Part 1 Histograms and polygons -...

Mathematics Stage 5

DS5.1.1 Data representation and analysis

Part 1 Histograms and polygons

Number: 43670 Title: DS5.1.1 Data Representation and Analysis (5.1)

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Part 1 Histograms and polygons 1

Contents – Part 1

Introduction – Part 1..........................................................3

Indicators ...................................................................................3

Preliminary quiz.................................................................5

Tallies and tables ..............................................................7

Histograms and polygons................................................11

Dot plots...................................................................................13

Cumulative frequency tables ...........................................15

Cumulative frequency diagrams......................................19

Mean, median, mode and range .....................................23

Suggested answers – Part 1 ...........................................29

Exercises – Part 1 ...........................................................35

2 DS5.1.1 Data representation and analysis


Introduction – Part 1

This part follows on from work begun in stage 4. You are referred to

DS4.2 Data analysis and evaluation for a review of statistical data.

This earlier work forms a basis from which this stage 5 material develops

and you need to be familiar with the main concepts presented there.

In this part you will group data to aid analysis and construct frequency

and cumulative frequency tables and graphs.

Indicators

By the end of Part 1, you will have been given the opportunity to work

towards aspects of knowledge and skills including:

• constructing a cumulative frequency table for ungrouped data

• constructing a cumulative frequency histogram and polygon (ogive).

By the end of Part 1, you will have been given the opportunity to work

mathematically by:

• constructing frequency tables and graphs from data obtained from

different sources

• reading and interpret information from a cumulative frequency table

or graph.Source: Extracts from outcomes of the Mathematics Years 7–10 syllabus

<www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_710_syllabus.pdf > (accessed 04 November 2003).© Board of Studies NSW, 2002.


Preliminary quiz

Before you start this part, use this preliminary quiz to revise some skills

you will need.

Activity – Preliminary quiz

Try these.

1 The dot plot shows the pulse rate (beats per minute) after exercise

for a group of runners.

130Post exercise pulse rates140 150 160 170

a How many runners were observed? ______________________

b Which pulse rate occurred most often? ____________________

c What is the range in values for these pulse rates? ___________

d What percentage of runners had a pulse rate of 160 or higher?

___________________________________________________

e Calculate the mean pulse rate, to the nearest integer.

___________________________________________________

2 Find the average of the following.

a 4, 7, 1, 0, 6 _________________________________________

b 5.6, 7.8, 3.1, 5.5 _____________________________________


3 What fraction is midway between 12

and 34

. (Average the numbers.)

_______________________________________________________

4 a Complete the frequency column for this frequency table.

Score Tally Frequency

0

1

2

3

4

5

6

7

b Comment on these two statements.

The table shows eight scores.

No I don’t think so. Thirtyscores are recorded.

___________________________________________________

___________________________________________________

___________________________________________________

Check your response by going to the suggested answers section.


Tallies and tables

In your Stage 4 course you collated loose data into tables, as it is easier to

make sense of it this way. This is especially true when you have a lot of

data. In this, and the next few sessions, you will review these ideas and

extend them.

In this session you will look at ways to organise data. It may be collected

then organised, or organised as it is collected. For the activities in this

session you will be using the following scores.

Forty homes in a town are chosen at random. A researcher asked each

householder how many children were in the home. The numbers of

children were written down as families were interviewed. They were:

1 2 0 1 2 3 2 2 1 2

2 4 3 2 0 2 0 2 2 3

7 1 2 2 1 2 4 1 2 1

2 3 1 1 0 4 3 3 1 2

The number of children in each home is called the score.


Activity – Tallies and tables

Try these.

1 Sort the scores from the previous page in increasing order then list

the scores in order. (Circle, or lightly cross out, each score as you

put it in the list below.) The first five have been done for you.

