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The mode, median and range
Calculate the mode, median and range for a set of data
Decide which average best represents the set of data
Key words
average
data
mode
median
range
An average gives information that is typical of a set of data .
The mode is a type of average. In a set of data it is the value or values that happen
most often:
1
1
2
2
2
3
4
4
6
9
The mode is 2 in this set of numbers as 2 is the value that occurs most often.
Data that is not about numbers (e.g. colours of car, exam grades) can only have the
mode as an average.
The median is another type of average. In an ordered set of data it is the middle
value:
1
2
2
4
3
6
9
The median is 3 in this set of numbers as it is the middle value.
If there is an even number of values in the set of data, the median is halfway between
the middle two values.
The range is not an average. It is the smallest value subtracted from the largest and
shows how spread out the data is:
1
2
2
3
4
6
9
The range is 8 in this set of numbers, because 9 1 8
Example
The ages in years of a group of people are:
12
10
7
35
35
What is a) the mode b) the median c) the range of their ages?
d) Which average would you use to represent the data? Explain why.
e) Another 10 year old joins the group. Calculate the new mode, median
and range.
Put the ages in order: 7 10 12 35 35
a) The mode is 35
There are more 35’s than
any other number.
b) The median is 12
c) The range is 28.
d) I would use the median, as it is halfway between the lower
and upper ages. The mode in this case is the highest value
so does not represent the younger people in the group.
e) There are now 2 modes: 10 and 35 years. The median is
11 years. The range is still 28.
46
Maths Connect 1R
12 is the middle number.
The range is the largest
smallest value
(35 7 28)
The ages in order are now
7 10 10 12 35 35.
The median is half way
between 10 and 12.
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Exercise 5.1 .............................................................................................
Ravi asks his friends what mark they got for their last piece of homework.
4
a)
b)
c)
d)
e)
4
5
6
6
7
8
8
8
9
9
What is the median mark?
What is the mode?
What is the range?
Ravi wants to include his mark of 7. What is the new median mark?
Which of the averages would you use to represent the data? Explain why.
The table shows the number of days that rain fell each month in London:
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Number of days
12
10
10
11
12
12
15
14
13
13
15
12
Find the mode, median and range of the data over the year.
Which of these cannot have a median value, only a mode?
a)
b)
c)
d)
e)
f)
g)
h)
Age
How long a person can hold his or her breath
Amount in £’s spent on trainers
Colour of school uniform
Weight
How long it takes to run 100 m
Type of pet owned
Favourite snack.
Information that is not to do
with numbers (called
categorical data) can only
have the mode as an average.
The table below shows the typical temperature each month for Summertown.
Month
°C
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
2
5
7
12
16
19
23
25
24
10
9
5
a) What is the range of the temperatures?
b) What is the median temperature?
Which consecutive whole numbers have a median of
8 and a range of 6?
3, 4, 5 are consecutive and have
a median of 4 and a range of 2.
Which consecutive whole numbers have a total of 14 and a median of 312?
Five dice are thrown. The numbers showing have a mode and median of 3 and a range of
2. Find all the possible solutions.
Lynn, Mac, Toby and Mair have a combined age of 70 years. The modal age is 33 years
and the median age is 18 years.
a) What are the ages of the 4 people?
b) Which average would you use to represent the data? Why?
Which consecutive whole numbers have a total of 160 and a range of 4?
Make a table using the headings below.
Data which can only have a mode
Data which can have a mode and median value
e.g. Hair colour
e.g. Height
Write a sentence describing the difference in the type of data for the two headings.
The mode,
median
and range 47
Number
1: Proportion
47
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Key words
The mean
mean
range
average
sum
Calculate the mean for a set of data
The mean is an average. Like the median, it can only be used for data that is
numerical. In everyday language when people say ‘average’ they are usually talking
about ‘the mean’.
the sum of all the data
The mean is calculated as: the number of items of data
Example
The best 8 long jump distances of an athlete were (in metres)
8.78
8.76
8.76
8.72
8.71
8.65
8.65
8.57
Find the mean distance.
the total distance jumped
The mean the number of jumps
8.78 8.76 8.76 8.72 8.71 8.65 8.65 8.57
8
69.6
8
8.7m
Exercise 5.2 ..........................................................................................
