probability and statistics-4.pdf

Arba Minch University Department of Statistics
College of Natural Sciences Probability and Statistics for Engineers
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Chapter One
1. Introduction
This chapter introduces the subject matter of statistics, the art of learning from data. It describes the two
branches of statistics, descriptive and inferential. The idea of learning about a population by sampling
and studying certain of its members is described. It is reasonable to start by thinking about this question,
relating it to your own experiences, and talking it over with friends. However, if you want to convince
others and obtain a consensus, it is then necessary to gather some objective information. We are using it
to indicate the modern approach to learning about a complicated question. Namely, one must collect
relevant information, or data, and these data must then be described and analyzed in such a way that
valid conclusion is made. This is the subject matter of statistics.
1.1 Definition and Classification of Statistics
The common usage of the word “statistics” has, therefore, two meanings. In one sense “statistics” is the
plural form which refers to the numerical facts and figures collected for a certain purposes. Statistics as
a numerical data: In this sense statistics is defined as aggregates of numerical expressed facts (figures)
collected in a systematic manner for a predetermined purpose.(in plural sense)
In the other sense;” statistics” refers to a field of study or to a body of knowledge or to a subject that is
concerned with systematic collection and interpretation of numerical data to make a decision. In this
sense the word statistics is singular. Statistics as a subject (field of study): in this sense statistics is
defined as the science of collecting, organizing, presenting, analyzing and interpreting numerical data to
make decision on the bases of such analysis.(in singular sense)
In this course, we shall be mainly concerned with statistics as a subject, that is, as a field of study.
Classification of statistics
Anyone can apply statistical techniques to, virtually, every branch of science and art. These techniques
are so diverse that statisticians commonly classify them into the following two broad categories
(Descriptive statistics and inferential statistics)

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Descriptive Statistics: it is an area of statistics which is mainly concerned with the methods and
techniques used in collection, organization, presentation, and analysis of a set of data without
making any conclusions or inferences. According to this definition the activities in the area of
Descriptive Statistics include:
 Gathering data
 Editing and classifying them
 Presenting data in tables
 drawing diagrams and graphs for them
 Calculating averages and measures of
dispersions.
Descriptive statistics doesn’t go beyond describing the data themselves.
Examples of activities of descriptive statistics:
 Recording a students’ grades throughout the semester and then finding the average of
these grades.
 From sample we have 40% employee suggest positive attitude toward the management of the
organization.
 Drawing graphs that show the difference in the scores of males and females.
 Of 50 randomly selected students at Water Supply and Environmental Engineering department
of Arba Minch University 28 of which are female. An example of descriptive statistics is the
following statement: "56% of these students are female."
All the above examples simply summarize and describe a given data. Nothing is inferred or
concluded on the basis of the above description.
Inferential Statistics: Inferential statistics is an area of statistics which deals with the method
of inferring or drawing conclusion about the characteristics of the population based upon the
results of a sample. Statistics is concerned not only with collection , organization , presentation
and analysis of data but also with the inferences which can be made after the analysis is
completed. In collecting data concerning the characteristics of a set of elements, or the element
can even be infinite. Instead of observing the entire set of objects, called the population, one observes
a subset of the population called a sample.

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Inferential statistics utilizes sample data to make decision for entire data set based on sample. Examples
of inferential Statistics are:-
1. Of 50 randomly selected students at biology department of Addis Ababa university 28 of which
are female. An example of inferential statistics is the following statement: "56% of students at
biology department of Addis Ababa University are female." We have no information about all
students at biology department of Addis Ababa University, just about the 50. We have taken that
information and generalized it to talk about all students at biology department of Addis Ababa
University.
2. “There is a definitive relationship between smoking and lung cancer”. This statement is the
result of continuous research of many samples taken and studied. Therefore, it is an
inference made from sample results.
3. As a result of recent reduction in oil production by oil producing nations , we can expect
the price of gasoline to double up in the next year.(It is an inference from sample survey).
4. As a result of recent survey of public opinion, most Americans are in favor of building
additional nuclear power plant.
5. If an instructor of Probability and Statistics calculates one of his/her classes to describe the
performance of that class, he/she is doing decretive statistics; where as if he/she uses this average
grade for all classes, he/she doing inferential statistics.
1.2 Stages in Statistical Investigation
Before we deal with statistical investigation, let us see what statistical data mean. Each and every
numerical data can’t be considered as statistical data unless it possesses the following criteria.
The data must be aggregate of facts
They must be affected to a marked extent by a multiplicity of causes
They must be estimated according to reasonable standards of accuracy
The data must be collected in a systematic manner for predefined purpose
The data should be placed in relation to each other

