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Form 4 Statistics
Chapter 4 Statistics
Section 4.1The mean, mode, median and Range
The idea of an average is extremely useful, because it enables you to compare one set of data with
another set by comparing just two values – their averages.
There are several ways of expressing an average, but the most commonly used averages are the mean,
mode, median and range.
The mean
The mean of a set of data is the sum of all the values in the set divided by the total number of values in
the set. That is:
This is what most people mean when they say ‘average’.
Example 1
The ages of 11 players in a football team are:
21 23 20 27 25 24 25 30 21 22 28
What is the mean age of the team?
Sum of all ages = 266
Total number in team = 11
Therefore,
Mean age = 266 11 = 24.2
Form 4 Statistics
Consolidation
Find the mean for each set of data:
a) 7 8 3 6 7 3 8 5 4 9
b) 47 3 23 19 30 22
c) 1.53 1.51 1.64 1.55 1.48 1.62 1.58 1.65
The Mode
The mode is the value that occurs the most in a set of data. That is, it is the value with the highest
frequency.
The mode is a useful average because it is very easy to find and it can be applied to non-numerical data.
For example, you can find the modal birthday month of the class.
Example 1
What is the mode of the following?
1, 1, 3, 7, 10, 13
Which is the most popular number?
Form 4 Statistics
Consolidation
1) 3 4 7 3 2 4 5 3 4 6 8 4
2) 100 10 1000 10 100 1000 10 1000 100 1000 100 10
3) rain sun cloud sun rain fog snow rain fog sun snow sun
4) The frequency table shows the colours of eyes of the students in a class.
Blue Brown Green
Boys 4 8 1
Girls 8 5 2
a. How many students are in class? ________________________________
b. What is the modal eye colour for:
i. boys ii. girls iii. the whole class
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c. After two students join the class the modal eye colour for the whole class is blue. Which
of the following statements are true?
Both students had green eyes
Both students had brown eyes
Both students had blue eyes
You cannot tell what their eye colours were
The Median
The median is the middle number of a set of data.
A way to remember this is the following:
MEDIAN
MIDDLE
Therefore the median is the middle number in a set of ordered numbers.
We can also use the following formula:
Median = ½ (n + 1)
Example 1
Let try and find the median of the following numbers
1, 3, 7, 10, 13
Therefore the median is 7
Form 4 Statistics
Special Case
Let us find the median of the following set of numbers
18, 19, 21, 25, 27, 28
If we have two median numbers all we have to do is find the mean of the two numbers.
Therefore;
Sum := 21 + 25 = 46
Mean := 46/2 = 23
As a result, 23 is the median.
To find the Median
1. Arrange the numbers in order from smallest to largest
2. Find the middle number
3. If you have two middle numbers find the mean of those two numbers
OR
You can use the formula to find the position of the MEDIAN number
Consolidation
1) 14 8 6 16 4 12 10 4 18 16 6
2) 10 6 5 7 13 11 14 6 13 15 4 15
Form 4 Statistics
3) The table shows the number of sandwiches sold in a shop over 25 days.
Sandwiches sold 10 11 12 13 14 15 16
Frequency 2 3 6 4 3 4 3
a. What is the modal number of sandwiches sold?
b. What is the median number of sandwiches sold?
The Range
The range for a set of data is the highest value of the set minus the lowest value.
The range is not an average. It shows the spread of the data. It is, therefore, used when comparing two
or more sets of similar data. You can also use it to comment on the consistency of two or more sets of
data.
Range = Highest value – Lowest Value
Consolidation
1) 3 8 7 4 5 9 10 6 7 4
Form 4 Statistics
2) 1 0 4 5 3 2 5 4 2 1 0 1
3) In a golf tournament, the club chairperson had to choose either Maria or Fay to play in the first
round. In the previous eight rounds, their scores were as follows.
Maria’s Scores : 75 92 80 73 72 88 86 90
Fay’s Scores: 80 87 85 76 85 79 84 88
a) Calculate the mean score for each golfer
b) Find the range of each golfer
c) Which golfer would you choose to play in the tournament? Explain why.
Support Exercise Pg 263 Exercsie 17A Nos 5, 7, 8, 9, 10 (Mean), 1, 2, 3, 4, 6
Form 4 Statistics
Section 4.2 The Use of Frequency Tables
When a lot of information has been gathered, it is often convenient to put it together in a frequency
table. From this table you can then find the values of the three averages and the range.
