Lecture 03 - Chapter 02 - Part 01 - Probability & Statistics by Muhammad Haroon

Probability & Statistics
(Lecture # 03)
Chapter # 02 (Part 01)
by
Muhammad Haroon
1
Cell: +92300-7327761 Email: mr.harunahmad2014@gmail.com

Representation of Data
Presentation of Statistical Data
The Raw Data which have been collected are usually very large in quantity. Therefore we
have to organize & summarize the collected data in a form that is easy to understand.
This is called presentation of statistical data.
2

Representation of Data
Different Methods of Presentation of Statistical Data
Following are the methods, which are used, in presentation of statistical data:
 Classification
 Tabulation
 Diagram
 Graphs
3

1.Classification
Classification is the process of arranging the data into relatively homogenous groups or
classes according to their resemblances and affinities. The best example of the
classification is the process of sorting letters in a post office. The letters are classified
according to the cities & further arranged according to sectors/streets.
Important types of classification are given below:
 One Way Classification
 Two Way Classification
 Manifold Classification
4

One Way Classification
When data are classified by one variable, it is called one-way or simple classification.
For example
Population of any location may be classified as rich, average income, poor according to
the income level.
5

Two Way Classification
When data are classified by two variables at the same time, it is called two-way
classification.
For example
We may classify the peoples based on education & income group.
6

Manifold Classification
When data are classified by many variables, it is called manifold classification.
For example
We may classify the peoples based on gender, education, income etc.
Data may also be classified according to quantitative, qualitative, temporal
(chronological) & geographical characteristic.
7

2.Tabulation
The systematic arrangement of data in the form of rows and columns for the purpose of
comparison & analysis is known as tabulation.
Types of Tabulation
 Simple Tabulation
 Double Tabulation
 Complex Tabulation
8

Simple Tabulation
When the data are presented relating to a single characteristic, it is called simple or one-
way tabulation.
For example
A table showing the scores of Pakistani batsmen in a cricket match against Australia:
9
Batsmen Scores
Saeed Anwar 101
Shahid Afridi 37

Double Tabulation
When the data are tabulated according to two characteristic at a time, it is called double
or two-way tabulation.
For example
A table showing the population of 5 division of Punjab by sex (Male & Female) in census
reports of 1981.
10
Division
Population ( in Lakhs)
Male Female
Bahawalpur 24 23
Rawalpindi 23 21
Gujranwala 39 37
Lahore 44 43
Multan 39 36

Complex Tabulation
When the data are tabulated according to several characteristic at a time, it is called
complex tabulation.
For example
A table showing population of Pakistan in each province classified by ages, region,
education etc.
11

Main Parts of Statistical Table
Generally, a statistical table has the following parts:
 Title
 Box-head
 Stub
 Body
 Prefatory Note
 Foot Note
 Source Note
12

Title
It is the main heading written in capital letters exposed at the top of the table. It should be
brief, concise & self-explanatory. It throws light on the whole table.
13

Box-Head
The heading of column in a table is called Box-head or Column caption. The caption
consists of the headings at top of the columns in a table explaining what each column
represents. Each column designation is called a caption heading.
14

Stub
The stub is the leftmost column or set of columns in a table, providing the designation for
each row in the table. Each row designation is called a stub item.
15

Body of the Table
It is the main and important part of the table. The body of the table consists of the
numerical information, Which is placed in appropriate cells governed by column and row
headings.
16

Prefatory Note
It provides explanation concerning the entire table or substantial part of it. It is placed just
below the title and in smaller or less prominent type.
17

Foot-Note
It is placed at the bottom of the table. It provides explanations concerning individual
figures or a column or row of figures.
18

Source Note
The source note is placed after the footnotes. It tells source from which the data have
been taken should be specified if they are not obtained from original source.
19