0 0 0 0 1

2 Range = highest score – lowest score. What is the range of the

scores? ________________________________________________


You can also organise the data into a table by taking a tally when scores

occur. You can place a tally mark ( ) for each recorded score next to

that score in the tally column. Draw every fifth stroke through the

previous four as shown as shown to form a group of five to make

counting easier.



Try these.

3 Complete the frequency distribution table for the above scores.

(The first row of the raw scores has been done for you.)


0

1

2

3

4

5

6

7

(Notice that the scores for 5 and 6 have been included, even though

they do not occur in this group. This is because they are possible

scores. The scores go as high as 7.)

4 Count tally marks for each score to find the frequency of each score.

Write them in the third column of the table.

5 Did you find that the frequency of the score 4 was three?

What does this mean?

_______________________________________________________

_______________________________________________________


A table that lists the frequencies of a set of scores is a frequency

distribution table. It shows the distribution of the scores.


From this investigation you can see:

• the scores range from 0 to 7, so the range is 7 – 0 = 7

• the most frequent score (the mode) is 2. (More families had 2

children than any other number)

• one family with 7 children is separated from the main body of

scores. Such a score that is separated from the main body of scores

is called an outlier.


Try these.

6 Use the data in the table used in the previous activity about family

sizes to answer the questions below.

a How many families surveyed had no children? _____________

b How many families had 5 children? ______________________

c What is the sum of the numbers in the frequency column? ____

d How many families were in the survey? __________________

e What is the highest score in the survey? __________________

f What is the greatest number of children in any family in this

survey? ____________________________________________

g Which score has the highest frequency? __________________

The symbol Σ (pronounced sigma) means the sum of. Σf is therefore the

sum of the frequencies that is calculated by adding all the frequencies up.


You have been practising recording loose data in frequency tables.

Now check that you can do these kinds of problems by yourself.

Go to the exercises section and complete Exercise 1.1 – Tallies and tables.


Histograms and polygons

Tables can give detailed and accurate information about any population.

You need to read the whole table to get a full picture of what it is telling

you. Graphs are a useful way of visually seeing things quickly.

In your Stage 4 course you have studied and made histograms to

represent data. This section revisits that learning.

Here is the frequency table about family sizes from the earlier activity.


0 4

1 10

2 16

3 6

4 3

5 0

6 0

7 1

Σf = 40

0

2

4

6

8

10

12

14

16

Freq

uenc

y

0 1 2 3 4 5 6 7Score (number of children)

Number of children in homes

On the right is a matching frequency histogram. In statistics it is usual

to let x represent a score and f represent the frequency. A histogram is a

statistical column graph. Each possible score is given even space across

the x-axis.

Each score adds one division to the column for that score so that the

height for each score gives its frequency. A properly drawn frequency

histogram should have a title.


This information can also be

shown using a frequency

polygon. This is a line graph

showing the frequencies of the

various scores.

Notice that the crosses for the

heights are still directly above the

scores numbers.

At each end of a polygon, lines

are drawn to the x-axis to show

these are all the scores.0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7

Number of children in homes f

x

Score, x, (number of children)

Activity – Histograms and polygons

Try these.

1 This histogram shows the distribution of marks achieved by a group

of year 9 students in a Maths test.

a Draw a frequency polygon to match the histogram.

3 4 5 6 7 8 9 10

2

0

4

6

8

10

Number of marks in a test

3 4 5 6 7 8 9 10

2

0

4

6

8

Number of marks in a testf

x

f

x

10

b What is the range of marks? ____________________________

c What is the most common mark? ________________________

d How many students got only 3 marks? ___________________


e How many students got 4 marks? ________________________

f How many students got 8 marks? ________________________

g What is the total number of students in all groups? __________


Dot plots

Another useful and quick way to display information is a dot plot.

This is very similar to a frequency histogram without the columns.

The frequency histogram in the activity you have just done is shown as a

dot plot below.

3 4 5 6 7 8 9 10

2

0

4

6

8

10

Number of marks in a testf

x 3 4 5 6 7 8 9 10

Number of marks in a test

x

You don’t need columns showing the frequency; just use a dot.