Example 1
A survey was done on the number of people in each car leaving a shopping centre. The results are
summarized in the table below.
Number of people in each car 1 2 3 4 5 6
Frequency 45 198 121 76 52 13
For the number of people in a car, calculate:
a) the mode b) the median c) the mean
a) The modal number of people in a car is easy to spot. It is the number with the largest frequency,
which is 198. Hence, the modal number of people in a car is 2.
b) The median number of people in a car is found by working out where the middle of the set of
numbers is located.
First, add up frequencies to get the total number of cars surveyed, which comes to 505.
Next calculate the middle position:
(505 + 1) 2 = 253
Now add the frequency across the table to find which group contains the 253rd item. The 243rd
item is the end of the group with 2 in a car. Therefore, the 253rd item must be in the group with
3 in a car. Hence, the median number of people in a car is 3.
Form 4 Statistics
c) To calculate the mean number of people in a car, multiply the number of people in the car by
the frequency. This is best done in an extra column. Add these to find the total number of
people and divide by the total frequency (the number of cars surveyed).
Number in Car Frequency Number in these cars
1 45 1 ╳ 45 = 45
2 198 2 ╳ 198 = 396
3 121 3 ╳ 121 = 363
4 76 4 ╳ 76 = 304
5 52 5 ╳ 52 = 260
6 13 6 ╳ 13 = 78
TOTAL 505 1446
Hence, the mean number of people in a car is
1446 505 = 2.9 (to 1 decimal place)
Consolidation
Find the i. mode, ii. median and iii. mean from each frequency table below.
1. A survey of the shoe size of all boys in one year of a school gave these results.
Shoe Size 4 5 6 7 8 9 10
Number of
Students
12 30 34 35 23 8 3
Mode:
Form 4 Statistics
Median:
Mean:
Shoe Size Frequency Total Shoe Size
4
5
6
7
8
9
10
TOTAL
2. A school did a survey on how many times in a week students arrived late at school. These are
the findings.
Number of times late 0 1 2 3 4 5
Frequency 481 34 23 15 3 4
Mode:
Form 4 Statistics
Median:
Mean:
Number of times late Frequency Total times late
0
1
2
3
4
5
TOTAL
Support Exercise Pg 267 Exercise 17B Nos 1 – 6
Section 4.3 Frequency Tables with Grouped Data
Sometimes the information you are given is grouped in some way. Normally, grouped data are
continuous data, which is data that can have any value within a range of values (e.g. height, mass, time).
In these situations, the mean can only be estimated as you do not have all the information.
Form 4 Statistics
Discrete data is data that consists of separate numbers, for example, goals scored, marks in a test,
number of children and shoe size.
In both cases, when using a grouped table to estimate the mean, first find the midpoint of the interval
by adding the two end-values and then dividing by two.
Example 1
Pocket money, p ($) 0 < p ≤ 1 1 < p ≤ 2 2 < p ≤ 3 3 < p ≤ 4 4 < p ≤ 5
Number of students 2 5 5 9 15
a) Write down the modal class.
b) Calculate an estimate of the mean weekly pocket money.
a) The modal class is easy to pick out, since it is simply the one with the largest frequency. Here the
modal class is $4 to $5.
b) To estimate the mean, assume that each person in each class has the ‘midpoint’ amount, then
build up the following table.
To find the midpoint value, the two end-values are added together and then divided by two.
Pocket money, p ($) Frequency (f) Midpoint (m) f ╳ m
0 < p ≤ 1 2 0.5 2 ╳ 0.5 = 1
1 < p ≤ 2 5 1.5 5 ╳ 1.5 = 7.5
2 < p ≤ 3 5 2.5 5 ╳ 2.5 = 12.5
3 < p ≤ 4 9 3.5 9 ╳ 3.5 = 31.5
4 < p ≤ 5 15 4.5 15 ╳ 4.5 = 67.5
Totals Σf = 36 Σ (m ╳ f) = 120
Form 4 Statistics
The estimated mean will be $120 36 = $3.33 (rounded to the nearest cent)
Note: You cannot find the median or range from a grouped table since you do not know the actual
values.
