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C H A P T E R
1CORE
Organising anddisplaying data
What is the difference between categorical and numerical data?
What is a frequency table, how is it constructed and when is it used?
What is the mode and how do we determine its value?
What are bar charts, histograms, stem plots and dot plots? How are they
constructed and when are they used?
How do you describe the features of bar charts, histograms and stem plots when
writing a statistical report?
1.1 Classifying dataStatistics is a science concerned with understanding the world through data. The first step in
this process is to put the data into a form that makes it easier to see patterns or trends.
Some dataThe data contained in Table 1.1 are part of a larger set of data collected from a group of
university students.
Table 1.1 Student data
Height Weight Age Sex Plays sport Pulse rate
(cm) (kg) (years) M male 1 regularly (beats/min)F female 2 sometimes
3 rarely
173 57 18 M 2 86179 58 19 M 2 82167 62 18 M 1 96195 84 18 F 1 71173 64 18 M 3 90184 74 22 F 3 78175 60 19 F 3 88140 50 34 M 3 70
Source: www.statsci.org/data/oz/ms212.html. Used with permission.
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2 Essential Further Mathematics Core
Variables
In a data set, we call the things about which we record information variables. An important
first step in analysing any set of data is to identify the variables involved, their units of
measurement (where appropriate) and the values they take. In this particular data set there are
sixvariables:
height (in centimetres)weight (in kilograms)
age (in years)
sex (M =male, F = female)plays sport (1 =regularly, 2 =sometimes, 3 = rarely)
pulse rate (beats/minute)
Types of variables: categorical and numericalHaving identified the variables we are working with, the next step is to decide the variable type.
Some variables representqualitiesor attributes. For example, F in theSexcolumn
indicates that the person is a female, while a 2 in thePlays sportcolumn indicates that the
person is someone who plays sport sometimes.
Variables that representqualitiesare calledcategorical variables.
Other variables representquantities. For example, a 179 in theHeightcolumn indicates
that the person is 179 cm tall, while an 82 in the Pulse ratecolumn indicates that they have a
pulse rate of 82 beats/minute.
Variables that representquantitiesare callednumerical variables.
Numerical variables come in two types: discrete and continuous.
Discretenumerical variables represent quantities that arecounted. The number of mobile
phones in a house is an example. Counting leads to discrete data values because it results in
values such as 0, 1, 2, 3 etc. There can be nothing in between. As a guide, discrete numericalvariablesarise when we ask the question How many?
Continuousnumerical variables represent quantities that aremeasuredrather than counted.
Thus, even though we might record a persons height as 179 cm, in reality that could be any
value between 178.5 and 179.4 cm. We have just rounded off the height to 179 cm for
convenience, or to match the accuracy of the measuring device.
Warning!!It is not the variable name itself that determines whether the data are numerical or categorical, it is
the way the data for the variable are recorded.For example:
weight recorded in kilograms, is anumericalvariable
weight recorded as 1 = underweight, 2 = normal weight, 3 = overweight, is acategorical
variable
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Chapter 1 Organising and displaying data 3
Exercise 1A
1What is:
a a numerical variable? Give an example. b a categorical variable? Give an example.
2There are two types of numerical variables. Name them.
3Classify each of the following variables as numerical or categorical. If the variable is
numerical, further classify the variable as discrete or continuous.
Recording information on:
a length of bananas (in centimetres)
b number of cars in a supermarket car park
c daily temperature in C
d eye colour (brown, blue, . . . )
e shoe size (6, 8, 10, . . . )
f the number of children in a family
g city of residence (NY, London, . . . )
h number of people who live in your city/area
i time spent watching TV (hours)
j the TV channel most watched by students
k salary (high, medium, low)
l salary (in dollars)
m whether a person smokes (yes, no)
n the number of cigarettes smoked per day
4Classify the data for each of the variables in Table 1.1 as numerical or categorical.
1.2 Organising and displaying categorical dataThe frequency table
With a large number of data values, it is difficult to identify any patterns or trends in the rawdata. We first need to organise the data into a more manageable form. A statistical tool we use
for this purpose is the frequency table.
The frequency table
Afrequency tableis a listing of the values a variable takes in a data set, along with how
often (frequently) each value occurs.
Frequency can be recorded as a
count: the number of times a value occurs, or
per cent: the percentage of times a value occurs (percentage frequency)
per cent =count
total count 100%
A listing of the values a variable takes, along with how frequently each of these values
occurs in a data set, is called afrequency distribution.
Example 1 Frequency table for a categorical variable
The sex of 11 preschool children is as shown (F = female, M =male):
F M M F F M F F F M M
Construct a frequency table to display the data.
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4 Essential Further Mathematics Core
Solution
1 Set up a table as shown. The variableSexhas two
categories: Male and Female.
2 Count up the number of females (6) and males (5).
Record this in the Count column.
Frequency
Sex Count Per cent
Female 6 54.5
Male 5 45.5
Total 11 100.0
3 Add the counts to find the total count, 11 (6 + 5).
Record this in the Count column opposite Total.
4 Convert the counts into percentages.
Record this in the Per cent column. For example:
percentage of females =6
11 100% = 54.5%
5 Finally, total the percentages and record.
There are two things to note in constructing the frequency table in Example 1.1 In setting up this frequency table, the order in which we have listed the categories Female
and Male is quite arbitrary; there is no natural order. However, if the categories had been,
for example, First, Second and Third, then it would make sense to list the categories in
that order.
2 The Total count should always equal the total number of observations; in this case, 11.
The percentages should add to 100%. However, if percentages are rounded to one decimal
place a total of 99.9 or 100.1 is sometimes obtained. This is due to rounding error. Totalling
the count and percentages helps check on your counting and percentaging.
How has forming a frequency table helped?
The process of forming a frequency table for a categorical variable:
displays the data in acompactform
tells us something about the way the data values are distributed(the pattern of the
data).
The bar chartOnce categorical data have been organised into a frequency table, it is common practice to
display the information graphically to help identify any features that stand out in the data. Astatistical tool we use for this purpose is the bar chart.
The bar chart represents the key information in a frequency table as a picture. It is designed
for categorical data. In a bar chart:
frequency (or percentage frequency) is shown on the vertical axis
the variable being displayed is plotted on the horizontal axis
the height of the bar (column) gives the frequency (count or percentage)
the bars are drawn with gaps between them to indicate that each value is a separate
category
there is one bar for each category.
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Chapter 1 Organising and displaying data 5
Example 2 Constructing a bar chart from a frequency table
Construct a bar chart for this frequency table
of climate types in various countries.Frequency
Climate type Count Per cent
Cold 3 13.0Moderate 14 60.9
Hot 6 26.1
Total 23 100.0
Solution
1 Label the horizontal axis with the variable
name, Climate type. Mark the scale off
into three equal intervals and label them Cold,Moderate and Hot.
2 Label the vertical axis Frequency. Scale
allowing for the maximum frequency, 14.
Fifteen would be appropriate. Mark the
scale off in fives.
3 For each interval, draw in a bar. There are
gaps between the bars to show that the
categories are separate. The height of the
bar is made equal to the frequency.