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A statistician should be involved at all the different stages of statistical investigation when planning to
conduct scientific research. This includes formulating the problem and then collecting, organizing
(classifying), presenting, analyzing and interpreting of statistical data.
Data Collection: This is a stage where we gather information for our purpose
o If data are needed and if not readily available, then they have to be collected.
o Data may be collected by the investigator directly using methods like interview,
questionnaire, and observation or may be available from published or unpublished sources.
o Data gathering is the basis (foundation) of any statistical work.
o Valid conclusions can only result from properly collected data.
Data Organization: It is a stage where we edit our data .A large mass of figures that are collected from
surveys frequently need organization. The collected data involve irrelevant figures, incorrect facts,
omission and mistakes. Errors that may have been included during collection will have to be edited.
After editing, we may classify (arrange) according to their common characteristics. Classification or
arrangement of data in some suitable order makes the information easer for presentation.
Data Presentation: The organized data can now be presented in the form of tables, diagram and graphs.
At this stage, large data will be presented in tables in a very summarized and condensed manner. The
main purpose of data presentation is to facilitate statistical analysis. Graphs and diagrams may also be
used to give the data a bright meaning and make the presentation attractive.
Data Analysis: This is the stage where we critically study the data to draw conclusions about the
population parameter. The purpose of data analysis is to dig out information useful for decision making.
Analysis usually involves highly complex and sophisticated mathematical techniques. However, in this
course only the most commonly used methods of statistical analysis are included in next chapters. Such
as the calculations of averages, the computation of majors of dispersion, regression and correlation
analysis are covered.
Data Interpretation: This is the stage where one draw valid conclusions from the results obtained
through data analysis. Interpretation means drawing conclusions from the data which form the basis for
decision making. The interpretation of data is a difficult task and requires a high degree of skill and
experience. If data that have been analyzed are not properly interpreted, the whole purpose of the
investigation may be defected and fallacious conclusion be drawn. So that great care is needed.

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1.3 Definition of Some Basic Statistical Terms
In this section, we will define those terms which will be used frequently.
Data: Data as a collection of related facts and figures from which conclusions may be drawn. In other
words data is simply a scientific term for facts, figures, information and measurement.
Population: A population is a totality of things, objects, peoples, etc about which information is
being collected. It is the totality of observations with which the researcher is concerned. The
population represents the target of an investigation, and the objective of the investigation is to draw
conclusions about the population hence we sometimes call it target population.
Example: population of trees under specified climatic conditions, population of animals fed a certain
type of diet, population of households, etc.
Census: a complete enumeration of the population. But in most real problems it cannot be realized,
hence we take sample.
Sample: A sample is a subset or part of a population selected to draw conclusions about the
population.
Sampling: The process of selecting a sample from the population.
Sample size: The number of elements or observation to be included in the sample.
Statistic: It is a value computed from the sample, used to describe the sample.
Parameter: It is a descriptive measure (value) computed from the population. It is the population
measurement used to describe the population. Example: population mean and standard deviation.
Sampling frame:-A list of people, items or units from which the sample is taken.
Variable: A certain characteristic whose value changes from object to object and time to time.
Sample size: The number of elements or observation to be included in the sample.
Census survey: It is the process of examining the entire population. It is the total count of the
population.
Censes survey (studying the whole population without considering samples) requires a great deal of
time, money and energy. Trying to study the entire population is in most cases technically and
economically not feasible. To solve this problem, we take a representative sample out of the population
on the basis of which we draw conclusions about the entire population.