Example 2
For the table of values given below, find:
i) the modal group
ii) an estimate for the mean.
x 0 < x ≤ 10 10 < x ≤ 20 20 < x ≤ 30 30 < x ≤ 40 40 < x ≤ 50
Frequency 4 6 11 17 9
i) The modal group:
ii) Mean:
x Frequency (f) Midpoint (m) f ╳ m
Total Σf = Σ f ╳ m =
Mean :
Consolidation
Form 4 Statistics
1) Jason brought 100 pebbles back from the beach and found their masses, recording each mass to
the nearest gram. His results are summarized in the table below:
Mass (m) 40 < m ≤ 60 60 < m ≤ 80 80 < m ≤ 100 100 < m ≤ 120 120 < m ≤ 140 140 < m ≤ 160
Frequency 5 9 22 27 26 11
i) Mode:
ii) Mean:
Mass Frequency (f) Midpoint (m) f ╳ m
Total
2) A gardener measured the heights of all his roses to the nearest centimeter and summarized his
results as follows:
Height (cm) 10-14 15-18 19-22 23-26 27-40
Frequency 21 57 65 52 12
a) How many roses did the gardener have?
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b) What is the modal class of the roses?
c) What is the estimated mean height of the roses?
Height (cm) Frequency (f) Midpoint (m) f ╳ m
10-14
15-18
19-22
23-26
27-40
Total
Support Exercise Pg 275 Exercise 17D Nos 1 – 5
Section 4.4 Drawing and Interpreting Bar Charts
A bar chart consists of a series of bars or blocks of the same width, drawn either vertically or
horizontally from an axis.
The heights and lengths of the bars always represent frequencies.
Example 1
Form 4 Statistics
The grouped frequency table below shows the marks of 24 students in a test. Draw a bar chart for the
data.
Marks 1-10 11-20 21-30 31-40 41-50
Frequency 2 3 5 8 6
Freq
uen
cy
8
7
6
5
4
3
2
1
0 1-10 11-20 21-30 31-40 41-50
Mark
Note:
Both axes are labeled
The class intervals are written under the middle of each bar
The bars are separated by equal spaces
By using a dual bar chart, it is easy to compare two sets of related data, as Example 2 shows.
Form 4 Statistics
Example 2
This dual bar chart shows the average daily maximum temperature for England and Turkey over a five-
month period.
Tem
per
atu
re (°F
)
100
90
80
70
60
50
40
30 Key
20 England
10 Turkey
0 April May June July August
Month
In which month was the difference between temperatures in England and Turkey the greatest?
Note:
You must always include a key to identify the two different sets of data.
Form 4 Statistics
Consolidation
1) For her survey on fitness, Samina asked a sample of people, as they left a sports centre, which
activity they had taken part in. She then drew a bar chart to show her data.
Freq
uen
cy
20
18
16
14
12
10
8
6
4
2
0
Squash Weight
Training Badminton Aerobics Basketball Swimming
Activity
a. Which was the most popular activity?
b. How many tool part in Samina’s survey?
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2) The frequency table below shows the levels achieved by 100 students in their A’ levels.
Grade F E D C B A
Frequency 12 22 24 25 15 2
a. Draw a suitable bar chart to illustrate the data.
b. What fraction of the students achieve a Grade C or Grade B?
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3) This table shows the number of point Mark and Joseph were each awarded in eight rounds of a
general knowledge quiz.
Round 1 2 3 4 5 6 7 8
Mark 7 8 7 6 8 6 9 4
Joseph 6 7 6 9 6 8 5 6
a. Draw a dual bar chart to illustrate the data.
b. Comment on how well each of them did in the quiz.
Support Exercise Handout
Form 4 Statistics
Section 4.5 Interpreting Pie Charts
Pictograms, bar charts and line graphs are easy to draw but they can be difficult to interpret when there
is a big difference between the frequencies or there are only a few categories. In these cases, it is often
more convenient to illustrate the data on a pie chart.
In a pie chart the whole of a data is represented by a circle (the ‘pie’) and each category of it is
represented by a sector of the circle. The angle of each sector is proportional to the frequency of the
category it represents.
So a pie chart cannot show individual frequencies, like a bar chart can, it can only show proportions.
Calculating the frequency that each sector represents
1) Measure angles from the pie chart
2) Find the fraction of the whole circle
3) Multiply this fraction with the total number of items to calculate the frequency.
Example 1
A chocolate firm asked 1440 students which type of chocolate they preferred. The pie chart showed the
following results:
Milk Chocolate - 150°
White Chocolate - 120°
Fruit and nut - 90°
This information can be recorded into a table and the frequencies for each type are calculated.
