11
Statistics Objectives ✔
learn about the nature and purpose of statistics
✔
construct frequency tables
✔
display data using bar charts, pictographs, pie charts, histograms and frequency polygons
✔
find the mean, median and mode for a set of data and know when to use them
✔✔
construct a cumulative frequency table and corresponding cumulative frequency curve
✔✔
find the mean and median from a set of grouped data
✔
determine and use measures of spread
What’s the point? What is an average? How can you tell? Statistics help you to interpret data and, for example, help a company to determine what quantities of a product they should stock.
Before you start You should know ...
Check in
1
1
How to read information from a graph. For example:
Here is a graph showing the height of water in a stream at certain times. ) m c ( r e t a w f o t h g i e H
180
Go up to the line then across
170 160 150 140 130 120 110 0
1
2
3
4
5
6
7
Time (h)
After 3 hours hou rs the height of water was 140 140 cm.
Use the graph in the example to find: (a)) the height of the water after (a 5 hours (b) the height of the water after 1.5 hours (c)) the time when the water was (c at a height of 165 cm.
23 2
STATISTICS
11.1
(c) School principals (c) (d)) Hospital administrators (d (e)) Politicians (e
Collec Col lectin ting g dat data a
In the modern moder n world huge huge amounts of data are a re collected every week by many different d ifferent people:
2 Person needing data
Purpose
Store manager
Stock control
Pollster
Find public opinion of an issue
Teacher
Determine student progress
To collect data you need a suitable data collection instrument. Simple instruments are: ● ● ●
●
For how long do you watch television at night?
There are four steps in carrying out a survey: 1. Ask a question 4. Use results to answer question
2. Collect data
Do you watch less television on a school night or a weekend night?
The mathematical topic that involves the collection, display and analysis of information is called statistics.
Exercise 11A For what purpose do you thin k the following people would need to collect data? (a)) Hotel managers (a (b) Tourist promotion officials
Do boys perform better bet ter in maths than tha n girls? girls?
(a)) Decide what data to collect to test this (a question. instru ment. (b) Design a suitable data collection instrument. (c)) Use your instrument to collect the data. (c (d)) Study your data to decide the answer to the (d question.
11.2
Organising Organi sing data
Data types
There are two basic types of data: d ata: ●
Often your results will wil l prompt you to make ma ke a followfollowup survey. In the case of television viewing a followup question might be:
1
3
●
3. Analyse data
●
(a)) Decide what data to collect to test this (a statement. instru ment. (b) Design a suitable data collection instrument. (c)) Use your instrument to collect the data. (c a nd decide whether the (d)) Look at your data and (d statement is true.
questionnaires tests checklists or tally sheets.
To find answers to many everyday questions a survey is often used. For example example,, you may wish to find fi nd out about television viewing in your school. A survey question might be:
The students in my class watch less than 2 hours television at night.
discrete and continuous.
Discrete data can only take ta ke definite values. For example: shoe sizes sizes – size 1, 1, size size 2, etc. gender – male, female colour – red, green, blue, etc. Continuous data can take any value. For example: height, weight, time. Frequency tables
When you have collected your data, you need to organise it. A good way to organise discrete data is in a frequency table using a tally t ally..
23 3
STATISTICS
Example 1
Example 2
Make a frequency table for the scores for a class of students in a test.
The heights of 25 boys, in centimetres, are: 103, 145, 138, 162, 149, 150, 175, 168, 138, 142, 161, 136, 125, 111, 143, 147, 159, 172, 165, 166, 133, 147, 152, 168, 171
3, 7, 6, 4, 2, 8, 8, 1, 10, 9 2, 5, 5, 6, 4, 7, 8, 6, 5, 8
Score
Tally
Construct a grouped frequency table to show the data. Use groups of 100–109, 110–119, p 170–179.
Frequency
1
1
2
2
3
1
Group
4
2
100–109
1
5
3
110–119
1
6
3
120–129
1
7
2
130–139
4
8
4
140–149
6
9
1
150–159
3
10
1
160–169
6
170–179
3
The table shows that only four students scored less than 4 marks and six scored 8 or more.
Exercise 11B 6, 7, 6, 5, 6, 9, 8, 7, 10, 6, 5, 8, 9, 10, 7, 6, 5, 9, 8, 7
2
Display the data in a grouped frequency table, using the same groups as Example 2.
2
The favourite colours of 16 pupils were noted during a survey:
Make a frequency table to display the data.
Construct a grouped frequency table to show the data. Use groups of 0 –4, 5 –9, 10–14, 15–19 and 20–24.
A 6-sided dice was rolled 30 times. Here are the scores: 1, 3, 2, 6, 5, 4, 5, 2, 5, 3, 6, 1, 3, 1, 4, 1, 4, 2, 1, 3, 6, 5, 2, 4, 3, 2, 1, 5, 1, 3 Display the scores in a frequency table.
A grouped frequency table is often used for both continuous and discrete data.
Rainfall (mm) in London, England is shown for the 30 days in November. 21, 20, 0, 12, 1, 11, 0, 3, 1, 4, 9, 1, 9, 0, 13, 3, 7, 18, 8, 4, 7, 18, 0, 4, 1, 2, 0, 12, 2, 16
red, blue, green, red, yellow, red, green, blue, blue, red, yellow, red, blue, blue, red, green
3
The heights of 25 girls, in centimetres, are: 101, 111, 159, 172, 132, 125, 113, 126, 138, 142, 158, 107, 109, 117, 125, 104, 129, 121, 143, 133, 168, 141, 121, 118, 141
Here are the shoe sizes of 20 adults:
Show the data in a frequency table.
Frequency
Exercise 11C 1
1
Tally
3
The times taken by 24 students to complete a mathematical puzzle are shown in minutes. 5, 22, 8, 13, 7, 9, 6, 8, 12, 10, 4, 9, 3, 19, 18, 9, 14, 2, 5, 15, 21, 11, 5, 17 Make a grouped frequency table for the data using groups of 0–4, 5–9, 10–14, 15–19 and 20–24.
23 4
STATISTICS
11.3
Displaying data
10 9
Barcharts and pictograms
8
One of the simplest ways of displaying discrete data is to use a bar chart.
y 7 c n e 6 u q e r 5 F
4
Example 3
3
The favourite singers of a class are:
Beenie Man
Shaggie
5
9
2
Vybz Kartel
Lady Saw
7
3
1 Blue
10
Favourite colour
2
Green Yellow Orange
(a) Which colour is the most popular? (b) Which colour is the least popular? (c) How many students does the bar chart represent? (d) Copy and complete this table using the information from the bar chart.
Display this on a bar chart.
y 8 c n e 6 u q e r F 4
Red
Blue
Red
Green
Yellow
Orange
Number of students
Beenie Shaggie Vybz Man Kartel
Lady Saw
2
The pictograph shows the number of soft drinks sold by a cafe last week. Soft drinks sold last week
You can also use a pictograph to display the information in Example 3. In a pictograph a picture is used to represent the data.