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Therefore, sampling survey
 Helps to estimate the parameter of a large population.
 Is cheaper, practical, and convenient.
 Save time and energy and easy to handle and analysis.
1.4 Applications, Uses and Limitations of Statistics
Application of Statistics
The scope of statistics is indeed very vast; and applicable in almost all fields of human endeavor. Apart
from helping elicit an intelligent assessment from a body of figures and facts, statistics is indispensable
tool for any scientific enquiry-right from the stage of planning enquiry to the stage of conclusion. It
applies almost all sciences: pure and applied, physical, natural, biological, medical, agricultural and
engineering. It also finds applications in social and management sciences, in commerce, business and
industry, applicable in some process e.g. invention of certain drugs, extent of environmental pollution
and industries especially in quality control area etc.
Uses of statistics
Today the field of statistics is recognized as a highly useful tool to making decision process by managers
of modern business, industry, frequently changing technology. It has a lot of functions in everyday
activities. The following are some of the most important uses of statistics.
 Statistics condenses and summarizes complex data. The original set of data (raw data) is normally
voluminous and disorganized unless it is summarized and expressed in few numerical values.
 Statistics facilitates comparison of data. Measures obtained from different set of data can be
compared to draw conclusion about those sets. Statistical values such as averages, percentages,
ratios, etc, are the tools that can be used for the purpose of comparing sets of data.
 Statistics helps in predicting future trends. Statistics is extremely useful for analyzing the past and
present data and predicting some future trends.
 Statistics influences the policies of government. Statistical study results in the areas of taxation, on
unemployment rate, on the performance of every sort of military equipment, family planning, etc,
may convince a government to review its policies and plans with the view to meet national needs.
 Statistical methods are very helpful in formulating and testing hypothesis and to develop new
theories.

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Limitations of Statistics
Even though, statistics is widely used in various fields of natural and social sciences, which closely
related with human inhabitant. It has its own limitations as far as its application is concerned. Some of
these limitations are:
 Statistics doesn’t deal with single (individual) values. Statistics deals only with aggregate values.
But in some cases single individual is highly important to consider in some situations. Example,
the sun, a deriver of bus, president, etc.
 Statistics can’t deal with qualitative characteristics. It only deals with data which can be
quantified. Example, it does not deal with marital status (married, single, divorced, widowed) but
it deal with number of married, number of single, number of divorced.
 Statistical conclusions are not universally true. Statistical conclusions are true only under certain
condition or true only on average. The conclusions drawn from the analysis of the sample may,
perhaps, differ from the conclusions that would be drawn from the entire population. For this
reason, statistics is not an exact science. Example: Assume that in your class there is 50 numbers
of students. Take those 50 student’s CGPA and analyses mean CGPA in AMU; that is assumed
3.00. This value is on average, because all individual has not CGPA 3.00. There is a student who
has scored above 3.00 and below 3.00, and this may not true in others Universities.
 Statistical interpretations require a high degree of skill and understanding of the subject. It
requires extensive training to read and interpret statistics in its proper context. It may lead to
wrong conclusions if inexperienced people try to interpret statistical results.
 Statistics can be misused. Sometimes statistical figures can be misleading unless they are
carefully interpreted. Example: The report of head of the minister about Ethio-Somalia terrorist
attack mission dismissed terrorists 25% at first day, 50% at second day, 75% at third day.
However, we doubt about the mechanisms how the mission is measured and quantified. This
leads miss use of statistical figures.