15
10
5
0Cold Moderate Hot
Frequency
Climate type
The modeOne of the features of a data set that is quickly revealed with a bar chart is the modeormodal
category. This is the most frequently occurring value or category. This is given by the
category with the tallest bar. For the bar chart above, the modal category is clearly Moderate.
That is, for the countries considered, the most frequently occurring climate type is Moderate.
However, the mode is only of interest when a single value or category in the frequency table
occurs much more often than the others. Modes are of particular importance in popularity
polls. For example, in answering questions such as Which is the most frequently watched TV
station between the hours of 6.00 and 8.00 p.m.? or What are the times when a supermarket
is in peak demand morning, afternoon or night?
What to look for in a frequency distribution of a categoricalvariable: writing a reportA bar chart, in combination with a frequency table, is useful for gaining an overall view of a
frequency distribution of a categorical variable, the so-called big picture.
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6 Essential Further Mathematics Core
Describing a bar chart
In describing a bar chart, we focus on two things:
the presence of a dominant category(or group of categories) in the distribution. This
is given by the mode. If there is no dominant category, then this should be stated.
theorder of occurrenceof each category and its relative importance.
In commenting on these features, it is usual to support your conclusions with percentages.
When quoting percentages, it is also advisable to indicate at the beginning the total number of
cases involved. Using the information in Example 2 to describe the distribution of climate
type, you might write as follows:
ReportThe climate types of 23 countries were classified as being, `cold', `moderate' or hot'. The
majority of the countries, 60.9%, were found to have a moderate climate. Of the remaining
countries, 26.1% were found to have a hot climate while 13.0% were found to have a coldclimate.
Stacked or segmented bar charts
Climate
Hot
Cold
Moderate
25
20
15
10
5
0
Frequ
ency
A variation on the standard bar chart is the
segmented or stacked bar chart. In a
segmented bar chart, the bars are stacked
on one another to give a single bar with
several components. The lengths of thesegments are determined by the frequencies.
When this is done, the height of the bar gives
thetotalfrequency. Segmented bar charts
should only be used when there are
a relatively small number of components; usually no more than four or five. Otherwise it
becomes difficult to distinguish the components. The segmented bar chart above was formed
from the climate data used in Example 2. Note that a legend has been included to identify the
segments.
Climate
Hot
Moderate
Cold
100
90
80
70
60
50
Percentage
40
30
20
10
0
In apercentage segmented bar chart,the lengths of each of the segments in the
bar are determined by the percentages.
When this is done, the height of the bar is
100. The percentage segmented bar chart
opposite was formed from the climate data
used in Example 2.
Percentage segmented bar charts are most
useful when we come to analyse the
relationship between two categoricalvariables, as we will see in Chapter 4.
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Chapter 1 Organising and displaying data 7
Exercise 1B
1 a In a frequency table, what is the mode?
b Identify the mode in the following data sets:
i Grades: A A C B A B B B B D Cii Shoe size: 8 9 9 10 8 8 7 9 8 10 12 8 10
2The following data identifies the state of residence of a group of people, where
1 =Victoria, 2 =SA and 3 = WA.
2 1 1 1 3 1 3 1 1 3 3
a Form a frequency table (with both counts and percentages) to show the distribution of
state of residencefor this group of people. Use the table in Example 1 as a model.
b Construct a bar chart using Example 2 as a model.
3Thesize(S =small, M =medium, L = large) of 20 cars was recorded as follows:
S S L M M M L S S M
M S L S M M M S S M
a Form a frequency table (with both counts and percentages) to show the distribution of
sizefor these cars. Use the table in Example 1 as a model.
b Construct a bar chart using Example 2 as a model.
4The table shows the frequency distribution ofSchool typefor a number of schools. The table
is incomplete.
Frequency
School type Count Percent
Catholic 4 20
Government 11
Independent 5 25
Total 100
a Write down the information missing from the table.
b How many schools are categorised
as Independent?
c How many schools are there in total?
d What percentage of schools are
categorised as Government?
e Use the information in the frequency table
to complete the following report.
Reportschools were classified according to school type. The majority of these schools, %,
were found to be schools. Of the remaining schools, were while
20% were schools.
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Chapter 1 Organising and displaying data 9
Example 3 Frequency table for discrete numerical data
The family sizes of 11 preschool children (including the child itself) are as follows:
3 3 4 4 5 3 2 4 3 5 3
Display the data in the form of a frequency table.
Solution
1 Set up a table as shown. In the data set, the
variablefamily sizetakes the values 2, 3, 4
and 5. List these values under Family size
in some order, here increasing.
Frequency
Family size Count Per cent
2 1 9.13 5 45.54 3 27.35 2 18.2
Total 11 100.1
2 Count up the number of 2s, 3s, 4s and 5s in
the dataset. For example, there are five 3s.
Record these values in the Count column.
3 Add the counts to find the total count, 11. Record this value in the Count column opposite
Total.
4 Convert the counts into percentages. Record them in the Per cent column. For example,
percentage of 3s =5
11 100% = 45.5%
5 Finally, total the percentages and record.
Grouping dataSome variables can only take on a limited range of values; for example, the number of children
in a family. Here, it makes sense to list each of these values individually when forming a
frequency distribution.
In other cases, the variable can take a large range of values; for example, age (0100).
Listing all possible ages would be tedious and would produce a large and unwieldy display. To
solve this problem, wegroupthe data into a small number of convenient intervals. There are
no hard and fast rules for the number of intervals but, usually, between five and fifteen intervals
are used. Usually, the smaller the number of data values, the smaller the number of intervals.Note that the intervals are defined so that it is quite clear into which interval each data value
falls. For example, you cannot define intervals as, 15, 510, 1015, 1520, . . . etc., as you
would not know into which interval to put the values, 5, 10, 15 etc.
Guideline for choosing the number of intervals
There are no hard and fast rules for the number of intervals to use but, usually, between five
and fifteen intervals are used.
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10 Essential Further Mathematics Core
Example 4 Grouping data
The ages of a sample of 200 people aged from 16 to 72 years are to be recorded. Group the
ages into six equal-sized categories that will cover all of these ages.
Solution
1 Write down the required number of intervals.
2 Determine interval width.
Ages range from 16 to 72, which covers
57 years. Six intervals will give intervals
of width57
6 =9.5.
Set the interval width to 10, the nearest
whole number above 9.5.
Number of intervals: 6
Interval width=57
6 = 9.5: use 10
Starting point: 153 Choose a starting point that ensures that
the intervals cover the full range of values.
15 would be a suitable starting point.
Intervals: 1524, 2534, . . . , 65744 Write down the intervals.
Once we know how to group data, we can form a frequency distribution for grouped data.
Example 5 A grouped frequency distribution for acontinuousnumerical variable
The data below give the average hours worked per week in 23 countries.
35.0, 48.0, 45.0, 43.0, 38.2, 50.0, 39.8, 40.7, 40.0, 50.0, 35.4, 38.8,
40.2, 45.0, 45.0, 40.0, 43.0, 48.8, 43.3, 53.1, 35.6, 44.1, 34.8
Form a grouped frequency table with five intervals.