Thursday
Shaggie Vybz Kartel
Friday Saturday
Lady Saw represents one student
Exercise 11D 1
Tuesday Wednesday
Beenie Man
Scale:
Monday
This bar chart shows the favourite colours of a group of students.
= 2 soft drinks
represent? (a) What does each (b) How many soft drinks did the cafe sell on Tuesday? (c) How many were sold on Friday? (d) On which day did the cafe sell most soft drinks? (e) How many soft drinks were sold in total during the week?
23 5
STATISTICS
(f) Copy and complete the table using information from the pictograph. Day
Mon
Tue
Wed
Thur
h Fri
Sat
Number of soft drinks sold
3
The table shows the favourite sports of some students at Portsmouth Secondary School. Volleyball
Football
Cricket
Netball
4
7
11
6
(a) Show the information on a bar chart. (b) Show the information on a pictograph. 4
The block graph shows the number of children per family, for the families in Donovan. Family size
Technology
Project 1 (a) Choose a paragraph in a book. Count the number of times each of the vowels a, e, i, o, u occurs. (b) Put your results in a table. Vowel
0
● ●
●
u
Which is the most common vowel? Which is the least common vowel? Why do you think this may be the case?
Project 2 1
2
3
4
5
6
7
Technology
You can use a spreadsheet to show a bar graph. For example, type the table from Question 3 of Exercise 11D into a spreadsheet.
●
o
Copy and paste your graphs into your report.
(a) How many families are there in Donovan? (b) How many children are there in Donovan? (c) Draw a pictograph for the information.
●
i
(c) Repeat for two more paragraphs. (d) Enter your tables into a spreadsheet and use it to display a bar graph for the number of vowels in each paragraph. (e) Write up your results using a word processing program. Be sure to answer questions such as ●
Number of childern in the family
h
e
Frequency
12
s e i 10 l i m a 8 f f o 6 r e b m 4 u N 2
a
Highlight the table and select the Chart option from the Insert menu. Choose the column bar chart from the pop-up window. Follow the instructions to give your bar chart a title and labelled axes.
(a) Carry out a survey in your class to find out the number of brothers and sisters each student has. (b) Copy and complete the table. Number of brothers and sisters
0
1
2
3
4
Frequency T
23 6
STATISTICS
ˇ
2
(c) Make a separate table for the number of brothers and another one for the number of sisters. (d) Display charts of your results in a spreadsheet program. (e) Using a word processing program, write a report of your findings and illustrate it with your graphs.
110 bus 20 car
3
A pie chart is another way to display your data. It is useful when you want to show the relative parts of a total.
4
$30 $10 $10
Record Savings
Show this information with a pie chart.
(a) Display the data on a pie chart. (b) If $3000 was spent on landscaping, what was the total budget?
The whole circle, 360°, represents $100 won. 30 Fraction spent on shirt = 100 .
30 100
=
108°
×
360°
The pie chart shows David’s expenditure.
Cap 36° 36° Cinema 72° Record
1
At one election they voted as follows: 10 Radical 20 Did not vote
Draw a pie chart to show this.
The pie chart shows the Cricket results of a survey of 124° favourite sports among Netball 1000 people. 81° (a) What fraction of 75° Basket r the people surveyed eFootball h ball t chose basketball as O 60° 20° their favourite sport? (b) How many people indicated cricket was their favourite sport?
6
The pie chart shows the budget of a certain island.
108° Savings
In the village of Newbury there are 240 people on the voting list.
150 Labour 60 Independent
5
Shirt 108°
Exercise 11E
The council in the town of St. Peter’s spent money from rates as follows: 45% on public services 25% on road maintenance 10% on transportation 10% on sewage improvements 5% on landscaping 5% on investments
$20 $30
So, the angle representing money spent on a shirt
French 2 Social Studies 6
(a) Display this data on a pie chart. (b) What percentage chose English as their favourite subject?
David won $100 in a lottery. He spent it as follows.
=
The 32 students of class 1B voted for their favourite subjects. The results were as follows: English 4 Maths 12 Science 8
Example 4
160 walk 10 cycle
Draw a pie chart to show this information.
Pie charts
Shirt Cap Cinema
The 300 students at Marigot High School travel to school as follows:
Services Education 22% 24% Other 9% Health 15%
Public works 30%
STATISTICS
(a) Draw the pie chart again indicating clearly the angle in each sector. (b) If $5 million is spent on health, what is the total budget? (c) Work out how much is spent on each area and draw a bar chart to illustrate the data. 7
The pie chart illustrates the sales of different makes of motor oil.
Shell 120°
Exxon y
55° Texaco West Indies Oil x
(a) What percentage of the sales does Texaco have? (b) If West Indies Oil accounts for 15% of the total oil sales, calculate the angles x and y. 8
A fruit importer checks the number of bad oranges in 100 boxes. This is what he found.
Number of bad oranges
0
1
2
3
4 or more
Number of boxes
55
32
10
3
0
(a) Draw a pie chart to show the data. (b) How many bad oranges were there altogether?
h
Technology
You can use a spreadsheet to show a pie chart. For example, type the data from Question 3 of Exercise 11E into a spreadsheet. ●
●
●
Highlight the table and select the Chart option from the Insert menu. Select the pie chart from the pop-up window. Follow the instructions to give your chart a title.
h
23 7
Technology
Project 3 (a) Make a survey in your class to find out which subject is most popular. (b) Using a spreadsheet, show your results in a pie chart. (c) Make separate pie charts for favourite subjects of girls and favourite subjects of boys. (d) Write up your results using a word processing program and illustrate your report with your pie charts. Histograms Grouped continuous data
The heights of 30 plants are shown in the table:
Height (cm)
Frequency
15–17
2
18–20
6
21–23
12
24–26
7
27–29
3
Each group of data is a class or interval. When a height is given as 21 cm (to the nearest cm) its true value lies between 20.5 cm and 21.5 cm. Therefore, each height in the class 21–23 has a true value between 20.5 and 23.5. These are the class boundaries of the 21–23 class. The class width = 23.5cm − 20.5cm = 3 cm. The class limits are 21 cm and 23 cm. Knowing the class boundaries you can draw a bar chart to show this continuous data. Bar chart showing the heights of plants
Notice the bars are drawn on the class boundaries.
12 10 y 8 c n e u 6 q e r 4 F
2 15 17 19 21 23 25 27 29 31 (cm) Height
23 8
STATISTICS
This type of bar chart is also called a histogram. ●
(a) Suppose the heights are rounded off to the nearest centimetre. In which class would you put a tick for a child whose height is: (i) 140.4 cm (ii) 149.2 cm (iii) 149.9 cm (iv) 161.3 cm (v) 164.8 cm (vi) 139.6 cm? (b) Into which class would you put a height of: (i) 159.5 cm (ii) 139.5 cm (iii) 144.5 cm? (c) What is the least height belonging to the class 155–159 cm? (d) Write down the class boundaries for these classes. (i) 135–139 cm (ii) 140–144 cm (iii) 145–149 cm (iv) 150–154 cm
In a histogram: ● there are no spaces between the bars ● the area of each bar represents the frequency.
