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GCSE Statistics
By: Hooria Batool
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[GCSE STATISTICS] By: Hooria Batool

Contents
Chapter 1: Information
...
6
Chapter 3 : Cumulative Frequency
...
15
Chapter 5: Graphical Representation
...
26
Chapter 7: Probability
...
39
Chapter 9: Index Number
...
47
Chapter 11: Scatter Diagram
...
51
Chapter 13: Sampling
...
Data might be
presented in an unprocessed form (raw form), for example, raw data from a
market research questionnaire representing weekly purchases of a particular
product made by a person is given below:
2,7,4,5,3,4,6,3,4,6,3,5,6,3,5,3,5,4,6,3,5,3
This data will be of little use to the users requiring the information
...

Statistics is defined as:
The practice or science of collecting and analyzing numerical data in large quantities,
especially for the purpose of inferring proportions in a whole from those in a
representative sample
...
Variable and Constant
A variable is an unknown value which changes with the circumstances
...
Constant, on the other hand, is a known fixed
amount, for example even numbers between 1-100 etc
...
1
...

For example:
1) Number of students in a class,
2) Number of cars in a parking lot,

[GCSE STATISTICS] By: Hooria Batool
3) Number of products in a shop etc
...
2
...

For example:
1) Height of a person
2) The weight of a box
3) The length of someone’s foot
2
...

For example:
a) The color of a piece of cloth
b) Religion of a person
Quantitative data is a data that can be expressed in numerical terms
...
However, not all numbers are continuous and measurable
...

3
...

E
...
Class: Limits, Width
...

Class Width: The difference between upper and lower class limits
Class Boundary: They are the points which separate the classes in a continuous data
...
The upper class boundary of a given class is obtained by averaging the upper
limit of the class and the lower limit of the next class
...
Tally Mark
Tally mark is a method of presenting data using groups of five
...

There are three main measures of central tendency:
1) Mean
2) Mode
3) Median

1
...
Mean(denoted by 𝑥̅ of a given set of data) can be found
by using the following formula:
𝑥̅ =

𝑆𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖𝑡𝑒𝑚𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑡𝑒𝑚𝑠

Formula for calculating the Mean of a grouped frequency distribution:
𝑥̅ =

∑ 𝑓𝑥
∑ 𝑓

Where,
f = frequency
x= variable
Example 1: Mean of an ungrouped frequency distribution
Find the mean of the following distribution:
x

f
2
3
4
5
6
7

34
24
53
42
23
25

[GCSE STATISTICS] By: Hooria Batool
Solution
x
2
3
4
5
6
7

f
34
24
53
42
23
25
201

fx
68
72
212
210
138
175
875

Mean: 875/201 =4
...
5
Calculation of mean using assumed mean:
Consider the following data:
x
23
33

f
2
4

[GCSE STATISTICS] By: Hooria Batool
43
53
63
73
83

7
4
7
3
2

Let us use 53 as our assumed mean (a)
...
689
Mean = 53-0
...
31

Characteristics of Mean:
1) It is effected by extreme values
2) If x is increased or decreased by a number, mean also changes by the same number
3) If x is multiplied by a constant, mean will also be multiplied by the same constant
2
...
It is the point of greatest density
...
e; the class with the highest frequency
...
5 +

(12−10)
×4
(12−10)+(12−7)

3
...

Example 1
Calculate median of the following data:
4,2,6,3,7,2
Solution
First of all, arrange the data:
X = 2,2,3,4,6,7
Median = Sum of frequencies+1/2 th value = (6+1)/2 = 3
...
5

= 15
...
e; 4

Example 3
Calculate median of the following data:

x
1-5
6 - 10
11 - 15
16 - 20

Frequency
2
7
8
3

Solution
Median :
Median =L +






(n/2) − cfb
fm

×w

L is the lower class boundary of the group containing the median
n is the total number of data
cfb is the cumulative frequency of the groups before the median group
fm is the frequency of the median group
w is the group width

For our example:






L = 11
n = 20
cfb = 2 + 7 = 9
fm = 8
w=5

Median = 11 + (20/2) − 9 × 5

[GCSE STATISTICS] By: Hooria Batool
8
= 11 + (1/8) x 5
= 11
...