Solution
1 Set up a table as shown. For five intervals and
data values ranging between 34.8 and 53.1,
use the intervals: 30.034.9, 35.039.9, . . . ,
50.054.9.
FrequencyAverage hoursworked Count Per cent
30.034.9 1 4.335.039.9 6 26.140.044.9 8 34.845.049.9 5 21.750.054.9 3 13.0
Total 23 99.9
2 List these intervals, in ascending order, under
Average hours worked.
3 Count the number of countries whose
average working hours fall into each of
the intervals. For example, six countries have
average working hours between 35.0 and 39.9.
Record these values in the Count column.
4 Add the counts to find the total count, 23.
Record this value in the Count columnopposite Total.
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Chapter 1 Organising and displaying data 11
5 Convert the counts into percentages. Record these in the Per cent column.
For example, for 35.039.9 hours,
percentage =6
23 100% =26.1%
6 Finally, total the percentages and record.
There are two things to note in the frequency table in Example 5.
1 The intervals in this example are of width five. For example, the interval 35.039.9, is an
interval of width 5.0 because it contains all values from 34.9500 . . . to 39.9499.
2 The modal interval is 40.044.9 hours; eight (34.8%) of the countries have working hours
that fall into this interval.
How has forming a frequency table helped?
The process of forming a frequency table for a numerical variable:
ordersthe datadisplays the data in acompactform
tells us something about the way the data values aredistributed(the pattern of the data)
helps us identify themode(the most frequently occurring value or interval of values).
The histogramThefrequency histogram, or histogram for short, is a graphical way of presenting the
information in a frequency table fornumericaldata. Later in the chapter, you will learn about
two other graphical displays for numerical data, the stem plot and the dot plot.
Constructing a histogram from a frequency table
In a frequency histogram:
frequency (count or per cent) is shown on the vertical axis
the values of the variable being displayed are plotted on the horizontal axis
for continuous data, each bar in a histogram corresponds to a data interval. For discrete
data, where there are gaps between values, the intervals start and end halfway between
values. Empty classes or missing discrete values have bars of zero height
the height of the bar gives the frequency (usually the count, but it can equally well be the
percentage).
Example 6 Constructing a histogram from a frequency table:continuousnumerical variable
Construct a histogram for this frequency table.Average hours worked Frequency (count)
30.034.9 1
35.039.9 6
40.044.9 8
45.049.9 5
50.054.9 3Total 23
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12 Essential Further Mathematics Core
Solution
1 Label the horizontal axis with the variable
name, Average hours worked. Mark in the
scale using the beginning of each interval
as the scale points: that is 30, 35, . . .
2 Label the vertical axis Frequency. Scale
allowing for the maximum frequency, 8.
Ten would be appropriate. Mark in the scale
in units.
3 Finally, for each interval, 30.034.9,
35.039.9, . . . , draw in a bar with the base
starting at the beginning of each interval
and finishing at the beginning of the next.
The height of the bar is made equal to the
frequency.
9
8
7
6
5
4
3
2
1
025 30 35 40 45 50 55 60
Average hours worked
Frequen
cy
Example 7 Constructing a histogram from a frequency table:
discretenumerical variable
Construct a histogram for this frequency table. Family size Frequency (count)
2 1
3 5
4 3
5 2Total 11
Solution
1 Label the horizontal axis with the variable name,
Family size. Mark the scale in units, so that it
includes all possible values.
2 Label the vertical axis Frequency. Scale to
allow for the maximum frequency, 5. Five
would be appropriate. Mark the scale in units.
3 Draw in a bar for each data value. The width of
each bar is 1, starting and ending halfway between
data values. For example, the base of the bar
representing a family size of 2 starts at 1.5 and
ends at 2.5. The height of the bar is made equal to
the frequency.
10
1
2
3
4
5
2 3 4 5 6
Family size
Frequency
Constructing a histogram from raw data
It is relatively quick to construct a histogram from a preprepared frequency table. However, if
you only have raw data (as you mostly do), it is a very slow process because you have to
construct the frequency table first. Fortunately, a graphics calculator will do this for us.
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Chapter 1 Organising and displaying data 13
How toconstruct a histogramusing the TI-Nspire CAS
Display the following set of marks in the form of a histogram.
16 11 4 25 15 7 14 13 14 12 15 13 16 14
15 12 18 22 17 18 23 15 13 17 18 22 23
Steps
1 Start a new document: Pressc and
selectNew Document(or use/ +N).
If prompted to save an existing
document, move cursor toNoand press
.
2 SelectAdd Lists & Spreadsheet.
Enter the data into a list namedmarks.
a Move the cursor to the name space
of column A (or any other column)
and type inmarksas the list name.
Press .
b Move the cursor down to row 1, typein the first data value and press .
Continue until all the data has been
entered. Press after each entry.
3 Statistical graphing is done through the
Data & Statisticsapplication.
Press/ + and selectAdd Data &
Statistics(or pressc, arrow to ,
and press ).
Note:A random display of dots will appear this is to indicate that data are available
for plotting. It is not a statistical plot.
a Presse to show the list of
variables. The variablemarksis
shown as selected. Press to
paste the variablemarksto that axis.
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14 Essential Further Mathematics Core
b A dot plot is then displayed as the
default plot. To change the plot to a
histogram press
b>Plot Type>Histogram
Note for CX only:To add colour (or change
colour) move cursor over the plot and press
/ +b >Color>Fill Color.
Your screen should now look like that
shown opposite. This histogram has a
column (or bin) width of 2 and a
starting point of 3.
4 Data analysis
a Move cursor onto any column,
will show and the column data will
be displayed as shown opposite.
b To view other column data values
move the cursor to another column.
Note:If you click on a column it will be selected.
To deselect any previously selected columns,
move the cursor to the open area and press .
Hint: If you accidentally move a column or data
point, press/ + to undo the move.
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Chapter 1 Organising and displaying data 15
5 Change the histogram column (bin) width to4and the starting point to 2.
a Press/ +b to get the contextual menu as shown (below left).Hint: Pressing/+b with the cursor on the histogram gives you access to a contextual menuthat enables you to do things that relate only to histograms.
b SelectBin Settings.
c In the settings menu (below right) change theWidthto4and theStarting Point(Alignment) to2as shown. Press .
d A new histogram is displayed with a column width of 4 and a starting point of 2 but
it no longer fits the viewing window (below left). To solve this problem press
/ +b>Zoom>Zoom-Datato obtain the histogram shown below right.
6 To change the frequency axis to a percentage axis, press/ +b>Scale>Percentand
then press .
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16 Essential Further Mathematics Core
How toconstruct a histogramusing the ClassPad
Display the following set of 27 marks in the form of a histogram.
16 11 4 25 15 7 14 13 14 12 15 13 16 14
15 12 18 22 17 18 23 15 13 17 18 22 23
Steps
1 From the application menu
screen, locate the built-in Statistics
application. Tap to open.
Tapping from the icon panel
(just below the touch screen) will
display the application menu if it is
not already visible.
2 Enter the data into a list named
marks.
