 \mid 3

means

13

.

Find the mode.

Explanation

Reading the plot:

13, \ 15, \ 17, \ 22, \ 22, \ 26, \ 29, \ 30, \ 34

. The value

22

appears twice; all others appear once. Mode

= 22

Q2·Straightforward

The stem-and-leaf plot below shows the times (in minutes) taken by

9

students to complete a puzzle.

\begin{array}{r|l} \text{Stem} & \text{Leaf} \\ \hline 1 & 1 \quad 4 \quad 6 \\ 2 & 0 \quad 3 \quad 7 \\ 3 & 2 \quad 5 \quad 8 \end{array}

Key:

1 \mid 1

means

11

.

Find the median.

Explanation

Reading the plot in order:

11, \ 14, \ 16, \ 20, \ 23, \ 27, \ 32, \ 35, \ 38

. There are

9

values, so the median is the

5

th value: Median

= 23

Q3·Straightforward

The table below shows the distribution of scores out of

10

for

12

students, as represented in a dot plot.

\begin{array}{|c|c|} \hline \text{Score} & \text{Frequency} \\ \hline 5 & 2 \\ 6 & 3 \\ 7 & 4 \\ 8 & 2 \\ 9 & 1 \\ \hline \end{array}

Find the mode.

Explanation

The score

7

has a frequency of

4

, which is the highest. Mode

= 7

Q4·Straightforward

The stem-and-leaf plot below shows the heights (in cm) of

9

plants.

\begin{array}{r|l} \text{Stem} & \text{Leaf} \\ \hline 3 & 2 \quad 5 \quad 8 \\ 4 & 1 \quad 4 \quad 7 \\ 5 & 0 \quad 3 \quad 6 \end{array}

Key:

3 \mid 2

means

32

.

Find the range.

Explanation

The minimum value is

32

(first leaf on stem

3

) and the maximum is

56

(last leaf on stem

5

). Range

= 56 - 32 = 24

Q5·Moderate

The table below summarises a histogram showing the number of hours

40

students spent studying in one week.

\begin{array}{|c|c|} \hline \text{Hours} & \text{Frequency} \\ \hline [0, 2) & 4 \\ [2, 4) & 8 \\ [4, 6) & 12 \\ [6, 8) & 10 \\ [8, 10) & 6 \\ \hline \end{array}

How many students are represented in total?

Explanation

Total

= 4 + 8 + 12 + 10 + 6 = 40

students.

Q6·Moderate

The table below summarises a histogram showing the number of hours

40

students spent studying in one week.

\begin{array}{|c|c|} \hline \text{Hours} & \text{Frequency} \\ \hline [0, 2) & 4 \\ [2, 4) & 8 \\ [4, 6) & 12 \\ [6, 8) & 10 \\ [8, 10) & 6 \\ \hline \end{array}

Find the relative frequency of the

[4, 6)

class. Give your answer as a decimal.

Explanation

The frequency for

[4, 6)

12

. Total frequency

= 40

. Relative frequency

= \dfrac{12}{40} = 0.3

Q7·Moderate

The table below summarises a histogram showing the number of hours

40

students spent studying in one week.

\begin{array}{|c|c|} \hline \text{Hours} & \text{Frequency} \\ \hline [0, 2) & 4 \\ [2, 4) & 8 \\ [4, 6) & 12 \\ [6, 8) & 10 \\ [8, 10) & 6 \\ \hline \end{array}

How many students studied for fewer than

4

hours?

Explanation

Fewer than

4

hours includes the classes

[0, 2)

and

[2, 4)

. Total

= 4 + 8 = 12

students.

Q8·Moderate

The table below summarises a histogram showing the number of hours

40

students spent studying in one week.

\begin{array}{|c|c|} \hline \text{Hours} & \text{Frequency} \\ \hline [0, 2) & 4 \\ [2, 4) & 8 \\ [4, 6) & 12 \\ [6, 8) & 10 \\ [8, 10) & 6 \\ \hline \end{array}

Which class interval is the modal class?

$[0, 2)$$[2, 4)$$[4, 6)$$[6, 8)$

Explanation

The frequencies are

4, \ 8, \ 12, \ 10, \ 6

. The highest frequency is

12

, which belongs to the class

[4, 6)

. Modal class

= [4, 6)

Q9·Challenging

A teacher records the individual test scores (out of

100

) for

30

students and wants to display the distribution of scores. Which display is most appropriate?

Pie chartStem-and-leaf plotLine graphBar chart

Explanation

The data is numerical and there are

30

individual values. A stem-and-leaf plot groups values by their tens digit, preserves each individual score, and clearly shows the shape of the distribution. A pie chart is for categories; a bar chart suits categorical or discrete data; a line graph is for data over time.

Q10·Challenging

A survey asks

120

people to name their favourite type of music. The researcher wants to show the proportion choosing each type. Which display is most appropriate?

HistogramStem-and-leaf plotDot plotPie chart

Explanation

Favourite music type is categorical data. A pie chart divides a circle into sectors, making it easy to compare proportions at a glance. A histogram and stem-and-leaf plot are for numerical data; a dot plot suits small numerical datasets.

Q11·Challenging

A scientist measures the heights (in cm) of

1500

seedlings, which range from

5

cm to

85

cm. Which display is most appropriate for showing the distribution?

Dot plotStem-and-leaf plotHistogramPie chart

Explanation

With

1500

continuous numerical values across a wide range, a histogram groups values into class intervals, making the distribution shape readable. A dot plot would be overcrowded; a stem-and-leaf plot becomes unmanageable with so many values; a pie chart is for categorical data.

Q12·Challenging

A weather station records the daily maximum temperature each day for one year. Which display is most appropriate for showing how temperature changed throughout the year?

Bar chartStem-and-leaf plotHistogramLine graph

Explanation

When data is collected at regular time intervals and the focus is on change over time, a line graph is most appropriate. Connecting the data points with a line makes trends, seasons and patterns easy to identify. A histogram and stem-and-leaf plot show distribution but not sequence; a bar chart does not emphasise continuity over time.