Score
1
2
3
4

Frequency
4
6
4+6+2=12
2
4+6+2+4=
4
16

Cumulative Frequecny
4
10
12
16

Cumulative frequency Polygon and curve
If we plot the cumulative frequencies on a graph and join them in a smooth curve, it will be called a
cumulative frequency curve
...


Figure 1 Frequency Polygon

Figure 2 Frequency Curve

[GCSE STATISTICS] By: Hooria Batool
Example
Tests of Maths and English were conducted in a class and the marks were recorded in the following
table:

Marks
1-10
11-20
21-30
31-40
41-50
51-60
51-70
71-80
81-90
91-100

Maths
1
2
1
2
3
7
11
13
9
1

English
1
1
2
4
5
8
9
10
6
4

Draw cumulative frequency graph of each subject
...

First Quartile (Q1), also known as lower quartile, is the middle number between the smallest number and
the median;
Second Quartile (Q2) is the median; and
Third Quartile (Q3), also known as upper quartile, is the middle number between the highest number and
the median (Q2)
...

Formulae
First Quartile
Second Quartile
Third Quartile
First Decile
x Decile
First Percentile
x percentile

𝑛+1
) th observation
4
𝑛+1
( ) × 2 th observation
4
𝑛+1
( 4 ) × 3 th observation
𝑛+1
( 10 ) th observation
𝑛+1
( 10 ) × 𝑥 th observation
𝑛+1
( 100 ) th observation
𝑛+1
( ) × 𝑥 th observation
100

(

Example
Find Q1, Q2,and Q3 of the following data:
2,4,3,5,6,7,9,3,5,8,1

Solution
First of all, we have to arrange the data in ascending order:

[GCSE STATISTICS] By: Hooria Batool
1,2,3,3,4,5,5,6,7,8,9
𝒏+𝟏
Q1 = ( ) th observation =(11+1/4)th observation = 3rd observation = 3
Q2 = (
Q3 = (

𝟒
𝒏+𝟏
)×
𝟒
𝒏+𝟏
)×
𝟒

𝟐 th observation =(11+1/4)X2 th observation = 6th observation = 5
𝟑 th observation =(11+1/4)X3 th observation = 9th observation = 7

Quartiles and Percentiles of Grouped data:
Quartile of a grouped data:
𝑘

𝑁−𝑓

𝑄 𝑘 = 𝐿 + [4 𝑓

𝑞𝑘

] × 𝐶, k=1,2,3

Where,
L is the lower limit of the quartile class (Qk)
...

f is the cumulative frequency immediately below the quartile class
...

Fqk is the frequency of class where Qk lies
Percentile of a grouped data:
𝑘

𝑃 𝑘 = 𝐿 + [100𝑓

𝑁−𝑓
𝑝𝑘

] × 𝐶, k=1,2,3…99

Where,
L is the lower limit of the quartile class (Pk)
...

f is the cumulative frequency immediately below Pk
...

Fpk is the frequency of class where Pk lies

[GCSE STATISTICS] By: Hooria Batool

Interquartile range
Interquartile range is the measure of statistical dispersion that is not affected by extreme values, and is
equal to the difference between upper and lower quartile
...
The area of each sector, arc length and central angle proportional
to the quantity it represents
Example
Following data is available for a retail shop

1st Qtr
2nd Qtr
3rd Qtr
4th Qtr

Sales in millions
8
...
2
1
...
2

Required: Construct a pie chart
...
2
3
...
4
1
...
8571
82
...
85714
360

Sales
8%
10%
1st Qtr
2nd Qtr
23%

3rd Qtr
59%

4th Qtr

[GCSE STATISTICS] By: Hooria Batool
Bar Charts:
(i)

Simple bar chart
Simple bar chart is a chart with rectangular horizontal or vertical bars with lengths
proportional to the values they represent
...