And you do not really need a grid either. Just make your dots about the

same size and evenly space them.

Dot plots work best for small numbers of scores.



Try these.

2 a Make a dot plot to represent the ages in years of students in a

small primary school. Use the data in the table below.

Age(years)

x 4 5 6 7 8 9 10 11 12 13

Frequency f 1 6 3 5 4 9 4 6 8 2

4 5 6 7 8 9 10 11 12 13

Age (years)

b How many students attend the school? ___________________

c Do you think the dot plot for this school is typical of all small

schools? Why?

___________________________________________________

___________________________________________________


You have been practising drawing histograms, polygons and dot plots.

Now check that you can solve these kinds of problems by yourself.

Go to the exercises section and complete Exercise 1.2 – Histograms and

polygons.


Cumulative frequency tables

In earlier work you learned how to construct frequency tables, and from

those to draw frequency histograms and polygons.

In this section you will extend the presentation of the frequency table to

produce a cumulative frequency table.

But first, consider this activity.

Activity – Cumulative frequency tables

Try these.

1 In a spellathon to raise funds for the school library, students were

required to spell 100 words. The number of errors made by each

student was recorded. This frequency distribution table shows the results.

Error, x Frequency, f

0

1

2

3

4

5

6

7

2

4

7

14

12

6

4

1

a Did any students make no mistakes? How many?

___________________________________________________

b What was largest number of errors made?

___________________________________________________


c What is the range of scores in this distribution? ____________

d What was the most common number of errors made? ________

e How many students made 2 errors? ______________________

f How many students made 2 errors or less?

___________________________________________________


Most of the questions above can be answered easily from the table.

To answer questions like f more easily, you can add another column to

the table. This column gives a running total of frequencies.

For each score this running total is called the cumulative frequency.

It tells you how many times you get this score or less.

• The cumulative frequency of the first score is simply the frequency

of that score.

• Add the next frequency to get the cumulative frequency of the

next score.

• Continue this all the way down the table.

Score, x Frequency, f Cumulativefrequency, cf

0

1

2

3

4

5

6

7

2

4

7

14

12

6

4

1

2

6

13

27

39

45

49

50

Σf = 50

2

2 + 4 = 6

6 + 7 = 13

13 + 14 = 27

27 + 12 = 39

39 + 6 = 45

45 + 4 = 49

49 + 1 = 50


If you have not made a mistake, you’ll find that the cumulative frequency

of the last score is the sum of all the frequencies. This is the total

number of scores. Can you see why?

You have simply been adding the frequencies from the top, writing down

a running total as you go.

Now, to answer the question: ‘How many scores are 2 or less?’

This is the number of scores of 0 or 1 or 2. Find the cumulative

frequency of the score 2. So 13 scores are 2 or less.

How many students had 5 errors or less? Look at the cumulative

frequency column. The answer is 45.

How many students had more than 5 errors? The answer to this is 5

since 45 had at most 5 errors, the rest therefore had more than 5 errors.

In more detail there are 50 students. Hence 50 – 45 = 5.

Five students had more than 5 errors.

In this case you would probably find it easier to add the numbers of

scores above 5:4 + 1 = 5. Doing it both ways gives you a check!



Try these.

2 A group of children had a test marked out of 10. The marks they

obtained are given in the frequency table below.


3 2

4 3

5 10

6 15

7 17

8 16

9 12

10 5

a Complete the cumulative frequency column.

b How many children were tested? ________________________

c Give the cumulative frequency of the score 8. ______________

d How many scored more than 8? _________________________

e How many children scored less than 5? ___________________


You have been practising drawing cumulative frequency tables.

Now check that you can solve these kinds of problems by yourself.

Go to the exercises section and complete Exercise 1.3 – Cumulative

frequency tables.


Cumulative frequency diagrams

Look at the spellathon data from a previous session.