To name the list:
a Highlight the heading of the
first list by tapping it.
b Pressk on the front ofthe calculator and tap the
tab.
c To enter the data, type the word
marksand pressE.
d Type in each data value and press
E or (which is found on thecursor button on the front of the
calculator) to move down to the
next cell.
The screen should look like the one
shown opposite.
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Chapter 1 Organising and displaying data 17
3 Set up the calculator to
plot a statistical graph.
a Tap from the toolbar. This
opens theSet StatGraphs
dialog box.
b Complete the dialog
box as given below.
Draw: selectOn
Type: select
Histogram( )
XList: selectmain \
marks( )
Freq: leave as1
c Taph to confirm your
selections.
Note:To make sure only this
graph is drawn, selectSetGraph
from the menu bar at the top and
confirm that there is a tick only beside
StatGraph1and no others.
4 To plot the graph:
a Tap in the toolbar.
b Complete theSet Interval
dialog box as follows.
HStart: type2(i.e. the
starting point of the first
interval)
HStep: type4(i.e. the
interval width)Tap OK to display histogram.
Note:The screen is split into two halves, with the graph displayed in the bottom half, as shown above.
Tapping from the icon panel allows the graph to fill the entire screen. Tap again to return
to half-screen size.
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18 Essential Further Mathematics Core
5 Tapping from the toolbar
places a marker (+) at the top of
the first column of the histogram
(see opposite) and tells us that
a the first interval begins
at 2(xc = 2)
b for this interval, the frequency
is 1(Fc = 1).
To find the frequencies and starting points of the other intervals, use the arrow ( ) to
move from interval to interval.
Exercise 1C
1The numbers of occupants in nine cars stopped at a traffic light were:
1 1 2 1 3 1 2 1 3
What is the mode of this data set? What does this tell us?
2The number of surviving grandparents for 11 preschool children is listed below.
0 4 4 3 2 3 4 4 4 3 3
Form a frequency table to show the distribution of the number of surviving grandparents.
3 a Write down the missing information in the
frequency table.
b How many families had only one child?
c How many families had more than one
child?
d What percentage of families had no
children?
e What percentage of families had fewer
than three children?
FrequencyNo. of childrenin family Count %
0 3
1 10 47.6
2 6 28.6
3
4 2 9.5
Total 21
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Chapter 1 Organising and displaying data 19
4 a Salaries of women teaching in a school range from $20 106 to $63579. Group the salaries
into five equal-sized categories that cover all teaching salaries.
b The number of students in VCE Further Mathematics classes ranges from 6 to 33. Group
the class sizes into six equal-sized categories that cover all Further Mathematics class
sizes.
c The amount of money carried by a sample of 23 students ranges from nothing to $8.75.Group the amount of money carried by the students into five equal-sized categories that
cover all amounts of money carried by the students.
05 10 15 20 25 30
5
10
15
20
25
30
35
Frequency(%)
Number of words in sentence
5The histogram opposite was formed by recording the
number of words in 30 randomly selected sentences.
a What percentage of these sentences contained:
i 59 words? ii 2529 words?
iii 1019 words? iv fewer than 15 words?
Give answers correct to the nearest per cent.
b How many of these sentences contained:
i 2024 words? ii more than 25 words?
c What is the mode (modal interval)?
6Use the information in the table opposite to
help you construct a histogram to display
population density. Use the histogram in
Example 6 as a model. Label axes and
mark in scales.
Population density Frequency(count)
0199 11
200399 4
400599 4
600799 2
800999 1
Total 22
7Use the information in the table opposite to
help you construct a histogram to display the
distribution of the number of rooms in the
houses of 11 preschool children. Use the
histogram in Example 7 as a model. Label
axes and mark in scales.
Number of rooms Frequency(count)
4 3
5 0
6 1
7 3
8 4
Total 11
8The pulse rates of 23 students are given below.
86 82 96 71 90 78 68 71 68 88 76 74
70 78 69 77 64 80 83 78 88 70 86
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20 Essential Further Mathematics Core
a Use a graphics calculator to construct a histogram so that the first column starts at 63 and
the column width is two.
b For this histogram:
i what is the starting point of the third column?
ii what is the count for the third column? Whatactualdata values does this include?
c Redraw the histogram so that the column width is five and the first column starts at 60.d For this histogram, what is the count in the interval 65 to
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Chapter 1 Organising and displaying data 21
Symmetric distributions
If a histogram is single-peaked, does the histogram region tail off evenly on either side of the
peak? If so, the distribution is said to besymmetric(see Histogram 1).
Histogram 1
lower tail peak upper tail10
86
4
2
0
Frequency
Histogram 2
peak peak10
86
4
2
0
Frequency
A single-peaked symmetric distribution is characteristic of the data that derive from
measuring variables such as peoples heights, intelligence test scores, weights of oranges in a
storage bin, or any other data for which the values vary evenly around some central value. The
histogram for average hours worked (see Example 6) would be classified as approximately
symmetric.
The double-peaked distribution (Histogram 2) is symmetric about the dip between the two
peaks. A histogram that has two distinct peaks indicates abimodal(two modes) distribution.
A bimodal distribution often indicates that the data have come from two different
populations. For example, if we were studying the distance the discus is thrown by Olympic
level discus throwers, we would expect a bimodal distribution if both male and female throwers
were included in the study.
Skewed distributionsSometimes a histogram tails off primarily in one direction. Such distributions are said to be
skewed.
If a histogram tails off to the right we say that it is positivelyskewed (Histogram 3). The
distribution of salaries of workers in a large organisation tends to be positively skewed. Most
workers earn a similar salary with some variation above or below this amount, but a few earn
more and even fewer, such as the senior manager, earn even more. The distribution of house
prices also tends to be positively skewed.
Histogram 3
peak long upper tail10
8
6
4
2
0
Frequency +ve skew
Histogram 4
long lower tail peak 10
8
6
4
2
0
Frequency ve skew
If a histogram tails off to the left we say that it is negativelyskewed (Histogram 4). The
distribution of age at death tends to be negatively skewed. Most people die in old age, a few in
middle age and even fewer in childhood.
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22 Essential Further Mathematics Core
OutliersOutliersare any data values that stand out from the main body of data. These are data values
that are atypically high or low. See for example, Histogram 5, which shows an outlier. In this
case it is a data value that is atypically low compared to the rest of the data values.
Histogram 5
outlier main body of data
108
6
4
2
0
Frequency
Outliers can indicate errors made collecting
or processing data; for example, a persons
age recorded as 365. Alternatively, they may
indicate data values that are very different
from the rest of the values. For example,
compared to her students ages, a teachers
age is an outlier.
Centre
050
Histograms 6 to 8
60 70 80 90 100 110 120 130 140 150
1
2
3
4
5
6
7
8
Frequency
Histograms 6 to 8 display the distribution
of test scores for three different classes
taking the same subject. They are identical
in shape, but differ in where they are
located along the axis. In statistical terms
we say that the distributions are centred
at different points along the axis.
But what do we mean by the centreof a
distribution? This is an issue we will return
to in more detail later. For the present we
will take centre to be themiddleof the
distribution.