The number of people attending Premiership football matches at the beginning of 2003
were:
38 096, 31 838, 23 413, 40 163, 40 034, 67 603, 52 147, 31 890
Find the mean attendance per match.
The temperature at dawn for ten days in March were:
5
3
1
2
0
4
2
1
4
3
Find the mean temperature.
Find the missing numbers shown by
Each set of numbers has a mean of 6.
48
a)
7
7
b)
5
5
c)
14
Maths Connect 1R
or
14
. They are all whole numbers, greater than 0.
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Kate asks her friends how
many times they were late
for school last term. She
puts her results in a
spreadsheet.
A
B
C
1
Number of times Ordered number of
late for school
times late for school
2
3
0
3
0
0
4
2
1
5
0
2
6
1
3
7
Total
8
Mean
a) Look at column B. What will Kate enter in cell B7 to find the total number of times her
friends were late?
b) What will Kate enter in cell B8 to find the mean number of times her friends were late?
c) Explain how she could use column C to find the median, mode and range of her data.
a) If the mean of four numbers is 10, what is the sum
of the four numbers?
b) If the mean of ten numbers is 17.3 what is the sum
of the ten numbers?
Natacha needs a mean average of 85% in her four
tests for an A* grade. So far her marks are 87% and
79%. What is the minimum average mark needed in
the final two tests to get an A*?
sum of the numbers
The mean amount of numbers
Start by finding the total sum she needs
in the 4 tests for a mean of 85%.
A set of numbers has a mean of 25 and a total sum of 350. How many numbers are in the
set?
The table below shows the countries with the greatest number of cars.
Which of the four countries has the most number of cars per head of population?
Country
Number of cars (millions)
Population (millions)
USA
147
277
Japan
43
127
Germany
40
86
Italy
30
58
Investigation
Make a list showing how many days there are in each month during a year that is
not a leap year. Find the mode and the median number of days in a month.
What is the mean number of days in a month in a leap year?
What is the mean number of days in a year over a period of four years?
The mean 49
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Key words
Frequency tables
frequency
event
frequency table
mean
Record data in frequency tables
Use frequency tables to calculate the mean
The frequency of an event is the number of times that event occurs.
A frequency table is a way of organising this data.
To find the mean , instead of adding up all the bits of data separately, we can find the
totals for each group.
Example
Thirty people were asked how many pets they owned. These are the results:
0
1
3
1
4
2
1
2
1
1
3
3
1
3
2
2
4
3
1
2
2
2
2
1
5
3
1
2
1
1
a) Draw a frequency table to show this data.
b) Use the table to work out the mean number of pets owned.
a)
Number of
pets owned
Tally
Frequency
0
1
2
3
4
5
1
11
9
6
2
1
30
Total number
of pets owned
01
1 11
29
36
42
51
Record the results in
a tally column.
0
11
18
18
8
5
9 people own 2 pets,
so 18 pets in total.
The number of pets
owned in total.
60
60
total number of pets owned
b) Mean 2
30
total frequency
The number of people asked.
(total frequency)
Exercise 5.3 ..........................................................................................
The frequency table below shows the number of letters for the first 50 words of a book
Complete the table to find the mean length of word.
50
Number of letters
Frequency
1
5
2
7
3
12
4
11
5
8
6
4
7
2
8
1
Maths Connect 1R
Number of letters frequency
First find the total
frequency and the
total number of
letters frequency.
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A teacher records the number of pupils who are absent each day for a month.
a) Put this information into a frequency table.
b) Find the mean number of absent pupils each day.
3 0 1 2 2 4 1 3 4 0
1 2 3 1 1 0 0 0 2 0
Jake is collecting information about how many complete books his friends read in one
week. He records the results for 15 friends. Unfortunately he has lost some of the
numbers. Can you help him?