0

1

2

3

4

5

6

7

2

4

7

14

12

6

4

1

2

6

13

27

39

45

49

50

Σf = 50

The numbers of errors were

tabulated and the cumulative

frequencies calculated.

You can now graph the

cumulative frequencies as

either a cumulative frequency

histogram or as a cumulative

frequency polygon.

Both the histogram and polygon can be drawn on the same grid.

The horizontal axis is the score axis. Record every score from lowest to

highest at equally spaced intervals.


0

5

10

15

20

25

30

35

40

45

50

55

Freq

uenc

y (n

umb

er o

f stu

den

ts)

Score (number of errors)0 1 2 3 4 5 6 7

Results of spellathon

The histogram is shown by the rectangles. They are shaded for clarity

but this is not needed as long as lines are clear.

The heights of the rectangles give the cumulative frequencies for each

successive score.

The vertical axis must be long enough to show the total number

of scores.

To draw the polygon start at the lower left hand corner of the rectangle

for the lowest score. Draw the diagonal of the first rectangle.

Join this point to the top right vertex of the next rectangle, and so on.

A cumulative frequency polygon is sometimes called an ogive.

Drawing a cumulative frequency histogram and polygon is very similar

to drawing a frequency histogram and polygon. But there are three

main differences.

• Because the cumulative frequency can only get larger (or stay the

same) in the table, columns in a cumulative frequency histogram are

the same length, or longer, from one score to the next.

In a frequency histogram, on the other hand, successive columns can

get longer or shorter depending on the frequency.


• The cumulative frequency polygon moves from top right hand corner

of one rectangle to the top right hand corner of the next rectangle.

This is different from a frequency polygon which moves from top

middle to top middle.

• In a frequency polygon, the line starts from the horizontal (score)

axis and ends back on that axis. But in a cumulative frequency

polygon the line starts from the score axis and ends at the top of the

last column. It does not return to the horizontal axis.

Activity – Cumulative frequency diagrams

Try these.

1 A sports squad line up in order of size. Here are their shirt sizes.

65 65 65 70 70 75 75 75 75 75 80 80 80 85 85

85 85 85 85 85 90 90 90 90 95 95 95 95 100 100

a Complete the table.

Size, x Frequency, f Cumulativefrequency, cf

65

70

75

80

85

90

95

100

Σf =


b Choose an appropriate scale to label the vertical cumulative

frequency axis.

Shirt size65 70 75 80 85 90 95 100

Shirt sizes of sports squad

x

Cum

ulat

ive

freq

uenc

y

c Draw a cumulative frequency histogram, in blue, for the data.

d Draw a cumulative frequency polygon over the histogram in red.

Remember to red the previous presented notes about cumulative

frequency polygons before beginning this.


You have been practising drawing cumulative frequency histograms and

polygons. Now check that you can solve these kinds of problems

by yourself.

Go to the exercises section and complete Exercise 1.4 – Cumulative

frequency diagrams.


Mean, median, mode and range

In your stage 4 Mathematics course you learned about measures that tell

you something about the middle of a set of scores. The three measures of

central tendency are the mean, median and mode. Another measure that

lets you know about the spread of scores is called the range.

• The mean is the average of a number of scores. The symbol for this

is x (x-bar), and is shown like this on your calculator.

To find the mean of a loose collection of scores (x), you can add them

together and then divide by the number of scores (n).

x =Σxn

.

For example, the mean of 6, 9, 12, and 15 is: x =6 + 9 +12 +15

4= 10.5

• The median is the middle for a set of numbers written in order.

For example, the median of 6, 9, 11, and 15 is the value midway

between 9 and 11. You can easily see the median to be 10, in this case.

Alternatively, you could average out these two values: 9 +112

= 10 .

For an odd number of values, such as 6, 9, 11, 14 and 15, the middle

value falls neatly onto a score. In this example it is 11.

• The mode is the most frequent score. That is, the score that occurs

most often.

For the examples above there is no mode as each score occurs once.