The middle of a symmetric distribution is reasonably easy to locate by eye. Looking at
Histograms 6 to 8, it would be reasonable to say that the centre or middle of each distribution
lies roughly halfway between the extremes; half the observations would lie above this point
and half below. Thus we mightestimatethat Histogram 6 (yellow) is centred at about 60,
Histogram 7 (light blue) at about 100, and Histogram 8 (dark blue) at about 140.
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Chapter 1 Organising and displaying data 23
For skewed distributions, it is more difficult to estimate the middle of a distribution by eye.
The middle is not halfway between the extremes because, in a skewed distribution, the scores
tend to bunch up at one end. However, if we
imagine a cardboard cut-out of the histogram,
the midpoint lies on the line that divides the
histogram into two equal areas (Histogram 9).
Histogram 9
line that divides
the area of the
histogram in half
150
1
2
3
4
5
20 25 30 35 40 45 50
Frequenc
y
Using this method, we would estimate the
centre of the distribution to lie somewhere
between 35 and 40, but closer to 35, so we
might opt for 37. However, remember that
this is only an estimate.
SpreadIf the histogram is single peaked, is it narrow? This would indicate that most of the data values
in the distribution are tightly clustered in a small region. Or is the peak broad? This would
indicate that the data values are more widely spread out. Histograms 10 and 11 are both single
peaked. Histogram 10 has a broad peak, indicating that the data values are not very tightly
clustered about the centre of the distribution. In contrast, Histogram 11 has a narrow peak,
indicating that the data values are tightly clustered around the centre of the distribution.
wide central region10
8
6
4
2
02 4 6 8 10 12 14 16 18 20 22
Frequen
cy
Histogram 10
narrow central region20
16
128
4
2 4 6 8 10 12 14 16 18 20 220
Frequen
cy
Histogram 11
But what do we mean by the spreadof a distribution? We will return to this in more detail
later. For a histogram we will take it to be the maximumrangeof the distribution.
Range
Range =largest value smallest value
For example, Histogram 10 has a spread (maximum range) of 22 (22 0) units, which is
considerably greater than the spread of Histogram 11, which has a spread of 12 (18 6) units.
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24 Essential Further Mathematics Core
Example 8 Describing a histogram in terms of shape, centre and spread
The histogram opposite shows the distribution of the
number of phones per 1000 people in 85 countries.
a Describe its shape and note outliers (if any).b Locate the centre of the distribution.
c Estimate the spread of the distribution.
0170 340 510 680 850 1020
5
10
15
20
25
30
35
Frequency(count)
Number of phones (per 1000 people)Solution
a Shape and outliers
b CentreCount up the frequencies fromeither end to find the middle interval.
c SpreadUse the maximum range to
estimate the spread.
The distribution is positively skewed.
There are no outliers.
The distribution is centred in the interval170340 phones/1000 people.Spread = 1020 0
= 1020 phones/1000 people
It should be noted that, with grouped data, it is difficult to precisely determine the location of
the centre of a distribution from a histogram. So, when working with grouped data, it is
acceptable to state that the centre of a distribution lies in the interval 170340. We will learn
how to solve this problem later in the chapter.
If you were using the histogram above to describe the distribution in a form suitable for astatistical report, you might write as follows.
ReportFor the 85 countries, the distribution of the number of phones per 1000 people is positively
skewed. The centre of the distribution lies somewhere in the interval 170340 phones/1000
people. The spread of the distribution is 1020 phones/1000 people. There are no outliers.
Exercise 1D
1Label each of the following histograms as approximately symmetric, positively skewed or
negatively skewed, and identify the following:
i the mode ii any potential outliers iii the approximate location of the centre
a
Frequency
Histogram A
20
15
10
5
0
b
Frequency
Histogram B
80
60
40
20
0
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Chapter 1 Organising and displaying data 25
c
Frequency
Histogram C
20
15
10
5
0
d 20
15
10
0
5
Histogram D
Frequency
2These three histograms show
the marks obtained by a group
of students in three subjects.
Frequency
Subject A Subject B
Marks
Subject C
1
2
3
4
5
6
7
8
9
10
02 6 10 14 18 22 26 30 34 38 42 46 50
a Are each of the distributions
approximately symmetric or
skewed?
b Are there any clear outliers?
c Determine the interval
containing the central mark
for each of the three subjects.
d In which subject was the
spread of marks the least? Use
the range to estimate the spread.
e In which subject did the marks vary most? Use the range to estimate the spread.
3Label each of the following histograms as approximately symmetric, positively skewed or
negatively skewed, and identify the following:i the mode(s) ii any potential outliers iii the approximate location of the centre
a
Histogram A
20
15
10
5
0
Frequency
b
Histogram B
Frequency80
60
40
20
0
c20
15
10
5
0
Frequency
Histogram C
d
Frequen
cy
Histogram D
15
20
10
5
0
e
Frequency
Histogram E
15
20
10
5
0
f
Frequency
Histogram F
80
60
40
20
0
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26 Essential Further Mathematics Core
4This histogram shows the distribution
of pulse rate (in beats per minute) for
28 students.
0
1
2
3
4
5
6
2
3
4
5
6
Pulse rate (beats per minute)
60 65 70 75 80 85 90 95 100 105 110 115
Frequency(count)
Use the histogram to complete the report
below, describing the distribution of
pulse rate in terms of shape, centre,
spread and outliers (if any).
ReportFor the students, the distribution of pulse rates is with an outlier. The
centre of the distribution lies in the interval beats per minute and the spread of the
distribution is beats per minute. The outlier lies in the interval beats per minute.
1.5 Stem-and-leaf plots and dot plotsStem plotsAstem-and-leaf plot, orstem plotfor short, is an alternative to the histogram. It is
particularly useful for displaying small to medium sized sets of data (up to about 50 data
values) and has the advantage of retaining all the original data values. This makes it useful for
further computations. A stem plot is also a very quick and easy way to order and display a set
of data by hand. Like a histogram, the stem plot gives information about the shape, outliers,
centre and spread of the distribution.
One of the stem plots advantages over a histogram in describing distributions is being able
to see all the actual data values. This enables the centre and the range of the distribution to be
located more precisely. It also enables the clear identification of outliers.
Constructing a stem plot
In a stem-and-leaf plot, each data value is separated into two parts: the leading digit(s) form
the stem, and the trailing digit becomes the leaf. For example, in a stem-and-leaf plot, the
data values 25 and 132 are represented as follows:
25 is represented by
132 is represented by
Stem Leaf
2 5
13 2
and so on.
To construct a stem plot, enter the stems to the left of a vertical dividing line, and the leaves
for each data point to the right. Usually we first construct an unordered stem plotby
systematically plotting each data point as listed in the data set. From the unordered
stem-and-leaf plot anordered stem plotis then easily obtained. In an ordered stem plot the
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Chapter 1 Organising and displaying data 27
leaves increase in value as they move away from the stem. It is usually the ordered stem plot
that we want, because an ordered stem plot makes it easy to find the key values.
Example 9 Constructing an ordered stem plot
University participation rates (%) in 23 countries are given below.2 6 3 1 2 2 0 3 6 1 2 5 2 6 1 3 9 2 6 2 7 3 0 1 1 5 2 1 7 8 2 2 3 3 7 1 7 5 5
Display the data in the form of an ordered stem plot.