A histogram with equal-width bars is a bar chart.
Example 5 The times taken by a group of children travelling to school are given in the table.
Time (minutes)
Frequency
5–9
2
10–14
8
15–19
10
20–24
6
25–29
4
Draw a histogram to show this information.
This is the completed frequency table for the 25 children.
Height (cm)
Frequency
135–139
1
140–144
3
145–149
4
y 8 c n e u q 6 e r F
150–154
7
155–159
5
160–164
4
4
165–169
1
Times taken travelling to school 10
Class boundaries are at 4.5, 9.5, 14.5, 19.5,..
2 5
10 15 20 25 Time (minutes)
30
Exercise 11F 1
2
A group of 25 children measure each other’s heights and record the answers by putting a tick on a chart:
Height (cm) 135–139 140–144
✓
145–149 150–154
✓✓✓
The histogram for this table must show the class boundaries, at 134.5 cm, 139.5 cm, p 169.5 cm. Below is part of the graph. Copy and complete it. Write a title for it. 5 n 4 e r d l i h 3 c f o r e b 2 m u N
1
155–159 160–164
✓
165–169
✓
130
135 140 145 Height (cm)
23 9
STATISTICS
3
The histogram shows the results of weighing 100 apples to the nearest gram. (a) How many apples are in the class 110–119 grams? (b) Why are the boundaries of this class drawn at 109.5 and 119.5? The results of weighing 100 apples
(a) Write down the class boundaries, starting at 19.5 mm. (b) Draw a histogram of the results. Use one small division to represent one unit, as in t he graph in Question 3. 5
50 40 y c n 30 e u q e r F20
Height (cm)
160
161
162
163
164
165
166
Frequency
4
5
6
9
16
22
27
Height (cm)
167
168
169
170
171
172
Frequency
25
18
11
6
3
2
(a) Redraw the frequency table, using intervals of 160–161, 162–163, p 172–173. (b) What are the boundaries of the interval 164–165 cm? (c) What is the boundary between the interval 166–167 cm and 168–169 cm? (d) Draw a histogram using the intervals in part (a).
10 0 90
100
110
120
130
140
150
160
Mass (grams)
(c) Draw a frequency table using the information in the graph. Use classes of 100–109, 110–119, p 150–159. (d) What is: (i) the lower boundary of the fourth class (ii) the upper boundary of the fifth class? (e) What is: (i) the minimum mass of an apple (ii) the maximum mass of an apple? 4
In a biology experiment, the leaves of a plant are measured and recorded to the nearest millimetre, as shown.
The heights of 154 boys, to the nearest centimetre, are:
6
The percentage marks of 100 students in a test were:
Marks %
No of students
0–19
5
20–29
6
30–39
13
40–49
22
50–59
24
60–69
16
Length (mm)
Frequency
20–24
1
70–79
8
25–29
4
80–89
6
30–34
8
35–39
18
40–44
25
45–49
24
50–54
17
55–59
2
60–64
1
(a) Draw up another frequency table using equal intervals of 20 marks. (b) Write down the boundaries for each interval. (c) Illustrate the information by a histogram.
24 0
STATISTICS
(a) What is the mid-interval value of the class 500–549 kg? (b) Draw a frequency polygon to show this information.
Frequency polygons
Frequency distributions can also be illustrated by a frequency polygon. Frequencies are represented by single points, at the centre of each interval (midinterval value). The points are joined by straight lines.
3
The lengths of insect larvae are measured to the nearest mm.
Example 6 Length (mm)
Frequency
20–24
15
25–29
33
30–34
58
35–39
50
40–44
4
The masses, in kilograms, of 24 children are: Mass (kg) 10–19 20–29 30–39 40–49 50–59 No. of children
1
2
6
12
3
Draw a frequency polygon to show this.
(a) How many insect larvae were measured? (b) What is the mid-interval value of the class 20–24 mm? (c) Draw a frequency polygon of the information.
The interval 10–19 goes from 9.5 up to 19.5. The centre of the interval is at 9.5 + 19.5 29 = = 14.5 2 2 So the frequency polygon is:
11.4 Averages – measures of central tendency
10
n r e d l i h c f o 5 r e b m u N
0
10 20 30 14.5 24.5
40
50
60
Sometimes, instead of looking at a large set of numbers it is more convenient to use a single number that is a good representation of all the data. This number is an average or a measure of central tendency. There are three commonly used averages:
Mass (k )
● ●
mean mode median.
Exercise 11G
●
1
Draw frequency polygons to illustrate the data in Questions 3 and 4 of Exercise 11F.
The choice of which measure to use will depend on the circumstances.
2
Here are the weights of cattle sold at a livestock market:
The mean
Mass (kg)
Frequency
450–499
16
500–549
130
550–599
42
600–649
12
The most frequently used average is the mean. It is found by adding up all the data and dividing by the number of values. sum of data ● Mean number of values
241
STATISTICS
The mean of a frequency distribution is sometimes written as © fx Mean = π f where x = value of each observation f = frequency and © is the Greek letter ‘sigma’ meaning ‘the sum of’.
Example 7 A batsman scored 35, 2, 71, 16, 8 runs in five innings. What is his mean score? Mean =
35 + 2 + 71 + 16 + 8
So in Example 8:
5
132 = 5 =
© f = the
26.4 runs
=
sum of the frequencies
40
© fx = the
You calculate the mean of a frequency distr ibution in the same way.
Example 8
sum of number of goals = 117
0
1
2
3
4
5
6
Frequency (No. of games)
3
7
6
5
12
7
0
What was the mean number of goals scored per match? To find the total number of goals scored you need to multiply the number of goals by the frequency. This is best done in a table:
1
Here are the total scores of two dice, thrown together twenty times: 9, 2, 8, 6, 10, 7, 7, 4, 5, 8, 9, 12, 3, 10, 8, 11, 7, 4, 6, 9. Calculate the mean score.
2
A biologist takes a sample of 10 grasses and measures the stem length. His results, in centimetres, are: 30, 28, 32, 29, 25, 27, 31, 39, 33, 26. Calculate the mean stem length.