Example

Constrict a multiple bar chart from the following data:
Retailer

A

B

C

D

Sales (millions)-2000
Sales (millions)-2001

5
4
...
8
5

8
...
3

3
5
...

Example

The following data shows the output of 3 countries from different sectors
...
3
B
2
...
5
D
4
...


Secondary
2
...
4
1
...


Example

The following data shows the output of 3 countries from different sectors
...
3
B
2
...
5
D
4
...


Secondary
2
...
4
1
...
3
B 2
...
5
D 4
...
8

Secondary
2
...
4
1
...
6

Tertiary
2
5
3
5
15

Primary%
29%
17%
24%
30%
100%

Secondary%
19%
35%
14%
32%
100%

Tertiary%
13%
33%
20%
33%
100%

100%
90%
80%
70%
60%
Tertiary
50%

Secondary

40%

Primary

30%
20%
10%
0%
A

B

C

D

[GCSE STATISTICS] By: Hooria Batool
Histograms
Whereas, bar charts are used to represent data which is discrete and ungrouped, histograms can be used
to represent data which is continuous and is summarized by grouped frequency distribution
...

(i)

Histogram with equal class width

Frequency

When the class intervals are of equal width, the height of each rectangle is equal to its
frequency
...

Frequency density = Frequency/class width
Example

Construct a histogram from the following data:
Class Interval
60-69
70-74
75-89

Frequency
60
40
30

[GCSE STATISTICS] By: Hooria Batool
Solution

Class width
10
5
15

Frequency Density
6
8
2

0 1 2 3 4 5 6 7 8 9 10

Frequency
60
40
30

Frequency Density

Class Interval
60-69
70-74
75-89

60 65 69 74
Class

89

Important to note!
1
...

2
...

3
...


Frequency Polygon:
Frequency Polygon refers to a figure obtained by joining the mid points of each of the class, with the help
of a scale
...


Figure 3 Frequency Polygon

[GCSE STATISTICS] By: Hooria Batool

Figure 4 Frequency Curve

[GCSE STATISTICS] By: Hooria Batool

Chapter 6: Standard Deviation and Variance
1
...
Variation/Dispersion
Variance measures how far a set of numbers is set out
...
2
...

Range = Highest value – Lowest Value

Advantages

Limitations

Scores exist in
data set

Value depends
on only 2 scores
...
3
...
5
̅
x-𝒙
-0
...
5
0
...
5
0
...
5
-0
...
5

X
4
2
5
3
5
7
4
6
Total

|𝐱 − ̅|
𝒙
0
...
5
0
...
5
0
...
5
0
...
5
10

Mean Deviation = 10/8 =1
...
34
1
...
66
1
...
66
2
...
66
3
...
66
Mean Deviation=

∑ 𝑓|x − ̅|
𝑥
∑ 𝑓

= 7
...
306
1
...
Standard Deviation
Standard
Deviation
measures
the
dispersion from the average (mean) or
expected value and is represented by the
Greek letter sigma, σ)
...
A
low Standard deviation means that the values are close to its mean
...
34
1
...
66
1
...
66
2
...
66
7
...


X

•If each value in a set of data is multiplied by a constant k, Standard deviation is, too, multiplied by k
...
35
∑ 𝑓𝑥 2
∑ 𝑓

Standard deviation = √

405

− 𝑥̅ 2 = √ 20 − 4
...
15

Standard Score
The z score is a measure of the number of standard deviations that an observation is above or below the mean
...
xn,
Z score =

𝑥−𝑥̅
𝜎

Where,
𝑥 ̅ = mean of the data; and

[GCSE STATISTICS] By: Hooria Batool
𝜎 = standard deviation
Properties of z-score

• Mean of z score is always 0
• Standard Deviation of z score is always 1
• Remains same if the numbers are scaled

• Distribution of z score is same as the distribution of original data

[GCSE STATISTICS] By: Hooria Batool

Chapter 7: Probability
Certain, likely, unlikely etc
...
But these words may not mean the same for every person
...
Probability is the measure of how likely an event is
...