Solution
1 The data set has values in the units, tens,
twenties, thirties, forties and fifties. Thus,
appropriate stems are 0, 1, 2, 3, 4, and 5.
Write these down in ascending order,
followed by a vertical line.
01234
52 Now attach the leaves.
The first data value is 26. The stem is 2
and the leaf is 6. Opposite the 2 in the stem,
write down the number 6, as shown.
The second data value is 3 or 03. The stem
is 0 and the leaf is 3. Opposite the 0 in the
stem, write down the number 3, as shown.
012 6345
0 312 6
345
Continue systematically working through
the data following the same procedure until
all points have been plotted. You will then
have the unordered stem plot, as shown.
0 3 1 9 1 7 8 31 2 3 5 72 6 0 5 6 6 7 1 2 3 6 0 745 5unordered stem plot
3 Ordering the leaves in increasing value asthey move away from the stem gives the
ordered stem plot, as shown.
0 1 1 3 3 7 8 91 2 3 5 72 0 1 2 5 6 6 6 73 0 6 745 5
ordered stem plot
Using a stem plot to describe a distribution
Stem plots are just like histograms, except that you can see all the data values. This enablesmore precise estimates to be made of the centre and spread.
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28 Essential Further Mathematics Core
Methods for determining the centre, spread and outliers from a stem plot
Centre (middle) Count up from either end of the distribution until you find the middle
value; the value that has an equal number of data values either side.
For an odd number of data values,n, the middle value is then + 1
2 th
value. Thus, the median will be an actual data value.
For an even number of data values,n, the middle value is then + 1
2 th
value. Thus, the median will lie between two data values.
Spread (range) Subtract the smallest data value from the largest data value.
Range =largest value smallest value
Outliers Data values that stand out from the main body of data are called outliers.
Their values can be read directly from the stem plot.
Example 10 Describing a stem plot in terms of shape, centre and spread
The ordered stem plot opposite shows the
distribution of test marks of 23 students.
a Name its shape and note outliers (if any).
b Locate the centre of the distribution.
c Estimate the spread of the distribution.
d Write down the values of any outliers.
Test marks
0
1 5 9 9 9
2 0 4 5 7 8 8 8
3 0 3 5 5 6 8
4 1 2 3 3 5
5
6 0Solution
a Shape
b CentreThere are 23 data values; the middle
value is the 12th value. Check by counting.
The distribution is approximately
symmetric with one outlier.
The distribution is centred at 30 marks.
c SpreadUse the range to estimate the spread.
d OutlierRead off the value of the outlier.
Spread = 60 15 = 45 marks
Outlier= 60 marks
If you were using the stem plot to describe the distribution in a form suitable for a statistical
report, you might write as follows.
ReportFor the 23 students, the distribution of marks is approximately symmetric with an outlier.
The centre of the distribution is at 30 marks and the distribution has a spread of 45
marks. The outlier is a mark of 60.
Split stemsIn some instances, using the simple process outlined above produces a stem plot that is too
bunched up to give us a good overall picture of the variation in the data. This is often the case
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Chapter 1 Organising and displaying data 29
when the data values all have the same first digit or the same one or two first digits. For
example, a group of 17 VCE students recently sat for a statistics test marked out of 20. The
results are as shown below.
2 12 13 9 18 17 7 16 12 10 16 14 11 15 16 15 17
Using the process described in Example 10 to form a stem plot, we end up with abunched-up plot like the one below.
0 2 7 9
1 0 1 2 2 3 4 5 5 6 6 6 7 7 8
When this happens, the stem plot scale can be stretched out by splitting the stems. Generally
the stem is split into halves or fifths. For example, for the interval 1019, the split stem system
works as follows.
1 (1011)
1 (1213)1 (1415)
1 (1617)
1 (1819)
1 (1014)
1 (1519)1 (1019)
Single stem Stem split into halves Stem split into fifths
In a stem plot with a single stem, the 1 represents the interval 1019.
In a stem plot with its stem split into halves, the top 1 represents the interval 1014,
while the bottom 1 represents the interval 1519.
In a stem plot with its stem split into fifths, the top 1 represents the interval 1011, the
second 1 represents the interval 1213, the third 1 represents the interval 1415, the
fourth 1 represents the interval 1617, while the bottom 1 represents the interval 1819.
Comparison of stem plots with different split stems
Using a split stem plot to display the test marks can show features not revealed by a standard
plot. This can be seen in the next plot with the stem split into fifths, indicating that a mark of 2
is an outlier.
0 2 7 9 0 2 0
1 0 1 2 2 3 4 5 5 6 6 6 7 7 8 0 7 9 0 2
1 0 1 2 2 3 4 01 5 5 6 6 6 7 7 8 0 7
0 9
1 0 1
1 2 2 3
1 4 5 5
1 6 6 6 7 7
1 8
Single stem Stem split into halves Stem split into fifths
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30 Essential Further Mathematics Core
Back-to-back stem plotsA back-to-back plot has a single stem with
two sets of leaves as shown.8 6 4 2 1 1 1 5 6 8 9 8
1 0 2 1 1 1 4
3 2 2 3 1 1 5The real value of back-to-back plots is
that they are a useful tool for comparing
the distribution of two sets of data values for the same variable.
Example 11 Using a back-to-back stem plot to compare two distributions
Use the back-to-back stem
plot to write a report
comparing the distribution
of the two sets of test
marks in terms of shape,
centre, spread and outliers.
Test 1 Test 2
0 8
9 9 1 9
8 6 6 5 0 2 0 4 5 7 8 8
9 8 7 6 5 5 0 3 0 3 5 5 6 8 9
8 7 5 3 3 2 2 0 4 1 2 3 3 4 55 0
Solution
ReportThe distribution of the Test 1 marks is negatively skewed while the distribution of the
Test 2 marks is approximately symmetric. The two distributions have similar centres;
36.5 and 35. The spread of the Test 1 marks is less than the Test 2 marks; 29
compared to 42. There are no outliers.
Dot plotsThe simplest way to display numerical data is to form a dot plot. A dot plot consists of a
number line with each data point marked by a dot. When several data points have the same
value, the points are stacked on top of each other. Like stem plots, dot plots are a great way of
displaying small data sets and have the advantage of being very quick to construct by hand.
They are best when the data values are relatively close together.
Example 12 Constructing a dot plot
The ages (in years) of the 13 members of a sporting team are:
22 19 18 19 23 25 22 29 18 22 23 24 22
Construct a dot plot.
Solution
1 Draw in a number line, scaled to include all
data values. Label the line with the variable
being displayed.
Age(years)
17 18 19 20 21 22 23 24 25 26 27 28 29 30
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Chapter 1 Organising and displaying data 31
2 Plot each data value by marking in a dot
above the corresponding value on the
number line.17 18 19 20 21 22 23 24 25 26 2 7 28 29 3 0
Age(years)
Interpreting a dot plot
Dot plots are interpreted in much the same way as stem plots. However, usually there is little
we can say about the shape of the distribution from the dot plot because there are not sufficient
data points for any pattern to be revealed.