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1.5 Types of variables and Scales of Measurements
Variables and Attributes
A variable in statistics is any characteristic, which can take on different values for different elements
when data are collected. A quantitative or qualitative characteristic that varies from observation to
observation in the same group is called a variable. In case of quantitative variables, observations are
made using interval scales whereas in case qualitative variables nominal scales are used.
Conventionally, the quantitative variables are termed as variables and qualitative variables are termed as
attributes. Example: attributes like gender, religion, marital status, coding etc. are attributes.
Types of Variables
A. Continuous Variables: - are usually obtained by measurement not by counting. These are variables
which assume or take any decimal value when collected. The variables like age, time, height,
income, price, temperature, length, volume, rate, time, amount of rainfall and etc. are all continuous
since the data collected from such variables can take decimal values.
B. Discrete Variables: - are obtained by counting. A discrete variable takes always whole number
values that are counted. Example: Variables such as number of students, number of errors per page,
number of accidents on traffic line, number of defective or non-defectives in production, etc.
Scales of Measurements
Normally, when one hears the term measurement, they may think in terms of measuring the length of
something (i.e. the length of a piece of wood) or measuring a quantity of something (i.e. a cup of flour).
This represents a limited use of the term measurement. In statistics, the term measurement is used more
broadly and is more appropriately termed scales of measurement. A scale of measurement refers to ways
in which variables or numbers are defined and categorized and/or is the assignment of numbers to objects or
events in a systematic fashion. Each scale of measurement has certain properties which in turn determine
the appropriateness for use of certain statistical analyses. The various measurement scales results from
the facts that measurement may be carried out under different sets of rules. Four levels of measurement
scales are commonly distinguished: nominal, ordinal, interval, and ratio; and each possessed different properties
of measurement systems.

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i. Nominal Scale:-Consists of ‘naming’ observations or classifying them into various mutually exclusive
categories. Sometimes the variable under study is classified by some quality it possesses rather than by
an amount or quantity. In such cases, the variable is called attribute. Example Religion (Christianity,
Islam, Hinduism, etc), Sex (Male, Female), Eye color (brown, black, etc.), Blood type (A, B, AB and
O) etc.
ii. Ordinal Scale: -Whenever observations are not only different from category to category, but can be
ranked according to some criterion. The variables deal with their relative difference rather than with
quantitative differences. Ordinal data are data which can have meaningful inequalities. The inequality
signs < or > may assume any meaning like ‘stronger, softer, weaker, better than’, etc.
Example: Patients may be characterized as (unimproved, improved & much improved), Individuals may
be classified according to socio-economic as (low, medium & high), Letter for grading system (A, B, C,
D, F), authority, career, etc.
Note: Qualitative variables can be either Nominal or Ordinal scales of measurements.
iii. Interval Scale: With this scale it is not only possible to order measurements, but also the distance
between any two measurements is known but not meaningful quotients. There is no true zero point but
arbitrary zero point. Interval data are the types of information in which an increase from one level to
the next always reflects the same increase. Possible to add or subtract interval data but they may not be
multiplied or divided. Example: Temperature of zero degrees does not indicate lack of heat. The two
common temperature scales; Celsius (C) and Fahrenheit (F). We can see that the same difference
exists between 10o
C (50o
F) and 20o
C (68O
F) as between 25o
c (77o
F) and 35o
c (95o
F) i.e. the
measurement scale is composed of equal-sized interval. But we cannot say that a temperature of 20o
c
is twice as hot as a temperature of 10o
c because the zero point is arbitrary.
iv. Ratio Scale: - Characterized by the fact that equality of ratios as well as equality of intervals may be
determined. Fundamental to ratio scales is a true zero point. All arithmetic operations are used to
calculate values of ratio scale. Most statistical data analysis procedures do not distinguish between the
interval and ratio properties of the measurement scales. Example: Variables such as age, height,
length, volume, rate, time, amount of rainfall, etc. are require ratio scale.
Note: Quantitative variables can be either Interval or Ratio scales of measurements.

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Chapter Two
2. Methods of Data Collection and Presentation
2.1. Sources of Data
Before one deal with methods of data collection it is necessary to see sources of data from which data
are collected, there are two sources of data that pervades the nature of data. These are Primary sources
and Secondary sources. The primary sources of data are objects or persons from which we collect the
figures used for first-hand information. The data obtained from sources are measurements observed or
recorded as a part of an original study or surveys being conducted are called primary data, or data
measured or collect by the investigator or the user directly from the source. The key point here is that the
data you collect is unique to you and your research and, until you publish, no one else has access to it.
Examples of primary data are weight, feeling, current air pressure, etc.
The secondary sources are either published or unpublished materials or records. Secondary data can be
literally defined as second-hand information and data or information that was either gathered by
someone else (e.g., researchers, institutions, other NGOs, etc.) or for some other purpose than the one
currently being considered, or often a combination of the two. Some of the sources of secondary data are
government document, official statistics, technical report, scholarly journals, trade journals, review
articles, reference books, research institutes, universities, hospitals, libraries, library search engines,
computerized data base and world wide web ( ). Examples of secondary data are age, birth date,
vital statistics from reports, etc.
Before use of secondary data investigator should examine:
 The type and objective of the situations.
 The purpose for which the data are collected and compatible with the present
problem.
 The nature and classification of data is appropriate to our problem.
 There are no biases and misreporting in the published data.
 Reliability, homogeneity, and completeness.