Graphical Representation
20

Frequency Distribution?
A frequency distribution is a tabular arrangement of the data that shows the distribution of
observations among different classes.
Construction of a Discrete Frequency Distribution
 Find range of the data i.e the difference between largest & smallest value.
 In the first column, place all possible value of the variable from the lower to highest
under tittle of the “variable”
 In the second column, a vertical bar called tally bar or tally mark is put against
occurred value of the variable. In marking tally it is custom that first four tallies are
recorded straight (////), & the fifth tally crossing four tallies (////). Continue this process,
until all the observation in the data are exhausted.
 In the final column, the number of bars corresponding to each value of the variable are
counted and placed under the column entitled “frequency (f). The total of the
frequency column must be equal to the total number of observations to confirm that all
21

Frequency Distribution…
the data have been accounted for.
When the discrete data are sufficiently large, then they are grouped in the same way as
the continuous data.
Construction a Continuous Frequency Distribution
The important steps involved in construction of continuous frequency distribution are
given below:
 Find range of the data i.e the difference between largest & smallest value.
 There is no hard and fast rule to decide that how many numbers of classes are
sufficient for classification of given data. However, it is decided based on range of the
data. The large will be the range; the greater will be the no of classes. Usually we
consider 5 to 20 classes sufficient to classify the data. H.A. Struges has given a
formula for determining the number of classes i.e c=1+3.322logn where n=number of
observation.
22

 The size of class-interval denoted by I or h is given:
h=
𝑅𝑎𝑛𝑔𝑒
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑙𝑎𝑠𝑠𝑒𝑠
Generally, the class-interval size should be multiples of 5 or 10.
 The lowest class usually starts with a number that is a multiple of class internal such
that it should cover the smallest value of the data.
 When the lowest class limit of the lowest class has been decided, then by adding the
class interval size to the lower-class limit, determine the upper-class limit. The
remaining lower and upper class limits may be determined according by adding the
size of the class interval.
 A vertical bar called tally bar or tally mark is put against occurred value of the variable.
In marking tally it is custom that first four tallies are recorded straight (////), & the fifth
tally crossing four tallies (////). Continue this process, until all the observation in the
data are exhausted.
23

 In the final column, the number of bars corresponding to each value of the variable are
counted and placed under the column entitled “frequency (f). The total of the
frequency column must be equal to the total number of observations to confirm that all
the data have been accounted for.
24

Class Limits
The class limits are defined as the value of the variable, which explain the classes, the
smaller value is called as lower class limit and the larger value is called as upper class
limit. They are stated in such a way that they are mutually exclusive.
For example
In the given table 20 – 24, 25 – 29 etc are called as class limits.
In class limits (20 – 24), the smaller value 20 is the
lower class limit and larger value 24 is the upper class limit.
25
Ages No. of Patients
20 – 24 1
25 – 29 4
30 – 34 8
35 – 39 11
40 – 44 15
45 – 49 9
50 – 54 2

Open-end Classes
If a frequency distribution has no lower class limit or no upper class limit of its any class is
called an open-end class.
26
Less than 25 1
25 – 29 4
30 – 34 8
35 – 39 11
40 – 44 15
45 – 49 9
Over 50 2

Class Boundaries
The class boundaries are the exact values, which break up one class from another class.
The smaller value of the class boundaries is called as lower class boundary & the larger
value as upper class boundary. The upper class boundary of a class matches with the
lower class boundary of the next class.
27
19.5 – 24.5 1
24.5 – 29.5 4
29.5 – 34.5 8
34.5 – 39.5 11
39.5 – 44.5 15
44.5 – 49.5 9
49.5 – 54.5 2

Class Marks/Mid Points
A class mark is average value of the lower & upper class limits or class boundaries. It is
also called as mid point.
Mid Point =
𝐿𝑜𝑣𝑤𝑒𝑟 𝑐𝑙𝑎𝑠𝑠 𝑙𝑖𝑚𝑖𝑡+𝑢𝑝𝑝𝑒𝑟 𝑐𝑙𝑎𝑠𝑠 𝑙𝑖𝑚𝑖𝑡
2
=
𝐿𝑜𝑣𝑤𝑒𝑟 𝐶.𝐵+𝑢𝑝𝑝𝑒𝑟 𝐶.𝐵
2
The mid point of 20 – 24 is 22
28