The more likely an event is, the higher its probability will be
...
If an event is
certain to happen, its probability shall be 1
...
It can be expressed as a percentage, fraction or decimal
...
If the coin is
unbiased, there is an equal chance of getting
both
...
5) of getting both
...
If a fair dice is
thrown, there are 6 possible outcomes: 1, 2, 3, 4, 5, and 6
...
If you calculate the probability of getting an even
number, there are 3 ways by which this event can occur (i
...

if you get 2, 4 or 6)
...
e
...
Governments
apply probabilistic methods in environmental regulation,
where it is called pathway analysis
...
Many consumer
products, such as automobiles and consumer electronics, use reliability theory in product design to
reduce the probability of failure
...
The score is calculated by adding the numbers you get on both dices
...
The number of total possible outcomes is 36
...


[GCSE STATISTICS] By: Hooria Batool
b)

P(Greater than 5 but less than 10) = 19/36
Activity
Continuing from the above example, find the probability of getting score:
a) Equal to 5
b) Less than or equal to 4
c) Greater than 10
d) Less than 3
e) Equal to 7

Instead of using a table to ascertain the total possible outcomes, tree diagram is used usually
...
Each branch of the tree represents an outcome
...
If the outcome of one event affects the other, then these events are called ‘Mutually
Exclusive’
...
Therefore, these
are ‘Mutually Exclusive Events’
...
So, these are ‘Independent Events’
...
Read the Example 1 again
...


In a tree diagram, we will present the coin example like this:

You can see that the mutually exclusive events are shown as branches of a same event
...

We multiply probabilities along the branches and add the probabilities down the columns
...
5X0
...
25

[GCSE STATISTICS] By: Hooria Batool

Now what is the probability that you get the same answer both times? This means that either you get
‘Heads’ or ‘Tails’ both times
...
5 X 0
...
5 X 0
...
5

As sum of all the probabilities is always equal to 1 and we have the probability of happing of a certain
event, we can calculate the probability of non-happening of that event
...
8, then the
probability that Jack will be late for work will be 1-0
...
2
...
Calculate the following probabilities
...


[GCSE STATISTICS] By: Hooria Batool

Chapter 8: Expectation
Expected Value
Expected value of a discrete random variable is the long-run average of recurrences of the experiment
that it is representing
...
5

E(X) = 3
...
For a discrete random
variable X, the variance shall be denoted by Var(X)
...

Properties of Expected Value

1

• Var (aX) = a2 Var (X)

2

• Var(b) = 0

3

• Var(aX +b) = a2 Var (X)

Questions

1
...
3b

2
0
...
1b

4
0
...
4b

Find the value of b
Caculate E(X)
Caculate the vaue of E(2X – 1)
Calculate Var (X)

Solution
a) Sum of all probabilities = 1
b+0
...
1b+0
...
4b = 1
b=0
...
632

0

1

2

3

4

5

0
...
144
0
...
096
0
...
048
0
...
048
0
...
192
0
...
632) – 1
=2
...
48
0
...
144
X^2
0
1
E(X^2)
0
0
...
528 – 2
...
096
0
...
384

0
...
144
9
0
...
048
0
...
768

0
...
96
25
4
...
4

[GCSE STATISTICS] By: Hooria Batool

Chapter 9: Index Number
Index Number
Index number is a method or technique that measures the relative change in price, quantity, value or
some other item of interest from one period to another
...

Index numbers can be used to:
1) Measure fluctuations during intervals of time, group differences of geographical position of
degree etc
...

3) Measure the purchasing power of money
...