Form 4 Statistics
Chocolate Angle Working Frequency
Milk 150°
600 students
White 120°
480 students
Fruit and Nut 90°
360 students
TOTAL 360° 1440 students
Consolidation
1) 300 passengers have boarded a train at Waterloo Station in London. The following angles where
given on a pie chart:
Southampton - 120° Bournemouth - 90°
Parkstone - 36° Branksome - 54°
Poole - 60°
Town Angle Working Frequency
TOTAL
Form 4 Statistics
2) 180 people were asked about their favourite type of music. Use the angles taken from a pie
chart, calculate the total number of people who chose each of the five types of music.
Rock - 144° R’n’B - 60°
Pop - 80° Dance - 40°
Classical - 36°
Music Angel Working Frequency
TOTAL
We are not always given the angle of the sector; we could be given the frequency and asked to find the
angle of the sector.
Calculating the angle that each frequency represents
1) Calculate the frequency of a particular sector from the pie chart
2) Find the fraction of the whole frequency
3) Multiply this fraction with the total degrees.
Form 4 Statistics
Example 2
The following table shows the eye colours of a group of 36 people. Find by how many degrees each
sector is going to be represented with.
Colour of Eyes Number of people Working Angle
Brown 12
120°
Blue 15
150°
Green 6
60°
Other 3
30°
Total 36 360°
Consolidation
1) The following table illustrates the choice of ice-cream flavor that customers had from an ice-
cream shop. Find the angles which will be used on the pie chart.
Flavour Frequency Working Angle
Vanilla 35
Strawberry 20
Chocolate 22
Raspberry Ripple 13
TOTAL
Form 4 Statistics
2)
The chart shows the sales of ice creams at a school party.
The total number of ice-creams sold was 600.
a) What fraction of the ice-creams sold where chocolate?
b) How many ice-creams sold where chocolate?
c) What fraction of the ice-creams is vanilla?
d) How many vanilla ice-creams were sold?
Form 4 Statistics
e) How many strawberry ice-creams were sold?
f) How many mint ice-creams were sold?
3) A store records how many CDs of different types of music they sold in one week.
They show the results in a pie chart.
There were 25 classical CDs sold and these had an angle of 45 on the pie chart.
a) What fraction, in its simplest form, of CDs sold were classical?
b) How many CDs were sold altogether that week?
c) There were 75 pop CDs sold that week. What angle would be used on the pie chart to show
sales of pop CDs?
Support Exercise Handout
Form 4 Statistics
Section 4.6 Drawing Pie Charts
Example 1
20 people were surveyed about their preferred drink. The replies are shown in the table below:
Drink Tea Coffee Milk Cola
Frequency 6 7 4 3
Show the results on a pie chart.
First we must know what the total number of people observed were if you are not told in the
question.
6 + 7 + 4 + 3 = 20 people
Second, we must work out the size of the angle which will represent the drink choice.
Tea :
Coffee:
Milk:
Cola:
Third, draw the pie chart sector by sector.
Form 4 Statistics
Tea, 108
Coffee,
126
Milk, 72
Cola, 54
Tea
Coffee
Milk
Cola
Note:
You should always label the sectors of the pie chart
You should always write the angle on each sector of the pie chart
Consolidation
1. Eight students are asked about how they travel to school.
Use the data in the following table to make up a pie chart.
Form 4 Statistics
Method of travel to school Number of students
Walk 3
Car 2
Bike 2
Bus 1
Working:
Pie Chart:
a) Which transport method is the most common?
b) From your pie chart, what proportion of students use ‘green’ transport?
Form 4 Statistics
2. Joseph asked 180 boys what was their favourite sport. Here are the results.
Sport Football Rugby Cricket Basketball Other
Frequency 74 25 18 37 26
a) Draw a pie chart for these results with radius 6cm.
Sport Football Rugby Cricket Basketball Other
Frequency 74 25 18 37 26
Angle
Joseph also asked 90 girls for their favourite sport. In a pie chart showing the results, the angle for
Tennis was 84°.
b) How many of the girls said that Tennis was their favourite sport?
Form 4 Statistics
3. 16 people were asked how many portions of fruit and vegetables they ate per day. The results
are shown below. Draw a pie chart for it:
Number of portions Frequency
None 3
Two 5
Five 6
Seven 4
a) You should eat at least 5 portions. What fraction of people is not eating healthily?
b) Can you write this fraction any other way?
Support Exercise Handout