Topics
7. Statistics
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Year 11

7.025 Display data

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Displaying data

Once we have organised the data, we need to present the data in a form that will be easy to read, understand and analyse.

Displaying data

Some common ways of displaying statistical data are listed below.

  • pie charts
  • histograms
  • bar charts and column graphs
  • dot plots
  • stem and leaf plots

The best type of display to be used will depend on the type of data and purpose of the investigation.

Another type of statistical graph, the box and whisker plot is used to display statistical summary data, and will be described in a later section.

Pie charts

A pie chart (pie graph or sector graph) is a chart for displaying categorical data that uses a circle divided into slices(sectors) to show relative sizes of data. This chart is useful to display and compare parts of data that make up a whole, such as proportion of voters voting for particular political parties.

Reading a pie chart

There are $360^\circ$360° in a circle. Using this fact we can:

  • Find the angle, ($\theta$θ), for segments knowing fraction of the whole the segment represents:

$\theta=\text{Fraction of group in sector}\times360^\circ$θ=Fraction of group in sector×360°

  • Find the number a segment represents:

$\text{Number in a sector}=\frac{\theta}{360}\times\text{Number in whole group}$Number in a sector=θ360​×Number in whole group

Practice question

Question 1

The sector graph represents the number of people taking leave from work at a particular company.

  1. If $5$5 people took leave in January, how many degrees represent $1$1 person?

  2. How many people took leave in November?

  3. How many people took leave between the beginning of November and the end of March?

  4. What percentage of the people took leave in December?

    Give your answer as a percentage, rounding to two decimal places.

 

Histograms, bar charts and column graphs

These graphs represent the frequency of data values as the length of horizontal bars or vertical columns.

Column graphs (also known as bar graphs) are usually used to display categorical data or discrete numerical data.

Histograms are similar to column graphs, with vertical columns used to display continuous numerical data. The main difference between a column graph and histogram is that histograms do not have spaces between the columns.

The reason that histograms do not have gaps between columns is that the class intervals are not separate categories. Instead, the columns represent the frequency of values observed in the class intervals. The width of the columns indicates the range of values in the class intervals. Below are some examples and a brief description of their differences.

Column graphs Histograms

  • Column graphs are used for displaying categorical or discrete numerical data.
  • Gaps are shown between column for categorical data to show separate categories.
  • Gaps are often also used for discrete numerical data to emphasise that the values are discrete but are not required.
  • Labels go underneath each column
  • Histograms are used to display continuous numerical data.
  • No gaps between columns
  • The axis forms a continuous number line and the scale usually indicates the boundary values of each group
 

Practice questions

Question 2

In a survey some people were asked approximately how many minutes they take to decide between brands of a particular product.

  1. Complete the table.

    Minutes Taken Tally Frequency
    1 |||| |||| ||| $\editable{}$
    2 |||| |||| |||| || $\editable{}$
    3 |||| |||| || $\editable{}$

  2. How many people took part in the survey?

  3. How much time did most people take to choose between brands?

  4. Complete the column graph using the results from the table above.

    MinutesNumber of People5101520123

Question 3

The histogram below shows the number of hours that students in a particular class had slept for the night before.

  1. How many students are in the class?

  2. How many students had at least $8$8 hours of sleep that night?

  3. What percentage of students had less than $6$6 hours of sleep?

 

Dot plots

Dot plots are a graphical way of displaying the distribution of numerical or categorical data on a simple scale with dots representing the frequency of data values. They are best used for small to medium size sets of data and are good for visually highlighting how the data is spread and whether there are any gaps in the data or outliers. We will look at identifying outliers in more detail in our next lesson.

In a dot plot, each individual value is represented by a single dot, displayed above a horizontal line. When data values are identical, the dots are stacked vertically. The graph appears similar to a pictograph or column graph with the number of dots representing the total count.

  • To correctly display the distribution of the data, the dots must be evenly spaced in columns above the line
  • The scale or categories on the horizontal line should be evenly spaced
  • A dot plot does not have a vertical axis
  • The dot plot should be appropriately labelled

 

Practice questions

Question 4

Here is a dot plot of the number of goals scored in each of Bob’s soccer games.

  1. How many times were five goals scored?

  2. Which number of goals were scored equally and most often?

    $1$1

    A

    $0$0

    B

    $4$4

    C

    $3$3

    D

    $2$2

    E

    $5$5

    F

  3. How many games were played in total?

Question 5

Christa is a casual nurse. She used a dot plot to keep track of the number of shifts she did each week for a number of weeks.