Base Year
A base year is the year used for comparison for the level of a particular index
...
The considerations that would go into
the selection of the base year could be briefly listed as follows:
1) Base year should be normal
2) Base year should be as recent as possible
3) Reliable data should be available for the selected base year

Characteristics of Index Numbers
1) Index numbers are expressed in percentage
2) Index numbers measure relative change
3) Index numbers measure changes that are not directly measurable
PROBLEMS IN THE CONSTRUCTION OF INDEX NUMBERS

1)
2)
3)
4)

Selection of items (Basket of goods)
Assignment of weights (importance)
Choice of appropriate average
Choice of base year

[GCSE STATISTICS] By: Hooria Batool
Types of Index Number
1) Price Index Numbers
These are designed to compare the changes in price of a particular commodity or number of
commodities over time periods or geographical locations
...
, and involves
both quantities and prices of items
...
e
...

Suppose that the cost of an item was $50 in 2000 and $70 in 2003
...
For this purpose, we have to express
year 2003 price as the percentage of year 2000 (Base Year) price:
70
× 100 = 140%
50
From this, we can say that price has increased by 140 %
...

Hence, the formula for calculating price relative is:
𝑃𝑟𝑖𝑐𝑒 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 (𝑃01 ) =
Where,
P0 = Price of base year, and
P1 = Price of current year

𝑃1
× 100
𝑃0

[GCSE STATISTICS] By: Hooria Batool
Example

From the data given below, calculate price relative for each commodity, taking year 1990 as base year:
Commodity
Sugar (Quintal)
Oil (Litre)
Wheat (Quintal)

Price-1995 (Rs)
1900
52
500

Price-1990 (Rs)
1500
44
390

Solution

Commodity
Sugar (Quintal)
Oil (Litre)
Wheat (Quintal)

Price-1995 (Rs)
(a)
1900
52
500

Price-1990 (Rs)
(b)
1500
44
390

Price Relative
c=a/b X 100
126
...
18
128
...
Find the index
number for year 2005, taking 2007 as base year
...

Where, the weights for a set of numbers x1 ,x2,x3…
...
wn respectively,
Weighted average =

𝑤1𝑥1+𝑤2𝑥2+𝑤3𝑥3+⋯
...
𝒘𝒏𝑰𝒏
𝑤1+𝑤2+𝑤3+⋯𝑤𝑛

Where I is a price relative
Example

Given two school classes, one with 20 students, and one with 30 students, the grades in each
class on a test were:
Morning class = 62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93,
98
Afternoon class = 81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91,
91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99
The straight average for the morning class is 80 and the straight average of the afternoon class is
90
...
However, this does
not account for the difference in number of students in each class; hence the value of 85 does not
reflect the average student grade (independent of class)
...

-source: Wikipedia

[GCSE STATISTICS] By: Hooria Batool
Crude Birth/ Death Rate
Crude birth/dearh rate means the rate of birth/death during 12 months per 1000 people of the
population
...
if these factors are taken into account, the rate will be specific death rate
Specific death/birth rate = deaths/births in a group of people/population of that group X 1000
Standardized Death and Birth rates:
Standardized death/birth rate = Total weighted death/ total standard population X 1000
=

∑ P 𝑠 S1
∑P 𝑠

Where,
Ps =Total Population ub a given age group in the standard population
S1= Specific Death/Birth rate in the same age group in the local population
Example
Following information is available for a year
Town A
Town B
Age
Population Death
Population
group
in ‘000s
rate/1000 in ‘000s
0-19
30
100
50
20-39
35
25
60
40-59
65
10
160
>60
20
30
30

Death
rate/1000
101
26
11
31

Calculate the death rate of town B on the basis of town A

Solution
Standard death rate

=

∑ P 𝑠 S1
∑P𝑠

= 30X101+35X26+65X11+20X32/150
= 35
...

Example
Construct a Pictograph of the following data
Month
Jan
Feb
Sales of
30
25
computers

March
40

Solution

Jan
Feb
March
April
Represents 5 computers
Advantages and Disadvantages of using Pictogram:

Advantages

Disadvantages

Universally
understandable

Representation
might be
difficult

Easy to read and
interpret

Can be
confusing

April
17

[GCSE STATISTICS] By: Hooria Batool

Chapter 11: Scatter Diagram
Scatter Diagram
Scatter diagram is a tool that is used to analyze the relationship between
two variables
...
The
data collected is plotted on a graph and if a relationship exists between the
two variables, the points will fall along a line or a curve
...