Number of books
Frequency
Number of books frequency
0
0
0
The frequency column
must add up to 15 as
his survey was for a
total of 15 friends.
1
2
2
3
4
12
4
5
2
Total 10
Total a) Copy and complete the frequency table.
b) Calculate the mean number of books read in one week.
The table below shows the number of children in each house in one particular street:
Number of children per house
Frequency
0
1
2
3
4
13
15
16
5
1
State whether each of the following statements is true or false:
a) The modal number of children is 4
The modal number is a different
b) The range of children is 4
way of asking what the mode,
c) The total number of children is 50
the most common value, is.
d) The number of houses in the street is 50
e) The mean number of children is 1.32
f) 37 houses contain children.
A primary school has 8 classes consisting of the following number of pupils:
25, 26, 27, 26, 29, 29, 27, 29
Class size
Frequency
a) Draw a frequency table for this data with the headings:
b) Find the mean number of pupils per class.
c) The mean class size must be 30 or less. What is the greatest number of pupils that can
join the school for this to be so?
Sunita plays hockey for her school team. The tally chart shows the number of goals she
has scored in matches this season.
a) Copy and complete the frequency
table.
b) Find her mean number of goals
per match.
c) Her mean score after the next game
changes to 1.6 goals per match.
How many goals did Sunita score
in the next game?
Number of goals scored
in a match
Tally
Frequency
0
1
2
3
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Key words
Interpreting diagrams
bar chart
compound bar chart
bar-line graph
pie chart
Interpret diagrams, graphs and charts
Draw conclusions and find simple statistics from diagrams
Bar charts , compound bar charts , bar-line graphs and pie charts can all be used
to represent data. Showing data in this way makes it easier to interpret and spot trends.
a) i) The range is 15 12 3
Compound bar chart showing the
number of pupils attending school
Boys
ii) The range is 15 14 1.
b) Yes. Putting the values in order gives 26 27 28 29 30, so the
median is 28 pupils.
c)
(28 27 29 30 26)
140
28
5
5
d) There is a total of 3 (1 from Wednesday and 2 from Thursday)
above the mean line. Below the line, there is also a total of
3 from the tops of the bars to the mean line (1 from Tuesday,
2 from Friday). As these are equal this shows that the
mean is 28.
e) No, as no value appears more than once.
52
Maths Connect 1R
Day
day
Fri
day
urs
Th
ay
esd
dn
We
Tu
esd
ay
ay
Girls
nd
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
Mo
a) What is the range of the
number of i) boys
ii) girls attending
school?
b) Is there enough
information on the
diagram to find the
median value?
c) The red dotted line
shows the mean value.
Show that the mean
number of students
attending school each
day is 28.
d) How does the diagram
help to show that the
mean is 28?
e) Is there a modal value?
Frequency
Example
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Exercise 5.4: ............................................................................................
Meena draws a bar-line graph showing the
number of goals scored by the school
hockey team.
a) Which of the three averages can be
calculated from the graph? Calculate
any averages that can be found.
b) Calculate the range of goals scored.
c) In week 7 the team scores 5 goals.
Calculate the new averages and the range.
Bar-line graph showing the
number of goals each week
6
5
4
3
2
1
0
week 1 week 2 week 3 week 4 week 5 week 6
The bar chart shows the grades for a class in
12
10
Frequency
their last mathematics test.
a) Explain why only the mode can be found
from the diagram.
b) Can the range be calculated?
c) How many pupils took the test?
Bar chart showing grades
for maths test
8
6
4
2
0
a) Which city had the month
d)
e)
f)
g)
B
C
Grade
D
E
Comparative bar chart showing the number
of days of rain for 2 cities in China
with the most days of rain?
12
Which month was this?
10
Which city had the month
8
with the fewest days of
6
rain?
4
Which month was this?