3
Two dice are thrown together 100 times. The following table is used to record the results and to calculate the mean:
Score
Frequency
x
f
2
1
2 12
No. of goals ( x)
Frequency ( f )
No. of goals × Frequency ( fx)
0 1 2 3 4 5 6
3 7 6 5 12 7 0
033= 0 137= 7 2 3 6 = 12 3 3 5 = 15 4 3 12 = 48 5 3 7 = 35 630= 0
3
4
4
7
5
8
6
12
7
15
40
117
8
16
9
16
10
12
11
7
12
2
Mean =
total number of goals total number of matches
117 = 40 =
2.93 (to 3 s.f.)
frequency
Exercise 11H
The numbers of goals scored by a football team over a 40-game period are: No. of goals
×
fx
24 2
STATISTICS
(a) Copy and complete the table. (b) What is the mean score? 4
7 Calculate the mean of the number x , from the following data:
A biologist takes a sample of 200 grasses to measure stem length, and obtains the following data:
Length x cm
Frequency
99.6
99.7
f
3
8
Frequency
f
Length x cm
25
1
33
15
Length (mm)
26
3
34
19
Frequency
27
4
35
22
28
6
36
26
29
8
37
30
30
10
38
20
31
12
39
8
32
14
40
2
17
32
100.0 100.1 22
8
8 The table shows the length of 100 rods: 196
197
198
199
200
9
18
31
22
20
9 These are the scores for 20 throws of a dice: Score
x
1
2
3
4
5
6
Frequency f
3
5
6
3
1
2
(a) What is the value of © f ?
80
81
82
83
84
85
86
1
5
11
18
8
4
3
No. of bulbs
99.9
(a) Calculate the mean length. (b) Calculate the mean length of the 80 rods that measure less than 200 mm.
A sample of 50 electric light bulbs was tested for length of life, and the results were:
Hours
99.8
f
(a) Check that π f = 200 (b) Draw up a table to calculate the mean. 5
x
(b) Calculate the value of
© fx © f
.
What name is given to this measure?
10 This frequency table gives the scores of a pair of dice, obtained in 100 th rows:
Calculate the mean length of life.
It only lasted 3 days!
well we could make bulbs that would last for years. But we wouldn’t stay in business long then!
Score
Frequency
Score
Frequency
2
0
8
15
3
3
9
21
4
7
10
11
5
8
11
8
6
8
12
7
7
12
Show that the mean score is a whole number.
6
In a game, a machine shows the numbers 0, 1, 2 or 3. An analysis of 100 games produces the results:
Number Frequency
0
1
2
3
25
55
15
5
Calculate the mean of the numbers displayed.
Means of grouped distributions Using the mid-interval value
In the case of grouped frequency tables the midinterval value is used to help find an estimate of the mean.
24 3
STATISTICS
For example, here is a frequency table recording the heights of 25 children:
Exercise 11I 1
Height (cm)
Frequency
140–144
1
Distance (km)
Number of students
145–149
3
Under 1
10
150–154
11
1–2
15
155–159
7
2–3
7
160–164
2
3–4
2
165–169
0
4–5
1
170–174
1
(a) What is the mid-interval value of the class interval 2–3 km? (b) Use mid-interval values to calculate the mean distance from school.
The table shows that 11 children had heights in the class interval 150–154 cm. This interval includes all heights between 149.5 cm and 154.5 cm. 149.5 and 154.5 are the interval boundaries. 149.5 + 154.5 = 152cm. The mid-interval value is 2 The mean height of the children can be calculated using the mid-interval value. This will g ive an approximation to the mean, as it assumes that al l 11 children have a height of 152 cm.
2
Mid-interval value (cm)
Frequency
0–9
10–19
20–29
30–39
Number
92
88
85
68
40–49
50–59
60–69
70–79
55
52
42
18
Age (years) Number
(a) What is the mid-interval value of the class interval 20–29 years? (b) Use mid-interval values to calculate the mean age of the population.
fx
f
3
x
142
1
142
147
3
441
152
11
1672
157
7
1099
162
2
324
167
0
0
172
1
172
25
3850
© f = 25 © fx = 3850 © fx
3850
© f
25
=
=
154
A census gives the following data for the ages of the population of a small village.
Age (years)
Use a table to calculate the mean height for the children:
Mean =
A group of students record the distances of their homes from school:
The age, in years, of 40 people in a certain village are:
Age (years)
0–9
10–19
20–29
30–39
Frequency
8
13
6
6
Age (years)
40–49
50–59
60–69
70–79
Frequency
3
1
2
1
What is the mean age of the villagers?
4
The heights of 60 children in a school were:
Height (cm)
100–109
110–119
120–129
130–139
Frequency
3
7
13
20
Height (cm)
140–149
150–159
160–169
170–179
Frequency
7
6
2
2
So the mean height is 154 cm. Estimate the mean height of the children.
24 4
5
STATISTICS
The mode
The marks in a test of 70 students were:
●
Marks Frequency Marks Frequency
0–9
10–19
20–29
30–39
40–49
2
5
10
13
21
50–59
60–69
70–79
80–89
90–99
6
6
3
2
2
Example 9 The shoe sizes of ten girls are
Estimate the mean mark.
6
The mode is the most common item in a distribution. It is the easiest average to find.
6, 4, 5, 4, 2, 1, 7, 6, 3, 6 What is the mode?
The masses of 100 school children were:
The most frequent shoe size is 6, so the mode is 6.
Mass (kg)
31–35
36–40
41–45
46–50
Frequency
6
8
22
31
Mass (kg)
51–55
56–60
61–65
66–70
Frequency
12
11
5
5
For a frequency distribution the mode (or modal class) has the highest frequency.
Example 10
Estimate the mean mass.
7
The ages of 50 people in a village are:
A biologist measures the lengths of 190 leaves:
Length (cm) Frequency Length (cm) Frequency
0–1.9
2–3.9
4–5.9
3
33
62
6–7.9
8–9.9
10–11.9
49
36
7
Number of marks
Paper A
Paper B
0–20
0
0
21–30
5
0
31–40
10
0
41–50
15
0
51–60
18
20
61–70
19
20
71–80
11
40
81–90
10
12
91–100
12
8
(a) What is the mid-interval value of the interval 21–30? (b) Draw separate tables to calculate the mean mark obtained in each paper. (c) Which paper was easier? Give reasons for your answer.
10–19
20–29
30–39
40–49
Frequency
12
9
7
7
6
50–59
60–69
70–79
80–89
90–99
4
3
1
1
0
Frequency
What is the modal class? The age group with the highest frequency is 0–9 years. The modal class is 0–9 years.
The table shows the marks obtained by 100 candidates in two mathematics papers.
Number of candidates
0–9
Age
Estimate the mean length.
8
Age
Exercise 11J 1
The number of books in 30 students bags are: 1, 2, 2, 1, 3, 1, 2, 4, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 0, 2, 4, 2, 4, 2, 6, 5, 2, 5, 8, 2 What is the modal number of books?
2
Find the modal class for Questions 3–7 of Exercise 11I.
3
Find the modal class for this distribution. 20 15
y c n e u10 q e r F
5 0 0
10
20
30
Age in years
40
STATISTICS
Sometimes you have to be careful which average you use.