But for 2, 3, 5, 5, 5, 5, 6, 6, 7, 9 the mode is 5.

• Range = highest score – lowest score.

For example, the set of scores 2, 3, 5, 5, 5, 5, 6, 6, 7, 9 has range = 9 – 2 = 7.


Activity – Mean, median, mode and range

Try these.

1 For the following sets of scores calculate the mean, median, mode

and range.

a 4, 6, 6, 8, 8, 9, 9, 9, 9, 10, 11, 12, 14

mean: _____________________________________________

median: ____________________________________________

mode: _____________________________________________

range: _____________________________________________

b 15, 9, 23, 16, 15, 18, 17, 15, 11, 20, 19, 16, 14, 17, 15, 18

When scores are presented like this not in a table or order they

are referred to as loose or raw data.

mean: _____________________________________________

median: ____________________________________________

mode: _____________________________________________

range: _____________________________________________


You can let your calculator calculate the mean ( x ) for you.

Look over your notes for stage 4, or consult with your teacher, if you do

not remember how to make your calculator do this.

Calculating measures like these is easy for a small number of loose

scores. It can become more difficult when there is a large collection

of scores.

Usually when there is a large number of scores they are generally

tabulated first. An example is shown on the next page.


To calculate the mean for a set of scores written as a table, you need to

add a new column frequency x score, and multiply the score by thefrequency. Then calculate the total, Σfx , for this column.

The mean is then found as x =ΣfxΣf

.

You calculator can also calculate the mean for tabulated values.

Follow through the steps in this example. Do your own working in the

margin if you wish.

Calculate the mean, median, mode and range for these

tabulated values.

Score, x Frequency, f Frequency × score, f × x

65

70

75

80

85

90

95

100

3

2

5

3

7

4

4

2

65 × 3 = 195

70 × 2 = 140

75 × 5 = 375

80 × 3 = 240

85 × 7 = 595

90 × 4 = 360

95 × 4 = 380

100 × 2 = 200

Σf = 30 Σfx = 2485

If you need to review how to calculate these values from tables

of data in detail, refer to the stage 4 unit DS4.2 on Data

analysis and evaluation, or ask your teacher.


Solution

The mean for these scores is x =248530

= 82.8 .

As there are 30 scores, the median is the value lying midway

between the 15th and 16th score. You can use a cumulative

frequency column, if one is available, to assist in keeping this

running total. Or else add values in the frequency column until

you arrive at the point between the 15th and 16th score.

3 + 2 + 5 + 3 = 13 (13th score) and 3 + 2 + 5 + 3 + 7 = 20

(20th score). So the median is 85.

The mode is 85 as this occurs most often (it occurs 7 times).

The range in values is 100 – 65 = 35.


Try these.

2 a Complete the f × x column in this table of data.

Score, x Frequency, f Frequency × score, f × x

3 2

4 3

5 10

6 15

7 17

8 16

9 12

10 5

Σf = Σfx =


b Use this table to calculate the mean, median, mode and range

mean: ______________________________________________

median: ____________________________________________

mode: _____________________________________________

range: _____________________________________________


You have been practising calculating mean, median, mode and range

from loose and tabulated data. Now check that you can solve these kinds

of problems by yourself.

Go to the exercises section and complete Exercise 1.5 – Mean, median,

mode and range.

You will need to be able to calculate these values in later sessions.


Suggested answers – Part 1

Check your responses to the preliminary quiz and activities against these

suggested answers. Your answers should be similar. If your answers are

very different or if you do not understand an answer, contact your teacher.

Activity – Preliminary quiz

1 a 30 runners b 148 beats/min c 168 – 132 = 36

d630

×1001

= 20%

e (sum of scores) ÷ (number of scores) = 150

2 a (4 + 7 + 1 + 0 + 6) ÷ 5 = 3.6 b 5.5

312 + 3

4

2=58

4 a Frequency column values are: 2, 4, 6, 1, 10, 0, 5, 2

b While there are 8 differently-valued scores (0, 1, …, 7) there are

actually 30 scores. There are two 0s, four 1s, six 2s and so on.