Class Interval/Class Width
The difference between the upper and lower class boundaries is called class interval or
class width. It may also obtain by finding the difference between two successive class
marks or class limits. It is denoted by h or i. The class interval of the class boundaries
19.5 – 24.5 is 5.
29

Class Frequency
The number of values falling in a specified class is called class frequency or frequency. It
is denoted by f.
30

Relative Frequency
The frequency of a class divided by the total frequency of all the classes is called the
relative frequency. i.e.
Relative frequency =
𝑓
𝑓
The total of relative frequencies is unity. A table showing relative frequency is called
relative frequency distribution.
31

Percentage Relative Frequency
If 100 multiply relative frequencies, we obtain percentage relative frequency. A table
showing percentage relative frequencies is called as percentage distribution.
32

Cumulative Frequency
The total frequency of all the classes less than the upper class boundary of a given class
is called as the cumulative frequency of that class.
33

Graphical Representation (G.R)
The visual display of statistical data in the form of points, lines, areas & other geometrical
form & symbols is called graphical representation. Such visual representation can be
divided into two main groups i.e. graphs & diagrams.
Graph is a visual representation of frequency distribution usually shown on graph paper,
while a diagram is any other one, two or three-dimensional form of visual representation
34

G.R Advantages
Some of the advantages of graphical presentation at given below:
 Graphs & diagrams present the numerical data in the visual form
 Graphical representation is more effective in attracting attention than any other
method of presentation
 Graphical representation helps in forecasting
 Estimates of certain values can be found graphically.
35

G.R Disadvantages
Some of the disadvantages of graphical presentation at given below:
 Accuracy is not possible through graphs
 Only approximate value can be shown in graphs while exact figure can be given in a
table.
 It requires a lot of time and effort to prepare a good chart or graph
36

Graphs of Frequency Distribution
What are types of graphs of frequency distribution?
Ans: There are various type of graphs of frequency distribution some important types of
graphs are given below:
 Frequency Histogram
 Frequency Polygon
 Frequency Curve
 Cumulative frequency polygon or Ogive.
37

Frequency Histogram
A frequency histogram or histogram is a set of adjacent rectangles for a frequency
distribution such that the area of each rectangle is proportional to the corresponding class
frequency.
The construction of histogram involves the following steps:
 We take the class boundaries along X-axis. (If the class intervals are not equal, the
height of the rectangle over an unequal class interval is to be adjusted.)
 Using an appropriate scale, we take the heights of the rectangles such that the areas
of rectangles are proportional to the corresponding frequencies
 We construct the rectangles to get the frequency histogram
38

Frequency Histogram
39

Frequency Polygon
A frequency polygon is a many sided closed figure that represents a frequency
distribution. It is constructed by plotting the mid points & corresponding frequencies (X, f),
& then connecting them by straight-line segments. The graph such drawn does not reach
the X-axis, so we add extra class marks at both ends of the distribution with zero
frequencies to obtain a closed figure. The frequency polygon is also obtained by
histogram by joining mid points of each rectangles by means of straight lines.
40

Frequency Polygon
41
Frequencies
Mid Points of Class Marks
Frequency Polygon

Frequency Curve
When a frequency polygon is smoothed, it approaches a continuous curve is called
frequency curve. There are many types of frequency curve. We will be discuss in the next
chapter.
42

Frequency Curve
43

Cumulative Frequency Polygon/Ogive
To Draw cumulative frequency polygon or Ogive we take upper class boundaries along
X-axis & cumulative frequencies along Y-axis. Then these points are joined by line
segments to obtain an ogive.
44

End
45

Lecture 03 - Chapter 02 - Part 01 - Probability & Statistics by Muhammad Haroon

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Lecture 03 - Chapter 02 - Part 01 - Probability & Statistics by Muhammad Haroon