The median ●
24 5
When the data is arranged in ascending or descending order, the median is the middle value.
Example 11
For example the heights of eleven boys, in centimetres, are:
Here are the weights of nine cricketers: 85 kg, 91 kg, 84 kg, 94 kg, 84 kg, 88 kg, 93 kg, 84 kg, 93 kg
150, 146, 158, 165, 168, 170, 158, 154, 162, 180, 181 Written in ascending order they are:
(a) Find their median weight. (b) Find the mode. (c) Which one is not a good average to use?
146, 150, 154, 158, 158, 162, 165, 168, 170, 180, 181
(a) First write them in order: 84, 84, 84, 85, 88 , 91, 93, 93, 94 middle value = median = 88 kg (b) mode = most common = 84 kg (c) The mode is not a good average to use for this data as 84 kg is also the lowest weight. The median height is the height of the middle boy—the sixth, that is, 162 cm. 146, 150, 154, 158, 158, 162 , 165, 168, 170, 180, 181 median
Exercise 11L 1
Sometimes there are two middle values, so we take the median to be halfway between them.
$23, $31, $1602, $58, $39, $31, $33, $23
(a) What was the mean amount saved? (b) Find the median. (c) Which of your answers to parts (a) and (b) is not a good indicator of the average savings? Why?
For example, if there were only eight boys: 146, 150, 154, 158, 158, 162, 165, 168 ¶ median = 158 2 158 = 158 +
So the median height is 158 cm.
2
Exercise 11K 1
Find the median of the numbers: (a) 2, 3, 5, 7, 8 (b) 6, 1, 4, 3, 9 (c) 4, 4, 1, 4, 6, 2
2
Find the median of each set of numbers: (a) 2, 5, 7, 9, 10, 11, 13 (b) 4, 3, 6, 2, 1, 8, 4 (c) 7, 2, 1, 7, 6, 9, 15, 13, 4, 9, 1 (d) 5, 8, 12, 15, 10, 12, 17, 13 (e) 3, 4, 9, 9, 6, 10, 12, 10, 8, 6, 10, 9
3
The masses of five people are 70 kg, 64 kg, 58 kg, 80 kg, 78 kg. What is the median mass?
Here are the amounts that eight friends have managed to save over the course of a year:
Here are the prices charged in eight different shops for a new watch strap: $9, $4, $3, $5, $6, $9, $3, $9
(a) (b) (c) (d)
3
Work out the median price. Find the mode. What is the mean price? Which of your answers to parts (a), (b) and (c) is not a good indicator of the average price of a new watch strap? Why?
The heights of six friends are listed: 174 cm, 101 cm, 162 cm, 183 cm, 191 cm, 178 cm
(a) Find the mean height. (b) Suggest a better average to use for this data. (c) What is the value of the average you suggested in part (b)?
24 6
STATISTICS
Exercise 11M
Finding medians from frequency distributions
1
When you have a frequency distribution you will need to construct a cumulative frequency table to determine the median value. The total of the frequencies up to a particular value is called the cumulative frequency.
The distribution of ages in Form 4 at Priory School are:
Age
5
6
7
8
9
No. of students
1
1
4
6
6
2
The ages of a class of 30 boys are: 13
14
15
16
2
6
18
4
No. of boys
14
15
16
17
8
13
21
18
4
What is the median age?
3
What is the median age? First construct a cumulative frequency table:
Age
Frequency
Cumulative frequency
13
8
8
14
13
13 + 8 = 21
15
21
8 + 13 + 21 = 42
16
18
8 + 13 + 21 + 18 = 60
17
4
8 + 13 + 21 + 18 + 4 = 64
4
4
5
6
7
8
9
10
No. of students
3
8
6
10
6
4
2
1
The histogram shows the number of brothers and sisters a class of 25 children has.
1
0
1
2
3
4
5
6
No. of brothers and sisters
What is the median number of brothers and sisters? To estimate the median from a grouped frequency distribution you will need to: ● ● ●
so the 32nd and 33rd youngest student are both 15 years old.
3
6
There are 64 students, so the median age is halfway between the 32nd and 33rd youngest student.
21st youngest student is 14 years old 42nd youngest student is 15 years old
Mark
n e r 5 d l 4 i h c f 3 o . o 2 N
8 students aged 13 21 students aged 14 or less 42 students aged 15 or less 60 students aged 16 or less 64 students aged 17 or less
From the cumulative frequency table you can see that
The marks of 40 students in a mathematics test were:
Find the median mark.
The cumulative frequency column shows that there are
That is, median age = 15 years.
4
Age
13
Frequency
Shoe size
What is the median size?
2
Example 12
The shoe sizes of 20 students are given in the table:
construct a cumulative frequency table plot points as a cumulative frequency curve read the median value off the curve.
24 7
STATISTICS
Quartiles
Example 13
To find the median you divide a set of data into two. To find quartiles you divide a set of data into four.
The table gives the masses of 100 apples:
The lower quartile is the value a quarter of the way through a set of data.
Mass (grams)
Frequency
100–109
2
110–119
15
120–129
45
The upper quartile is the value three quarters of the way through a set of data.
130–139
27
For the 100 apples in Example 13:
140–149
7
150–159
4
lower quartile = 121g upper quartile = 133g
Find the median mass.
Exercise 11N
First, construct the cumulative frequency table:
1
Mass (grams)
Cumulative Frequency
125–129
3
Less than 119.5
17
130–134
12
Less than 129.5
62
135–139
20
Less than 139.5
89
140–144
34
Less than 149.5
96
145–149
25
Less than 159.5
100
150–154
4
155–159
1
160–164
1
upper quartile lower quartile 121g median
127g
lower quartile 133g
median
Finally, read off graph median = 127g
lower quartile 20
2
0 100
110
120
130
140
150
160
If you arrange 100 apples in order of mass, the median falls between the 50th and 51st apples. When using a cumulative frequency graph, it is accurate enough to read off the 50th value as the median. The graph shows that the median mass of the apples is 127 g.
I’m really an average sort of mouse
(a) Draw a cumulative frequency table for this information. (b) What is the greatest length that belongs to the interval 130–134 mm? (c) Draw a cumulative frequency graph of the information. (d) Using a cumulative frequency of 50, estimate the median length of the mice from your graph.
100
90
Frequency
2
The results of weighing 100 apples: cumulative frequency graph
y c n e u 60 q e r f e v i t a l u 40 m u C
Length (mm)
Less than 109.5
Next, plot cumulative frequency against the upper class boundary for the mass:
80
The lengths of 100 mice, measured to the nearest millimetre, were:
100 pigs were weighed to the nearest kilogram:
Mass (kg)
Number of pigs
70–74
6
75–79
13
80–84
24
85–89
30
90–94
16
95–99
11
24 8
STATISTICS
Draw a cumulative frequency table and use it to draw a cumulative frequency graph. Use your graph to estimate: (a) the number of pigs lighter than 82 kg (b) the number of pigs heavier than 88 kg (c) the median mass.