From the dot plot in Example 12, we see that the distribution of ages is centred at 22 years
(the middle value) with a spread of 11 years (29 18 = 11).
Which graph?One of the issues that you will face is choosing a suitable graph to display a distribution. The
following guidelines might help you in your decision-making. They are guidelines only,because in some instances there may be more than one suitable graph.
Type of data Graph Qualifications on use
Categorical Bar chart
Segmented bar chart Not too many categories (4 or 5 maximum)
Numerical Histogram Best for medium to large data sets (n 40)
Stem plot Best for small to medium sized data sets (n 50)
Dot plot Suitable only for small data sets (n 20)
Exercise 1E
1The data below give the urbanisation rates (%) in 23 countries.
54 99 22 20 31 3 22 9 25 3 56 12
16 9 29 6 28 100 17 9 35 27 12
a Construct an ordered stem plot.
b What advantage does a stem plot have over a histogram?
2For each of the following stem plots (A, B and C):
a name its shape and note outliers (if any)
b locate the centre of the distribution
c determine the spread of the distribution
d write down the values of outliers (if any)
Stem plot A Stem plot B Stem plot C
0 0 0 1 1 2 6 7 7 9 0 0 0 1 3
1 2 2 3 5 5 5 5 6 1 1 3 6 9 1
2 0 1 4 7 2 0 0 1 5 6 8 8 2
3 2 2 3 2 2 2 4 5 9 9 9 3 2
4 0 4 1 2 4 4 6 4 0 2 4
5 2 5 2 3 5 1 1 3 5 8 86 6 2 6 0 0 4 4 4 7 7 8 9
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32 Essential Further Mathematics Core
3The data below give the wrist circumference (in cm) of 15 men.
16.9 17.3 19.3 18.5 18.2 18.4 19.9 16.7 17.1 17.6 17.7 16.5 17.0 17.2 17.6
a Construct a stem plot for wrist circumference using:
i stems 16, 17, 18, 19 ii these stems split into halves
b Which stem plot appears to be more appropriate for the data?
c Use the stem plot with split stems to help you complete the report below.
ReportFor the men, the distribution of their wrist circumference is . The centre of
the distribution is at cm and it has a spread of cm. There are no outliers.
4The data below give the weight (in kg) of 22 students.
57 58 62 84 64 74 57 55 56 60 75
68 59 72 110 56 69 56 50 60 75 58a Construct a stem plot for weight using:
i stems 5, 6, 7, 8, 9, 10 and 11 ii these stems split into halves
b Use the stem plot with a split stem to write a brief report on the distribution of the
weights of the students in terms of shape (and outliers), centre and spread. Use the report
from Question 3 as a model.
5The number of possessions (kicks, mark, handballs, knockouts etc.) recorded for players in a
football game between Carlton and Essendon is shown below.
Carlton Essendon
10 44 32 44 19 35 11 5 24 28 21 32 21 59 21 12 19 26 23 22 29 34
22 34 36 20 14 25 16 19 32 32 14 29 8 22 21 26 44 19 21 22
a Display the data in the form of anorderedback-to-back stem plot.
b Complete the following report comparing the two distributions in terms of shape (and
outliers), centre and spread.
ReportThe distribution of the number of possessions is for both teams. The two
distributions have similar centres, at and possessions, respectively. The spread ofthe distribution is less for Carlton, possessions, compared to possessions for
Essendon.
6The following data give the number of children in the families of 14 VCE students:
1 6 2 5 5 3 4 4 2 7 3 4 3 4
a Construct a dot plot.
b What is the mode?
c What is:i the centre? ii the spread?
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Chapter 1 Organising and displaying data 33
7The following data give the life expectancies in years of 13 countries:
76 75 74 74 73 73 75 71 72 75 75 78 72
a Construct a dot plot.
b What is the mode?
c What is:
i the centre? ii the spread?
8Data have been collected for each of the following variables. The data are to be displayed
graphically. In each case, decide which is the most appropriate graph. Select from bar chart,
histogram, stem plot or dot plot. Sometimes more than one sort of graph is suitable.
a number of passengers in a bus 1000 buses in sample
b amount of petrol purchased (in litres) 30 petrol purchases
c type of petrol purchased (super, unleaded, premium)
d prices of houses sold in Melbourne over a weekend
e the number of medals won by countries winning medals at the Olympicsf state of residence of a sample of 200 Australians
g number of cigarettes smoked in a day (a sample of 120 people)
h resting pulse rates of 7 students
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Re
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34 Essential Further Mathematics Core
Key ideas and chapter summary
Types of data Data can be classified asnumericalorcategorical.
Frequency table Afrequency tableis a listing of the values a variable takes in a data set,
along with how often (frequently) each value occurs.Frequency can be recorded as a: count: the number of times a value occurs; for example, the number
of females in the data set is 32 per cent: the percentage of times a value occurs; for example, the
percentage of females in the data set is 45.5%.
Categorical data Categorical dataarise when classifying or naming some quality or
attribute; for example, place of birth, hair colour.
Bar chart Bar chartsare used to display the frequency distribution of categorical
data.
For a small number of categories, the distribution of a categorical
variable is described in terms of the dominant category(if any), the
orderof occurrence of each category and itsrelative importance.
Describing
distributions of
categorical variables
Mode Themodeis the value or group of values that occurs most often
(frequently) in a data set. For example, for the data 2 1 1 3 3 2 5 1 6 1 1
2 1 1, the mode is 1, because it is the data value that occurs most often.
Numerical data Numerical dataarise from measuring or counting some quantity; for
example, height, number of people etc.Numerical data can be discrete or continuous.Discrete dataarise when
youcount.Continuous dataarise when youmeasure.
Histogram Ahistogramis used to display the frequency distribution of a numerical
variable; suitable for medium to large sized data sets.
Stem plot Astem plotis an alternative graphical display to the histogram; suitable
for small to medium sized data sets.
The advantage of the stem plot over the histogram is that it shows the
value of each data point.
Dot plot Adot plotconsists of a number line with each data point marked by a
dot; suitable for small sets of data only.
The distribution of a numerical variable can be described in terms of:Describing thedistribution of a
numerical variable
shape: symmetric or skewed (positive or negative)? outliers: values that appear to stand out centre: the midpoint of the distribution (median) spread: one measure is the range of values covered
(Range = largest value smallest value)
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Chapter 1 Organising and displaying data 35
Skills check
Having completed this chapter you should be able to:
differentiate between numerical and categorical data
interpret the information contained in a frequency table
identify and interpret the mode
construct a bar chart or histogram from a frequency table
decide when it is appropriate to use a histogram rather than a bar chart and vice
versa
construct a histogram from raw data, using a graphics calculator
construct a dot plot and a stem plot from raw data, using split stems if required
locate the mode of a distribution from a histogram, stem plot, dot plot or bar chart
recognise a symmetric, positively skewed and negatively skewed histogram or stem
plot
identify potential outliers in a distribution from its histogram or stem plot
write a brief report to describe the distribution of a numerical variable in terms of
shape, centre, spread and outliers (if any)
write a brief report to describe the distribution of a categorical variable in terms of
the dominant category (if any), the order of occurrence of each category and their
relative importance.