2
How many days of rain
0
J F M A M J J A S O N D
did each city have in total
Month
during the year?
Which city has the greatest range of days of rain?
What is the mean number of days of rain for each of the two cities?
Number of days
b)
c)
A
Pupils in Year 7 were asked which
sport was their favourite.
a) Which sports were liked equally
by both boys and girls?
b) Which sport was preferred by
girls? By boys?
c) How many pupils were asked
in total?
d) How many girls were asked?
e) How many boys were asked?
Guangzhou
Beijing
Comparative bar chart showing favourite sport
Girls
Boys
Swimming
Cricket
Table tennis
Hockey
Tennis
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Number of pupils
Choose a bar graph or pie chart from a newspaper, magazine or travel brochure. Make
up some questions about the information it shows for a partner to answer. Include
questions about any averages that can be found.
Interpreting diagrams 53
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Key words
Describing probabilities
event
outcome
probability
chance
impossible
certain
Understand and use the probability scale from 0 to 1
Know that the probability of an event not happening is
(1 probability of the event happening)
An event (such as rolling a dice) can have different outcomes . When an ordinary
six-sided dice is rolled, the possible outcomes are 1, 2, 3, 4, 5 or 6.
Probability is a measure of chance . A probability can be expressed using words or
numbers. When probability is expressed as a number it must be between 0 and 1.
1
2
0
unlikely
impossible
even
1
likely
certain
Probability can be written as a fraction, a decimal or a percentage.
the number of ways the outcome can happen
Probability of an outcome the total number of possible outcomes
Example
In a game of scrabble there are eight letters remaining:
e
a
t
v
t
i
s
This means that each
letter has the same
chance of being chosen.
t
A letter is chosen at random.
What is the probability it is a) t b) a vowel c) s or v d) b e) i f) not an i?
a)
b)
c)
3
8
3
8
2
8
or
An event either happens or it doesn’t
happen. The probability of an event
happening the probability of it not
happening equals 1,
1
7
so 1 8 (probability of an i) 8
1
4
d) 0
1
8
7
8
e)
f)
Exercise 5.5: .........................................................................................
Decide if each of the following is a fair way to start a game using a dice.
If it is not fair, which player is most likely to start each time?
54
Player 1
Player 2
a)
Odd
Even
b)
6
Not a 6
c)
Less than 4
4 or more
d)
1 or 6
2, 3, 4 or 5
Maths Connect 1R
Starting a game is fair
if both players have
the same chance of
starting.
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All the letters from the word BRILLIANCE are written on separate pieces of paper and
put in a bag. One letter is taken out without looking. What is the probability that the
letter will be a) a vowel b) L c) Z d) A, B or C e) I f) Not I?
Show all your answers as a fraction, a decimal & a percentage.
A bag of counters contains 20 counters: 14 red and 6 blue. A counter is chosen at random.
It is red. What is the probability that the next counter chosen is red if
a) the counter is replaced into the bag
b) the counter is not replaced?
The probability of a train being late is 0.16. What is the probability that it will not be late?
Mandy has accidentally put two old batteries back into a packet that also contains six
new ones. She picks out a battery.
a) What is the probability that it will work?
b) What is the probability that it will not work?
c) How many batteries should she take out to be certain that at least one will work?
Copy and complete the spinner so that the probability of getting an
1
1
odd number is 2 and the probability of getting a 1 is 3.
A bag contains 20 cubes of four different colours.
The probability of choosing each of the colours is shown.
How many of each colour cube are there?
Colour
Probability
Red
1
4
Green
1
5
Yellow
9
20
White
1
10
A bag contains a number of counters. The probability of choosing a green counter is 0.2.
a) Explain why it is not possible for there to be eight counters.
b) Harry takes two counters from the bag. They are both green. What is the smallest
possible number of counters in the bag?
A bag contains ten blue, nine white and six red cubes.
a) A cube is taken out of the bag at random. Find the probability of each of the different
colours being chosen as a decimal.