Strong Positive
Correlation

Strong Negative
Correlation

Weak Positive
Correlation

Weak Negative
Correlation

Complex
Correlation

No Correlation

Line of Best Fit using Semi-Average method
1
...

3
...


Divide the original data into two equal parts
...

Take average of each half
Plot the mid-point and the two semi-averages on graph
...
This line is called line of best fit
...

Solution
Year
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009

Sales(m)
40
32
50
53
48
69
70
65
82
78

Semi Total

Semi Averages

223

44
...
8

Midpoint of all data = 587/10 = 58
...
6
Midpoint of second semi total = 364/5 = 72
...
7273x + 1

2

0
0

2

4

6

8

10

12

[GCSE STATISTICS] By: Hooria Batool

Chapter 12: Moving Average
Some sets of data shows trend depending on time of year, day and
other factors
...


This graph shows the sales of coffee maker over a period of
four years
...
This is called time
series graph
...

The first step involves calculating Moving Average
...
e adjusting them so that they
may coincide with relevant year
...
5
292
...
5
365
420

290
308
...
25
351
...
5

600

500

y = 28
...
56

400
Series 1

300

Linear (Series 1)
200

100

0
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

[GCSE STATISTICS] By: Hooria Batool

Chapter 13: Sampling
Population and Sample

Population consists of all items or people with a characteristic one wishes to understand
...

Sampling Process

Define
Population

Define
Sampling
frame

Specify
Sampling
Method

Determine
Sample
Size

Implement
Plan

Sampling Frame
Sampling frame is a list of all those within a population from which a
sample might be drawn

Population
Sampling Frame

Methods of Sampling
Sample

1
...
Systematic Sampling
In systamtic sampling, an item is selected at random from the sampling frame, and then
every nth item is selected
...

For example, if a sample is to be selectd of 50 students from class with total strength of
200 students, a random pupil is selected and then every 25 th (200/50) student is
selected
...
Stratified Sampling
In this method, the population is divided
into
strata (groups) and sample is selected using
simple random sapling from each stratum
...
Quota Sampling
In this method, population is divided into
subgroups (as in stratified sampling) and then
judgment is used to select the subjects or units
from each segment based on a specified proportion
...
Cluster Sampling
Cluster sampling is a sampling technique where the entire population is divided into
groups, or clusters and a random sample of these clusters are selected
...
To make sure that the
sample is a true representative of the population, the sample is selected from each grade
depending on the strength of each grade
...

7

82

17

72

80

34

50

76

27

59

67

48

3

55

19

4

93

40

20

90

100

13

46

70

94

44

49

14

21

53

29

34

8

84

71

27

Step 1: Select a number randomly
e
...
e; every 4th number will be selected
Step 3: Determine Sample
Starting from 3rd number and then every 4th number will be selected
17, 50, 67, 19, 20, 46, 49, 29, 71
Exam Question
A sample of size 9 is to be taken from the adult residents of three streets
...

Name of street
Jamaica Drive
Liberia Avenue
Malawi Road

Number of
residents
24
12
18

adult Random
allocated
00-23
24-35
36-53

numbers

Using this information and the two-digit random number table below, you are to use different
methods to select the sample of size 9 from the residents
...


[GCSE STATISTICS] By: Hooria Batool

(i)

(ii)

a) Starting at the beginning of the first row of the random number table, and moving
along the row, select a simple random sample of the required size
...

xc
A systematic sample is to be selected by starting at the beginning of the second row of the
table, and moving along the row
...

b) Write down the number of the first resident selected
c) Write down the numbers of the other eight residents selected for the systematic
sample
...

(iii)

(a) State how many residents in each street would be selected for such a sample
(b) Starting at the beginning of the third row of the table, and moving along the
row, select a sample stratified by street
...




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