Multiple-choice questions
The following information relates to Questions 1 to 3
A survey collected information about the number of cars owned by a family and the car
size (small, medium, large).
1 The variablesNumber of carsowned and carSizeare:
A both categorical variables B both numerical variables
C a categorical and a numerical variable respectively
D a numerical and a categorical variable respectively
E neither numerical nor categorical variables
2 To graphically display the information about car size you could use a:
A dot plot B stem plot C histogram
D segmented bar chart E back-to-back stemplot
3 TheNumber of carsowned is:
A a continuous numerical variable B a discrete numerical variable
C a continuous categorical variable D a discrete categorical variable
E none of the above
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36 Essential Further Mathematics Core
The following information relates to Questions 4 to 6
A number of teenagers were asked to
nominate their favourite leisure
activity. Their responses have been
organised into a frequency table, asshown. Some information is missing.
Frequency
Leisure activity Count Percentage
Sport 73 29.2
Listening to music 70
Watching TV 19.2
Other 59 23.6
Total 250
4 The percentage of students who said that listening to music was their favourite
leisure activity is:
A 17.5 B 28.0 C 29.2 D 50.0 E 70.0
5 The number of students who said watching TV was their favourite leisure activity
is:
A 19 B 48 C 62 D 125 E 70.0
6 For the students surveyed, the most popular leisure activity is:
A sport B listening to music C watching TV
D other E cant tell
Questions 7 to 11 relate to the histogram shown below
This histogram displays the test scores of a class
of Further Mathematics students.
6 8 10 12 14 16 18 20 22 24 26 28
Test score
6
5
43
2
1
0
Freque
ncy
7 The total number of students in the class is:
A 6 B 18 C 20 D 21 E 22
8 The number of students in the class who
obtained a test score less than 14 is:
A 4 B 10 C 14 D 17 E 28
9 The histogram is best described as:
A negatively skewed B negatively skewed with an outlier
C approximately symmetric
D approximately symmetric with outliers E positively skewed10 The centre of the distribution lies in the interval:
A 810 B 1012 C 1214 D 1416 E 1820
11 The spread of the students marks is:
A 8 B 10 C 12 D 20 E 22
12 For the stem plot shown opposite, the modal interval is:
A 2024 B 2529 C 2029
D 25 E 29
1 0 2
1 5 5 6 9
2 3 3 4
2 5 7 9 9 9
3 0 1 2 4
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Chapter 1 Organising and displaying data 37
The following information relates to Questions 13 and 14
This percentage segmented bar chart
shows the distribution of hair
colour for 200 students.
0
10
2030
4050
60
7080
90100
Percentage
Blonde
Brown
Black
Red
Other
13 The number of students withbrown hair is closest to:
A 4 B 34 C 57
D 68 E 114
14 For these students, the most common
hair colour is:
A black B blonde C brown D red E other
15 The ages of 11 primary school children were collected. The best graph to display
the distribution of ages of these children would be a:
A bar chart B dot plot C histogram
D segment bar chart E stem plot
Extended-response questions
1 One hundred and twenty-one students were
asked to identify their preferred leisure activity.
The results of the survey are displayed in a
bar chart.
30
25
20
15
10
5
0
Sport TV
Preferred leisue activity
Music
Movies
Reading
Other
Percen
tage
a What percentage of students nominated
watching TV as their preferred leisure
activity?
b What percentage of students in total
nominated either going to the movies or
reading as their preferred leisure activity?
c What is the most popular leisure activity for
these students? How many students rated this
activity as their preferred leisure activity?2 The number of people killed in natural and non-natural disasters in 1997 by world
region is shown in the table below.
a Construct a bar chart.
b In which region was the:
i greatest number of people killed?
ii least number of people killed?
Region Number killed
Europe 874
Africa 8 327
Asia 10 551
Oceania 457
The Americas 1581
includes Australia (41)
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38 Essential Further Mathematics Core
3 A group of 52 teenagers was asked
Do you agree that the use of marijuana
should be legalised? Their responses
are summarised in the table opposite.
Frequency
Legalise Count Per cent
Agree 18
Disagree 26
Dont know 8Total 52
a Construct a properly labelled and scaledfrequency bar chart for the data.
b Complete the table by calculating the
appropriate percentages, correct to one decimal place.
c Use the percentages to construct a percentage segmented bar chart for the data.
d Use the frequency table to help you complete the following report.
Report: In response to the question, `Do you agree that the use of marijuana
should be legalised?', 50% of the 52 students . Of the remaining
students, % agreed, while % said that they .
4 The table below gives the distribution of the number of children in 50 families.
Number of children Frequency
in family Count Per cent
0 5 10
1 6
2 19 38
3 7 14
4
5 2 46 3 6
7 0 0
8 1 2
Total 50 100
a Is the number of children in a
family a numerical or categorical
variable?
b Write down the missing information.
c What is the mode?
d Determine the number of
families with:
i three children
ii two or three children
iii less than three children
e Determine the percentage of
families with:
i six children
ii more than six children
iii less than six children
5 Students were asked how much they
spent on entertainment each month. The
results are displayed in the histogram.
Use this information to answer the
following questions.
90 100 110 120 130 140
Amount ($)
10
8
6
4
2
0
Frequency
a How many students:
i were surveyed?
ii spent $100105 per month?
b What is the mode?
c How many students spent $110 or more per month?
d What percentage spent less than $100 per month?
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Chapter 1 Organising and displaying data 39
e i Name the shape of the distribution displayed by the histogram.
ii Locate the interval containing the centre of the distribution.
iii Determine the spread of the distribution using the range.
6 This stem plot displays the ages (in years) of a group of women.
a What was the age of the youngest woman?
Note: 17 2 = 17.2 years
17 2 3 4
17 5 6 6 8 8 9 9
18 0 1 3 3 3 4
18 5 5 5 5 5 5 6 7 8 8 8 9
19 1 2 2 3 3
19 8
20
20 6
b In terms of age, one of the women is a
possible outlier. What is her age?
c How many women were aged between
17.0 and 17.4 years, inclusive?
d How many women were 19 years old
or older?
e What is the modal age category?
f What percentage of women were younger
than 20 years old?
g i Name the shape of the distribution
of ages, noting outliers.
ii Locate the centre of the distribution.
iii Determine the spread of the distribution.
7 The distribution of the waiting times of 37 cars
stopped by a traffic light is as shown in
the histogram opposite. Use the histogram to
write a report on the distribution of waiting
times in terms of shape, centre, spread
and outliers.
5 10 15 20 25 30 35 40 45 50 55
Waiting time (seconds)
10
8
6
4
2
0
Freque
ncy
8 Use a graphics calculator to construct histograms for the following sets of data.
a Use intervals of width 5 starting at 90.
Monthly expenditure on entertainment (in dollars)
110 115 105 98 118 114 125 95 114 104 97 130 122
112 107 135 121 94 108 118 106 121 125 107 109 93
b Use intervals of width 8 starting at 32.
Life span (in years)
58 65 68 74 73 73 75 71 72 61 67 66 37