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2.2. Methods of Data Collection
Planning to data collection requires: identify source and elements of the data, decide whether to consider
sample or census, if sampling is preferred, decide on sample size, selection method,. decide
measurement procedure, and set up the necessary organizational structure. Then, data can be collected
by one of the following methods based on nature of items of information, objectives of the enquiry,
operational feasibility and cost. In primary data collection, you collect the data by yourself using
methods such as interviews, observations, laboratory experiments and questionnaires.
1) Observation: It involves recording the behavioral patterns of people, objects and events in a
systematic manner. The technique ranges from single visual observation to those requiring special
skills like direct observation/examination. Example: measuring height, weight, temperature,
chemical component in water, etc. And it may include laboratory experiment; conducting laboratory
experiments on fields of chemical, biological sciences and so on.
2) Questionnaire: It is a popular means of collecting data, but is difficult to design and often require
many rewrites before an acceptable questionnaire is produced. In this method a set of questions are
administered (provided) to respondent either physically or through mail (Email, Postal, etc).
Schedule through enumerations is the method in which investigator approach to the informant with
prepared questionnaire and got replies to the questions.
Advantages:
 Can be used as a method in its own right or as a basis for interviewing or a telephone survey.
 Can be posted, e-mailed or faxed.
 Can cover a large number of people or organizations and wide geographic coverage.
 Relatively cheap and avoids embarrassment on the part of the respondent.
 Respondent can consider responses, and there is no interviewer bias.
Disadvantages:
Historically low response rate (although inducements may help).
Time delay whilst waiting for responses to be returned
Several reminders may be required and it assumes no literacy problems.
No control over who completes, and it is not possible to give assistance if required.
Respondent can read all questions beforehand and then decide whether to complete or not.

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3) Interviewing: is a technique that is primarily used to gain an understanding of the underlying
reasons and motivations for people’s attitudes, preferences or behavior. Interviews can be
undertaken on a personal (face to face)or via telephone (indirect method). They can be conducted at
work, at home, in the street or in a shopping center, or some other agreed location.
Advantages:
Serious approach by respondent resulting in accurate information and good response rate.
Completed and immediate.
Interviewer in control and can give help if there is a problem.
Possible in-depth questions.
Can use recording equipment.
Characteristics (motives and feelings) of respondent assessed – tone of voice, facial expression,
hesitation, etc.
If one interviewer used, uniformity of approach.
Used to pilot than other methods.
Disadvantages:
Need to set up interviews.
Time consuming.
Geographic limitations.
Can be expensive.
Normally need a set of questions.
Respondent bias – tendency to please or impress, create false personal image, or end
interview quickly.
Embarrassment possible if personal questions.
Transcription and analysis can present problems (subjectivity).
If many interviewers, training required.

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4) Extract from Records/Documentary Sources: it is method of collecting information (secondary
data) from published or unpublished sources. Secondary data also collected from diaries. A diary is
a way of gathering information about the way individuals spend their time on professional activities.
In management research it can provide information about work patterns and activities
Advantage of secondary data
Secondary data may help to clarify or redefine the definition of the problem as part of the
exploratory research process.
Time saving
Does not involve collection of data
Provides a larger database as compared to primary data
Disadvantage of secondary data
Lack of availability
Lack of relevance
Inaccurate data
Insufficient data
5) Focus Group discussion
6) Experimental design
7) New Product Registration and other methods in various application areas.
2.3. Methods of Data Presentation
So far you know how to collect data. So what do we do with the collected data next? Now you have to
present the data you have collected so that they can be of use. Thus, the collected data also known as
‘raw data’ are always in an unorganized form and need to be organized and presented in a meaningful
and readily comprehensible form in order to facilitate further statistical analysis.Raw data: recorded
information in its original collected form, whether it is counts or measurements, is referred to as raw
data.Classification is a preliminary and it prepares the ground for proper presentation of data.
The objectives of data classification (organization) are:-
 To avoid/eliminate unnecessary details and repetitions.
 To bring clear point of similarity and differences; i.e.to arrange objects or data in groups or
classes according to their resemblance and disparities.
 To enable one to form mental picture of data.
 To facilitate data presentation.