This gives a total of 30 scores. (You can find this by adding the

frequency column.)


1

0 0 0 0 1 1 1 1 1 1

1 1 1 1 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 4 4 4 7

2 Range = 7 – 0 = 7


3 and 4 (table below)


0 4

1 10

2 16

3 6

4 3

5 0

6 0

7 1

5 The frequency of a score tells you how often that score occurs.

The scores here are the numbers of children in a family.

The frequency of the score 4 is 3 tells you there are 4 children in

each of 3 of the families interviewed.

6 a 4 families surveyed had no children.

b No families had 5 children.

c The sum of the frequencies is 40.

d 40 homes were surveyed.

e The highest score in the survey is 7.

f The greatest number of children in any family in this survey is 7,

g The score with the highest frequency is 2. More families have

2 children than any other number. (Remember the most

frequent score is called the mode. The mode in this survey is 2.)



1 aNumber of marks in a test

0

2

4

6

8

10

f

3 4 5 6 7 8 9 10 x

b range = 7

c The most common mark has the highest frequency. It is 6.

d 1 student got only 3 marks.

e 4 students got 4 marks.

f 6 students got 8 marks.

g Students are grouped by their scores. To find the total number

of students, add the numbers in each group. Then find the sum

of these frequencies. 1 + 4 + 7 + 9 + 8 + 6 + 3 + 2 = 40.

There are 40 students altogether.

2 a

4 5 6 7 8 9 10 11 12 13

Ages of students in small primary school

b By counting the dots, or adding the frequencies, you get a total

of 48 students.


c You can expect a dot diagram for such a small number of

children to have a variation in the numbers in each age group.

The fact that the ages are in years and not months can affect a

regular distribution of ages. For example, of the children

recorded as being 9 years old, some may have only just left the

eight year olds and others may be almost 10. For a large group

of children you might expect the numbers of children in each

age group to be about equal. Of those going to school, you

wouldn’t expect many four years olds. Most 13 year olds would

probably be in high school.


1 a Yes, 2 students had no mistakes.

b The largest number of errors is 7

c Range = 7 – 0 = 7

d The most common number of errors is 3. (14 students made

3 errors each.)

e 7 students made 2 mistakes.

f To find the number of students with less than 2 errors you must

add the number of students who made 0, 1 or 2 errors.

As 2 + 4 + 7 = 13, then 13 students made 2 errors or less.

Score, x Frequency, f Cumulative frequency, cf

3 2 2

4 3 5

5 10 15

6 15 30

7 17 47

8 16 63

9 12 75

2 a

10 5 80

b 80 children were tested c 63


d 80 – 63 = 17. So 17 children got more than 8.

(You could also check: 12 + 5 = 17)

e 5 children scored less than 5. (This is the same as asking for

how many scored 4 or less.)

Activity – Cumulative frequency diagrams

1 a

Size, x Frequency, f Cumulative frequency, cf

65

70

75

80

85

90

95

100

3

2

5

3

7

4

4

2

3

5

10

13

20

24

28

30

Σf = 30

b, c, and d (columns in blue; line in red)

5

0

15

10

25

20

30

Shirt size of Sports Squad

60 65 70 75 80 85 90 10595 100

Shirt size

Cum

ulat

ive

freq

uenc

y



1 a mean, x =11513

= 8.85 ; median = 9 (count through to the 7th score);

mode = 9 (this score occurred 4 times); range = 14 – 4 = 10

b Arrange the scores in order first:

9, 11, 14, 15, 15, 15, 15, 16, 16, 17, 17, 18, 18, 19, 20, 23

mean = 25816

=16.1; median = 16 (this value lies between the

8th and 9th scores); mode = 15; range = 23 – 9 = 14.