1
5
b) A cube is removed from the bag. The probability of a blue cube is now 12, a white is 3
1
and a red is 4. What colour was removed from the bag? Explain your answer.
Investigation
Draw a probability line showing the probability of six different events happening
tomorrow.
Tomorrow I will:
Impossible
Certain
Go to the
moon
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Key words
Probabilities of events
Find all the possible outcomes of one event
Find all the possible outcomes of two events happening together
Record all possible outcomes using a table
random
outcome
event
If a letter is chosen from the alphabet at random (every letter has the same chance of
being chosen) there are many possible outcomes .
Different outcomes may have different probabilities. For example, the probability of
getting a vowel is less than that of getting a consonant.
When two or more events happen together, such as tossing a coin and throwing a
dice, it is useful to organise all of the outcomes in a table so that no outcomes are
missed.
Example 1
If a letter is chosen at random from the word REVERSE there are four
possible outcomes choosing R E, V and S.
A letter is chosen at random.
What is the probability it is a) an R? b) an E? c) a V?
d) a vowel? e) a consonant?
2
d) probability of a vowel 7
3
e) probability of a consonant 7
a) probability of an R 7
b) probability of an E 7
3
4
1
c) probability of a V 7
Example 2
Helen spins a 3-sided spinner and throws a dice.
a) List all the possible outcomes using a table.
b) Use the table to find the following
probabilities: i) a red on the spinner
ii) a 6 on the dice
iii) a 3 on the dice and white on the spinner
iv) not a blue
v) white on the spinner and a 5 or 6 on the dice.
a)
Blue
1 1, blue
2 2, blue
3 3, blue
4 4, blue
5 5, blue
6 6, blue
56
Maths Connect 1R
b) i)
Red
1, red
2, red
3, red
4, red
5, red
6, red
White
1, white
2, white
3, white
4, white
5, white
6, white
ii)
iii)
iv)
v)
6
18
3
18
1
18
12
18
2
18
1
or 3
1
or 6
2
or 3
1
9
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Exercise 5.6 .............................................................................................
Twenty cards are numbered from 1 to 20. If a card is chosen at random:
What is the probability of choosing an odd number?
What is the probability of choosing a number less than 5?
What is the probability of choosing a number divisible by 3?
What is the probability of choosing a number in the range of 11 to 15 inclusive?
a)
b)
c)
d)
A school sells two colours of tickets for a raffle.
Pink tickets are numbered from 1 to 50. Blue tickets are numbered from 1 to 100.
All of the tickets are sold. If the tickets are drawn at random find the probabilities of drawing:
a) a number from 1 to 50
b) an odd number
c) a blue ticket
d) a number with two or three digits.
Draw a table to show the outcomes when these
two spinners are spun together.
Spinner 1
4
1
3
2
Spinner 2
Draw a table to show the outcomes when a 5-sided
spinner and a coin are spun together.
Use the table to find the following probabilities:
a) The spinner landing on blue.
Spinner
b) The coin landing on tails.
c) The spinner not landing on white or red.
d) The spinner landing on green and the coin landing on heads.
e) The spinner landing on blue or black and the coin landing on tails.
f) The spinner not landing on red and the coin not landing on heads.
Coin
Copy and complete the tables below to show the total when two different pairs of dice
are thrown and the numbers showing are added.
Pair 2: dice from 0–5
Pair 1: ordinary dice 1–6
1
1
2
2
3
4
5
6
0
0
0
2
1
3
2
4
3
5
4
6
11
1
2
3
5
Which pair of dice give the greatest probability of getting a total of:
a) i) 4
ii) 8
iii) 6
iv) an even amount
b) List the probabilities for your answers to part a).
4
5
9
v) less than 6?
Investigation
Two normal 1–6 dice are thrown and the numbers showing are multiplied together.
Player A wins if the answer is even, otherwise player B wins.
Play this game with a partner 30 times, recording who wins each time in a frequency table.
Draw a table to investigate the different outcomes. Is the game fair?
Probabilities of events 57