3
Frequency
0–19
5
20–29
6
30–39
13
40–49
22
50–59
24
60–69
16
70–79
8
80–100
6
16
17
18
19
20
21
22
23
24
25
Frequency
3
5
10
16
24
21
15
9
5
2
Draw a cumulative frequency table and use it to read off the median height to the nearest centimetre. The table shows the heights of 31 men who apply for jobs in a police department.
Height (cm)
Number of applicants
Under 170
4
Ϫ175 Ϫ180 Ϫ185 Ϫ190
10
5
7
33 22 28 27 22
37 31 20 29 28
16 7 23 14 27 36 36 41 14 43
27 25 25 17 36
The scores of 65 candidates are shown in the cumulative frequency table: 20 or less
3
60 or less
55
30 or less
10
70 or less
60
40 or less
20
80 or less
64
50 or less
46
90 or less
65
Find, to the nearest whole number: (a) the median score (b) the upper and lower quartiles.
8
Height (cm)
5
49 33 38 43 28 27 17 22 23 26 28 16 41 6 35
(a) Draw a frequency table using intervals of 0.5 to 10.5, 10.5 to 20.5, etc. (b) Draw a cumulative frequency graph of the scores. (c) Use the graph to obtain an estimate of the median score and the values of the quartiles.
This table gives the heights of 110 plants:
7
In an aptitude test, the scores were: 22 44 38 17 37 13 19 36 16 31
(a) Draw a cumulative frequency table using the limits less than 19.5, less than 29.5, etc. Be careful, because the intervals are not equal. (b) Use the table to plot a cumulative frequency graph. (c) Read off the values of the quartiles and the median from your graph. (d) What pass mark allows 60% of the candidates to pass? (e) What percentage of the candidates pass, if the pass mark is 41?
5
6
The marks of 100 candidates in a test were:
Mark
4
(a) If the 31 applicants stand in order of height, which ones represent the median and the upper and lower quartiles? (b) Use a cumulative frequency graph to find an approximation for the median height.
100 people took part in a walkathon. Their times were recorded and grouped to give the following table where t is the time in minutes and f is the frequency. t
f
41–50
2
51–60
11
61–70
18
71–80
28
81–90
21
91–100
12
101–110
5
111–120
3
24 9
STATISTICS
(a) Make a cumulative frequency table and draw a cumulative frequency curve. (b) How many people took less t han 75 minutes? (c) How many people took more than 95 minutes? (d) Anyone who finished in less than 65 m inutes received a prize. How many people won prizes?
In the two groups above: Range of group 1 = 10 − 0 = 10 Range of group 2 = 6 − 4 = 2 This indicates that the scores in group 1 are widely spread and those in group 2 are not very widely spread. The interquartile range
The interquartile range, IQR, measures the spread of the middle half of the data.
●
11.5 Measures of dispersion
IQR
Look at the scores of two groups of six students in a test.
Group 1
1
0
0
10
9
10
Group 2
4
6
7
5
3
5
A student claims that both groups did equally well since: 1 + 0 + 0 + 10 + 9 + 10 =5 Mean group 1 = 6 4+6+7+5+3+5 Mean group 2 = = 5 6 However, the frequency distributions show that the performances of the two groups are very different. The mean does not completely describe the data. The scores in group 1 are much more dispersed or spread out than the scores in group 2. A single number can be found that gives a measure of this spread. There are three commonly used measures of dispersion: ● ● ●
range interquartile range standard deviation.
The range
This is a simple statistic. The range is defined as
Range
highest value
lowest value
lower quartile
The semi-interquartile range, SIQR, is half the interquartile range:
●
SIQR =
upper quartile 2 lower quartile 2
Example 14 The table shows the scores of 20 students in a science test:
Score
4
5
6
7
8
9
10
Frequency
1
0
4
5
3
4
3
Calculate the interquartile range. ●
Write the data in ascending order: 4 6 6 6 6 7 7 7 7 7 8 8 8 9 9 9 9 10 10 10
●
Divide the data into four groups of five 4 6 6 6 6 7 7 7 7 7 8 8 8 9 9 9 9 10 10 10 lower median upper quartile quartile
Lower quartile = Upper quartile =
Generally the larger the value of any of these statistics the more spread out the data. You will be covering the first two of these in this book.
●
upper quartile
6+7 2 9+9 2
=
6 12
=
9
IQR = 9 2 6 12 = 2 12 To find the IQR for grouped frequency distributions a cumulative frequency curve has to be drawn.
25 0
STATISTICS
This can be seen in the histogram where masses of 46 kg and 54 kg are not typical of the group.
Example 15 The results of weighing a group of 25 students are shown in the histogram and cumulative frequency graph.
Exercise 11O 1
9
Work out the interquar tile range for this table of data.
8
Score
1
2
3
4
5
6
7
Frequency
2
3
4
1
3
4
s t n e d 6 u t s f 5 o s r e 4 b m u 3 N
2
134 cm, 152 cm, 143 cm, 148 cm, 159 cm, 129 cm, 138 cm, 142 cm, 137 cm, 131 cm, 151 cm, 146 cm. Calculate the interquartile range for the data.
2
3
1 0 46 47
48
49 50
51 52
53
The histogram shows the masses of a group of students.
54
6
Mass (kg)
s t n 5 e d u 4 t s f o r 3 e b m2 u N
Cumulative frequency graph y 25 c n e u 20 q e r f 15 e v i t 10 a l u m 5 u C
Here are the heights of twelve girls:
upper quartile
lower quartile
1
0
0 45 46 47 48 49 50
46 47
51 52 53 54 55
51 52
53 54
(a) What is the range for the group? (b) Draw a cumulative frequency curve, and from the graph find: (i) the quartiles (ii) the interquartile range. (c) What information about the masses of the group of students can you derive from your answers to part (a) and part (b(ii))?
Find the range and interquartile range. From the histogram it can be seen that the range is 54.5 kg 2 45.5kg = 9 kg. From the cumulative frequency graph it can be seen that the interquartile range is 51kg 2 49.6kg = 1.4kg.
The range of 9 kg tells you that there are some extreme results.
49 50
Mass (kg)
Mass(kg)
In Example 15 the interquartile range of only 1.4 kg tells you that the majority of data is closely distributed about the median.
48
4
A lab technician checks the accuracy of two balances, A and B, using a standard 100 g mass. He weighs the mass a hundred times on each balance, and records the readings. The frequency table for the results is shown.
STATISTICS
(d) Use the graphs to estimate the median and the interquartile range in each case. (e) Using your results in par t (d), describe how the histograms of the results would differ. (f) Draw histograms of the results for papers A and B. Compare them. Are they the shapes you expected?