2 a f × x column values are: 6, 12, 50, 90, 119, 128, 108, 50;

Σf = 80; Σfx = 563.

b mean = 563 ÷ 80 = 7.0; median = 7 (between the 40th and 41st

score); mode = 7 (occurs 17 times); range = 10 – 3 = 7.


Exercises – Part 1

Exercises 1.1 to 1.5 Name ___________________________

Teacher ___________________________

Exercise 1.1 – Tallies and tables

1 The number of strokes taken by a group of golfers to sink the ball in

the first hole were recorded and the results shown by the tallies in

this table.


2

3

4

5

6

Σf =

a Complete the table of frequencies.

b How many golfers sank the ball in two strokes? ____________

c What was the largest number of strokes needed to sink the ball?

_______________________________________________________

d How many players needed five strokes? ___________________

e What was the most common number of strokes needed to sink

the ball? ____________________________________________

f How many golfers were in the group? ____________________(The answer to this is the same as the value of Σf , the sum of

the frequency column.)


2 A social club organised a children’s Christmas party. To be able to

provide age appropriate gifts, a survey of the children’s ages was

taken. Their ages are shown below.

5 4 9 5 8 7 3 7 4 6

9 9 6 2 3 3 5 5 7 3

5 9 4 4 5 2 4 4 5 2

a What is the age of the youngest child?_____________________

b What is the age of the oldest child? _______________________

c What is the range of the ages? ___________________________

d List the ages in your score column, and then complete the table

for these scores.

Age of children (years)


Σf =

e What does the frequency column tell you? _________________

___________________________________________________

f What is indicated by Σf ? ______________________________

g How many four year olds are expected at the party?__________

h What is the mode score?________________________________

What does this mean?__________________________________


Exercise 1.2 – Histograms and polygons

1 Here are the shirt sizes (in cm) for a sports squad.

85 65 75 70 80 95 75 100 85

90 90 75 85 75 65 80 85 75

90 85 95 80 95 100 95 65 75

70 80 90

a What are the smallest and largest sizes? ___________________

b Draw a dot plot to represent these shirt sizes.

c What shirt size is the mode? ____________________________

2 A class of students measure their temperatures.

They record their results in degrees Celsius (°C).

36.7 36.8 36.5 37.0 36.8 37.2 37.1 36.9 36.8

38.6 36.8 37.2 36.5 26.5 37.0 36.7 36.9 37.0

37.0 36.8 36.6 37.1 37.1 36.7

a Which temperature is an outlier by comparison with the others?

___________________________________________________

b Draw a dot plot for this set of temperatures. (Omit the outlier.)

36 37 38 39 4036.5 37.5 38.5 39.5


c One of the students is sick with quite a high temperature.

What is this temperature? ______________________________

d What would you say is normal body temperature, to the nearest

degree? ____________________________________________

3 A biologist counted the number of insects in equal-sized plots of

land measured from the bank of a river. The results are shown in the

frequency table.

Distance (m) Number ofinsects

10

15

20

25

30

35

40

45

8

14

22

31

26

14

10

30 5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

35Insect counts

Distance from river (m)

Freq

uenc

y (n

umb

er o

f ins

ects

)

On the axes provided, draw both a frequency histogram and a

frequency polygon for this data. Label your graph showing which is

the histogram and which the polygon.


4 The frequency polygon shows the results of a maths quiz.

0 1 2 3 4 5 6 7 8 9 10

2

4

6

8

10

12

14

Quiz results

Test mark

Freq

uenc

y (n

umb

er o

f stu

den

ts)

11 12

16

18

Test mark Frequency

2

3

a Draw a frequency histogram on the

graph above for this data.

b Use the graph information to

complete the frequency table.

c How many students are displayed in

these graphs?

d How many students scored more

than 6?

Σf =

e What percentage of students scored full marks (that is, 10 out of 10)?

___________________________________________________


5 A cat club kept track of the number of kittens in litters of Persian

cats over a period of time. These are their results.