A dot plot has its horizontal axis labeled "Shifts per Week" and marked at intervals of 1, ranging from 3 to 8. Above each number of shifts on the horizontal axis, a vertical stack of red dots represent the number of weeks. At 3 shifts, 1 red dots are stacked. At 4 shifts, 5 red dots are stacked. At 5 shifts, 4 red dots are stacked. At 6 shifts, 7 red dots are stacked. At 7 shifts, 2 red dots are stacked. At 8 shifts, 2 red dots are stacked.

  1. Over how many weeks did Christa record her shifts?

  2. For how many weeks did she work $5$5 shifts?

    $\editable{}$ weeks

  3. How many weeks did she work less than $6$6 shifts?

    $\editable{}$ weeks

  4. When Christa works at least $6$6 shifts a week, she buys a weekly train ticket. What proportion of the time did she buy a weekly train ticket?

 

Stem and leaf plot

A stem and leaf plot, or stem plot, is used for organising and displaying numerical data. It is appropriate for small to moderately sized data sets. The graph is similar to a column graph on its side. An advantage of a stem and leaf plot over a column graph is the individual scores are retained and further calculations can be made accurately.

In a stem and leaf plot, the right-most digit in each data value is split from the other digits, to become the 'leaf'. The remaining digits become the 'stem'.

The values in a stem and leaf plot should be arranged in ascending order (from lowest to highest) from the centre out. To emphasise this, it is often called an ordered stem and leaf plot.

The data values $10,13,16,21,26,27,28,35,35,36,41,41,45,46,49,50,53,56,58$10,13,16,21,26,27,28,35,35,36,41,41,45,46,49,50,53,56,58 are displayed in the stem and leaf plot below.

  • The stems are arranged in ascending order, to form a column, with the lowest value at the top
  • The leaf values are arranged in ascending order from the stem out, in rows, next to their corresponding stem
  • A single vertical line separates the stem and leaf values
  • There are no commas or other symbols between the leaves, only a space between them
  • In order to correctly display the distribution of the data, the leaves must line up in imaginary columns, with each data value directly below the one above
  • A stem and leaf plot includes a key that describes the way in which the stem and the leaf combine to form the data value

 

Practice questions

Question 6

Which of the following is true of a stem-and-leaf plot?

Stem Leaf
$0$0 $7$7
$1$1  
$2$2  
$3$3 $1$1 $3$3 $3$3 $3$3
$4$4 $1$1 $2$2 $3$3 $4$4 $9$9
$5$5 $1$1 $2$2 $4$4 $5$5 $5$5
$6$6 $0$0
 
Key: $1$1$\mid$∣$2$2$=$=$12$12
A stem-and-leaf plot is displayed. The plot is divided into two columns: "Stem" on the left, and "Leaf" on the right. The "Stem" column lists the digits in the order 0, 1, 2, 3, 4, 5, and 6, starting with 0 at the topmost column. Each digit in the "Stem" column is paired with aligned with a group of digits in the "Leaf" column. For stem 0, the leaf is 7. For stems 1 and 2, there are no leaves. For stem 3, the leaves are 1, 3, 3, and 3. For stem 4, the leaves are 1, 2, 3, 4, and 9. For stem 5, the leaves are 1, 2, 4, 5, and 5. For stem 6, the leaf is 0. Below the plot is a row named "Key," which explains the notation. On the "Key" row, it is written that 1 | 2 = 12.

  1. The scores are ordered.

    A

    A stem-and-leaf plot does not give an idea of outliers and clusters.

    B

    It is only appropriate for data where scores have high frequencies.

    C

    The individual scores cannot be read on a stem-and-leaf plot.

    D

Question 7

The stem-and-leaf plot below shows the age of people to enter through the gates of a concert in the first $5$5 seconds.

Stem Leaf
$1$1 $1$1 $2$2 $4$4 $5$5 $6$6 $6$6 $7$7 $9$9 $9$9
$2$2 $2$2 $3$3 $5$5 $5$5 $7$7
$3$3 $1$1 $3$3 $8$8 $9$9
$4$4  
$5$5 $8$8
 
Key: $1$1$\mid$∣$2$2$=$=$12$12
years old

  1. How many people passed through the gates in the first $5$5 seconds?

  2. What was the age of the youngest person?

    The youngest person was $\editable{}$ years old.

  3. What was the age of the oldest person?

    The oldest person was $\editable{}$ years old.

  4. What proportion of the concert-goers were under $20$20 years old?

Outcomes

2.3.1.5

select and justify an appropriate graphical display to describe the distribution of a numerical dataset, including dot plot, stem-and-leaf plot, column chart or histogram

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