Frequency Reading (g)
for A
for B
99.97
12
3
99.98
48
8
99.99
29
19
100.00
11
28
100.01
0
23
100.02
0
10
100.03
0
7
100.04
0
2
6
(a) Construct a cumulative frequency table for each balance. (b) Draw a cumulative frequency graph for each balance. (c) For each, find: (i) the median (ii) the range (iii) the interquartile range. (d) Which machine is more accurate? Explain your choice. 5
25 1
The table shows the marks obtained by 100 candidates on two mathematics papers.
Paper B
0–20
0
0
21–30
5
0
31–40
10
0
41–50
15
0
51–60
18
20
61–70
18
20
71–80
11
36
81–90
10
16
91–100
13
8
(a) Plot the cumulative frequency curves of the marks for each paper. What is the range for each? (b) What pass mark would allow 70% of the candidates to pass paper A? (c) What pass mark would allow 70% of the candidates to pass paper B?
Frequency
Under 55
60
55–64
124
65–74
147
75–84
86
85–94
55
95 and over
28
Exercise 11P – mixed questions
Number of candidates Paper A
Mass (kg)
From a graph of cumulative frequency, estimate the median and the interquartile range. Then calculate the semi-interquartile range.
1 Number of marks
This table gives the masses of 500 men to the nearest kilogram.
The heights of 50 plants of a certain species were measured to the nearest centimetre and grouped to give this table.
Height (cm)
No. of plants
15–17
3
18–20
9
21–23
15
24–26
14
27–29
7
30–32
2
(a) What are the boundaries of the interval 24–26 cm? (b) What is the maximum height of the plants? (c) Draw a histogram to illustrate the information. 2
An organisation gives an aptitude test to all applicants for employment. The results of 100 tests are shown in the table.
25 2
STATISTICS
5
Frequency
1–10
5
11–20
8
Score
f
21–30
11
1–5
1
31–40
12
6–10
4
41–50
20
11–15
4
51–60
16
16–20
7
61–70
13
21–25
13
71–80
7
26–30
9
81–90
5
31–35
7
91–100
3
36–40
3
41–45
1
46–50
1
(a) Draw a histogram to illustrate this information. (b) What percentage of the applicants scored less than 60.5? (c) What percentage of the applicants scored between 50.5 and 80.5? 3
4
Draw the frequency polygon for the heights of the 50 children recorded in the table.
The table shows the distance travelled by 70 cars on the same amount of petrol.
Distance (km) No. of cars
150–159
160–169
170–179
180–189
190–199
5
8
18
25
14
Find the mean distance travelled by the cars.
f
130–134
1
135–139
7
140–144
16
Mass (kg)
Frequency
145–149
15
51–55
4
150–154
5
56–60
6
155–159
4
61–65
10
160–164
2
66–70
13
71–75
9
76–80
4
81–85
4
7
Draw the corresponding histogram. 50 40 30 20 10 100
6
Height (cm)
Here is a frequency polygon.
50
The scores of 50 students in a biology test are shown in the table. Calculate the mean score.
Score
150
200
250
300
The masses, in kilograms, of 50 people are recorded in the table.
(a) What is the greatest mass in the interval 61–65 kg? (b) Draw a cumulative frequency table for the information. (c) Draw the corresponding cumulative frequency graph. (d) Use the graph to estimate: (i) the number of people weighing less than 63 kg
STATISTICS
(ii) the number of people weighing more than 74 kg (iii) the median weight. 8 The table shows the marks obtained by 100 students in a biology class test. Score 1–10 11–20 21–30 31–40 41–50 51–60
f
6 9 10 18 32 25
9 Use a graph to estimate the median mass of 300 ten-week old rats, given these results:
Frequency
(a) State the range for both subjects. (b) In which subject do the students show more of a mixed ability? (c) Plot both cumulative frequency curves on the same axes, and state: (i) the pass mark in each subject that would allow 70% of the students to pass the test (ii) the pass mark that would allow the same number of students to pass the test in each subject. What number of students is this? 11 In a survey, the masses of students were recorded. The table gives the data.
(a) Draw a cumulative frequency curve. (b) Use the curve to estimate: (i) the upper quartile (ii) the lower quartile (iii) the mark that would allow 55% of the students to pass the test. (c) Calculate the interquartile range.
Mass (g)
25 3
38–39
40–41
42–43
44–45
46–47
139
89
52
20
0
10 The table shows the marks of 100 candidates in mathematics and geography tests. Score
Mathematics
Geography
1–10
0
5
11–20
0
7
21–30
10
8
31–40
12
11
41–50
21
19
51–60
35
13
61–70
16
12
71–80
6
11
81–90
0
8
91–100
0
6
Mass (kg)
f
20–29
16
30–39
26
40–49
34
50–59
44
60–60
40
70–79
24
80–89
12
90–99
4
(a) Draw a cumulative frequency table for the data. (b) Draw a cumulative frequency curve. (c) Use the curve to estimate: (i) the median mass of the students (ii) the number of students who weighed between 42 kg and 76 kg (iii) the percentage of students who weighed no more than 62 kg.
25 4
11
STATISTICS
Consolidation 444 + 447 + 600 + 1057 + 1824 + 918 + 770 + 620 + 936
Example 1
=
The mass in grams of 20 bars of soap made at a factory are:
50 7616
=
50
=
152.3 cm
134 137 132 134 135 135 134 133 135 136 136 134 134 137 136 132 133 134 134 135
Exercise 11
Construct a frequency table for this data.
1
Roll a die 30 times. Construct a frequency table for the data you obtain.
2
The masses of 24 children in kilograms are:
Mass (g)
Tally
Frequency
132
2
133
2
134
7
135
4
136
3
137
2
53 48 49
3
The life time of 100 electric lights bulbs is shown in the table. Draw a histogram to show this information. Frequency
801–900
8
901–1000
12
1001–1100
51
10
1101–1200
23
0
1201–1300
6
800
58 55 57
56 54 52
47 53 63
Three weeks after planting, the heights in centimetres of 50 seedlings were: 0 –2.9
3.0 –5.9
6.0 –8.9
9.0 –11.9
12.0 –14.9
Number of seedlings
3
12
15
16
4
(a) Draw a histogram to show this data. (b) Estimate the mean height of a seedl ing. (c) Draw a cumulative frequency graph for this data. (d) Find the median of the distribution.
900 1000 1100 1200 1300 Life time (hours)
Application 11
4
148
149
150
151
152
153
154
155
156
3
3
4
7
12
6
5
4
6
What is the mean height of the boys? Mean height = =
60 44 53
50
The table shows the heights in centimetres of 50 boys.
Frequency
51 65 54
Height (cm)
y c 40 n e u 30 q e r 20 F
Example 3
Height
44 53 53
(a) Construct a suitable grouped frequency table to show the data. (b) Draw a histogram to show the data.