Kittens inlitter

1 2 3 4 5 6 7 8 9 10 11

Frequency 2 6 9 11 14 10 8 4 3 2 1

On the axes provided draw a frequency histogram and polygon for

this data. (Don’t forget to label the axes and give the graph a title.)

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0


Exercise 1.3 – Cumulative frequency tables

1 A school sells sports tops in sizes 8 to 20. The table shows sizes and

the numbers of each size sold.

Size, x Frequency, f Cumulative frequency, cf

8 16

10 25

12 38

14 46

16 32

18 15

20 8


b How many tops were sold altogether? ____________________

c How many tops sold were size 16 or larger? _______________

2 The table shows the number of children in a group of families.

Number ofchildren

Number offamilies, f

Cumulativefrequency, cf

0 3

1 10

2 16

3 6

4 3

5 0

6 2


b How many families were surveyed? ______________________

c How many families have 3 or fewer students? ______________

d What fraction of families have less than two children?

___________________________________________________


3 The results of a survey of the numbers of claims made by drivers for

motor accident are partly tabulated.

Number ofaccidents

Number ofdrivers

Cumulativefrequency

0 48

1 79

2 92

3

4 2 100

Σf =

a Complete the table.

b How many drivers were surveyed? ______________________

c How many drivers had 3, or fewer, accidents? _____________

4 The number of tries scored by the Endeavour football team in a

season are:

4 0 3 5 1 2 3 4 2 2

2 2 3 4 3 0 5 6 4 2

a Draw a frequency distribution table for this data and include a

cumulative frequency column.

b What is the mode number of tries scored? _________________

c Calculate the mean number of tries, correct to one

decimal place.

___________________________________________________


Exercise 1.4 – Cumulative frequency diagrams

1 State two differences between a frequency histogram and polygon

and a cumulative frequency histogram and polygon.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

2 The results of a survey of the numbers of claims made by drivers for

motor accident are tabulated.

Number ofaccidents

Number ofdrivers

Cumulativefrequency

0 48 48

1 31 79

2 13 92

3 6 98

4 2 100

Use this information to

draw a cumulative

frequency histogram and

polygon on the same axes.

(Give your diagram a

title.)

Σf = 100

Number of accidents0 1 2 3 4 5

Cum

ulat

ive

freq

uenc

y


3 Use this cumulative frequency histogram and polygon to complete

the table below.

Score1 2 3 4 5

Cum

ulat

ive

freq

uenc

y

6 70

10

20

30

40


1

2

3

4

5

6

7

Σf =


Exercise 1.5 – Mean, median, mode and range

1 Calculate the mean, median, mode and range for these loose scores.

a 18, 19, 20, 22, 23, 23, 25, 25, 25, 26, 27, 27, 30, 30, 32

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

b 56, 45, 38, 46, 67, 66, 53, 49, 53, 60, 56, 58, 53, 52

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

c 7.6, 7.1, 7.4, 7.5, 7.6, 7.8, 7.6, 7.2, 7.0

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

2 Why does this set of scores not have a mode?

105, 98, 68, 85, 60, 47, 88, 75, 92, 61, 75

_______________________________________________________

3 Which of mean, median and mode must always be a score? Why?

_______________________________________________________

_______________________________________________________


4 A study was made of the young born to the cougar (mountain lion)

and the following table constructed.

Number ofyoung, x

Frequency, f Frequency × score,f × x

1

2

3

4

5

6

3

5

7

12

10

6

Σf = Σfx =

a Complete the values in the table.

b Use the table to calculate the mean number of young born,

correct to one decimal place.

___________________________________________________

c What is the mode number of young born? _________________

d What is the median number of young born? _______________

e Two more births were later reported, one with 2 young, the other

with 3 young. Which of the values mean, median and mode will

not change?

___________________________________________________


f Use the frequency distribution table to draw a labelled frequency

histogram and polygon for the data.

1 2 3 4 5 6

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