Example 2
Life time (Hours)
42 52 49
© fx © f
1 3 3 148 2 + 1 3 3 149 2 + 1 4 3 150 2 + 1 7 3 151 2 + 1 12 3 152 2 + 6 3 153
+
5 3 154
+
4 3 155
3 + 3 + 4 + 7 + 12 + 6 + 5 + 4 + 6
+
6 3 156
Conduct a survey to find out how many hours your class spends watching television each week. (a) Draw a grouped frequency table to show the data. (b) Draw a histogram to show the data. (c) Find the mean time spent watching television each week. (d) Draw a cumulative frequency curve from your data. (e) Estimate the median number of hours watched each week.
25 5
STATISTICS
5
Do girls in your class spend more time each week on their homework than boys? (a) Conduct a survey to find out the answer to this question. (b) Display your results on: (i) a histogram (ii) a cumulative frequency graph. (c) Estimate the mean and median times spent per week on homework by both boys and girls.
6
At the 2008 Olympic Games in Beijing, the distances thrown by the women’s discuss finalists are shown below. Distance (metres) Frequency
56Ϫ 58Ϫ 4
3
0Ϫ
2Ϫ
4Ϫ
7
10
6
Ϫ
8Ϫ
3
2
(a) Estimate the mean distance a discuss was thrown by a finalist. (b) Draw a cumulative frequency graph of the data and use it to estimate the length of the median throw.
Companion CD Want some extra practice? Go to Chapter 11 on your companion CD for fur ther exercises, animations, and full worked solutions.
Summary You should know ...
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1
1
How to show discrete data in a frequency table. For example: this list of scores: 0, 1, 1, 5, 2, 4, 3, 1, 2, 4, 0, 5, 1, 5, 4
Here are the numbers of catches taken by 20 cricketers during a season: 6, 7, 2, 0, 4, 5, 3, 1, 6, 5, 1, 9, 7, 1, 2, 3, 8, 9, 7, 6
can be shown in a frequency table:
Show the data in a frequency table.
Score
0
1
2
3
4
5
Frequency
2
4
2
1
3
3
This means there were three 5s in the list.
2
How to display data in bar charts, pictographs and pie charts. For example: the data above can be shown as: Bar chart
Pictograph
4
0
y c 3 n e u q e r 2 F
1
2
From your frequency table for Question 1, draw a: (a) bar chart (b) pictograph (c) pie chart.
2 3 4
1 0
1
2 Score
3
4
5
5 T
25 6
STATISTICS
ˇ
Pie chart 5
0 the angle representing a score of 1
1
4
= 3
3
2
4 " 360° = 96° 15
How to find the interval boundaries of a given interval. For example: in a table in which the class intervals are 100–104, 105–109, 110–114 p the boundaries of the second class are 104.5 and 109.5.
3
The table shows the mass in kilograms of some boys.
Mass
f
45–49
7
50–54
8
55–59
11
60–64
3
State the: (a) lower boundary of the second class (b) upper boundary of the third class.
4
A histogram has no spaces between the bars. The area of each bar represents the frequency. For example: the information in this table:
4
Draw a histogram to show the information in this table.
Weight (kg) Height (cm)
Frequency
10–19
1
5–9
3
20–29
4
10–14
5
30–39
9
15–19
6
40–49
11
20–24
4
50–59
15
60–69
27
70–79
21
80–89
16
90–99
7
can be shown in a histogram. The class boundaries of this bar are 19.5 and 24.5
6 y5 c n e 4 u q3 e r F2 1 5
10
Frequency
15 20 Height (cm)
25 T
25 7
STATISTICS
ˇ
5
How to draw a frequency polygon using the mid-interval values.
5
For example: Height (cm)
The number of spectators attending a football match are shown in the table.
120–129
130–139
140–149
150–159
Age (years)
f
3
1
7
4
11–20
30
21–30
50
31–40
70
41–50
60
51–60
40
No. of children
7 n r e 6 d l i 5 h c f 4 o r 3 e b 2 m u 1 N 0
Mid-interval value = 154.5 cm
Draw a frequency polygon for the data. 120
130
140
150
160
Height (cm)
6
Mean =
sum of data
6
number of values
Mode is the most common value in a distribution. Median is the middle value when the data is arranged in order.
Find the: (a) mean (b) mode (c) median.
For example: 3, 3, 4, 7, 8
Mean =
3+3+4+7+8
= 5 5 Mode = most common value = 3 Median = middle value = 4
7
How to draw a cumulative frequency table from a frequency table. For example: Height (cm)
120–129
130–139
140–149
150–159
3
1
7
4
No. of children
the cumulative frequency table would be:
Height (cm)
Cumulative frequently
less than 129.5
3
less than 139.5
4
less than 149.5
11
less than 159.5
15
Here is a list of height of plants: 4 cm, 6 cm, 9 cm, 10 cm, 5 cm, 3 cm, 2 cm, 1 cm, 8 cm, 11 cm, 9 cm, 4 cm
7 Mass (kg)
f
40–49
4
50–59
13
60–69
20
70–79
10
80–89
3
Draw a cumulative frequency table from the frequency table given.
T
25 8
STATISTICS
ˇ
8
How to use the mid-interval values to estimate the mean fr om a grouped frequency table.
8
For example: id-interval value of the first interval is 7
Height (cm)
Frequency
5–9
3
10–14
5
15–19
4
Mark
1 3 3 7 2 + 1 5 3 12 2 + 1 4 3 17 2 12 © f = 12.4 (to 3 s.f.)
Mean =
The marks in a test of 50 students were:
© fx
Frequency
0–19
4
20–39
12
40–59
21
60–79
8
80–99
5
Use the mid-interval values to estimate the mean mark.
=
So the mean height is 12.4 cm.
9
You can draw a graph of cumulative frequency and use it to find the median, upper quartile, lower quartile, interquartile range and semi-interquartile range. For example: Mass (kg)
40–49
50–59
60–69
70–79
Frequency
4
13
20
10
3
Cumulative frequency
4
17
37
47
50
80–89
Median = 63 kg L.Q. = 56 kg U.Q. = 70 kg
50
y c n 40 e u q e r 30 f e v i t 20 a l u m10 u C 0
Interquartile range = 70 kg − 56 kg = 14 kg Semi-interquartile range = 14 kg 2 = 7 kg 40 50
60 L M . Q e . d i a n
70 80 U . Q .
90 Mass (kg)
9
The table shows the amount of pocket money received weekly by a group of children.
Pocket money ($)
Frequency
10–19
5
20–29
7
30–39
18
40–49
24
50–59
14
60–69
12
70–79
10
80–89
6
90–99
4
(a) Draw a cumulative frequency table and graph. (b) Use the graph to find the: (i) median (ii) lower quartile (iii) upper quartile. (c) What is the interquartile range? (d) What is the semiinterquartile range?