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After collecting and organize data the next important task is effective presentation of bulk volume data.
The major objectives of data presentation are:-
 To presenting data in visual display and more understandable
 To have great attraction about the data
 To facilitate quick comparisons using measures of location and dispersion.
 To enable the reader to determine the shape and nature of distribution to make statistical
inference.
 To facilitate further statistical analysis.
There are three methods of data presentation, namely: Tables, Diagrams, and Graphs are commonly
used to summarize both qualitative and quantitative data.
2.3.1. Tabular presentation of data
Tables are important to summarize large volume of data in more understandable way. Based on the
characteristics they present tables are:
i. Simple (one way table): table which present one characteristics for example age distribution.
ii. Two way table: it presents two characteristics in columns and rows for example age versus sex.
iii. A higher order table: table which presents two or more characteristics in one table.
In statistics usually we use frequency distribution table for different type of data, however sometimes we
use the above tables to summarize raw data and to present results of statistical analysis.
Frequency Distribution: is the table that shows data classified in to a number of classes according to
some criteria with corresponding frequencies; in other words a frequency distribution is the organization
of raw data in table form, using classes and frequencies. Where, Frequency is the number of values in a
specific class of the distribution.
There are three basic types of frequency distributions, and there are specific procedures for constructing
each type. The three types are categorical, ungrouped and grouped frequency distributions.

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A. Categorical Frequency Distribution
The categorical frequency distribution is used for data which can be placed in specific categories such
as nominal or ordinal level data. For example, for data such as political affiliation, religious affiliation,
blood type, or major field of study categorical frequency distribution is appropriate.
The major components of categorical frequency distribution are class, tally and frequency (or
proportion). Moreover, even if percentage is not normally a part of a frequency distribution, it will be
added since it is used in certain types of graphical presentations, such as pie chart.
Steps of constructing categorical frequency distribution
1. You have to identify that the data is in nominal or ordinal scale of measurement
2. Make a table as show below
3. Put distinct values of a data set in column A
4. Tally the data and place the result in column B
5. Count the tallies and place the results in column C
6. Find the percentage of values in each class by using the formula %
100
n
f
x
Where, f is frequency, and n is total number of values.

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Example 2.1: Twenty-five army inductees were given a blood test to determine their blood type.
The data set is given as follows:
A B B AB O
O O B AB B
B B O A O
A O O O AB
AB A O B A
Construct a frequency distribution for the above data.
Solution
Our data type is nominal so we use categorical frequency distribution type to present data. After we have
followed the above six steps the following frequency distribution was obtained.
B. Ungrouped Frequency Distribution
It is a table of all the potential raw values that could possible occurs in the data along with the number
of times each actually occurred. In other words ungrouped frequency distribution is the distribution that
use individual data values along with their frequencies. It is often constructed for small set of data on
discrete variable (when data are numerical), and when the range of the data is small. However,
sometimes it is complicate to use ungrouped frequency distribution for large mass of data, as result we
use grouped frequency distribution.
The major components of this type of frequency distributions are class, tally, frequency, and cumulative
frequency.

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Cumulative frequencies (CF):- are used to show how many values are accumulated up to and including a
specific class. We have less than and more than cumulative frequencies.
Less than Cumulative Frequency (LCF):-is the total sum of observations below specified class including
that class
More than Cumulative frequency (MCF):- is the total sum of observations above specified class
including that class.
Steps of constructing ungrouped frequency distribution
The steps are almost similar with that of categorical frequency distribution
 First find the smallest and largest raw score in the collected data.
 Arrange the data in order of magnitude and count the frequency.
 To facilitate counting one may include a column of tallies.
 Put respective frequency, relative frequency, and cumulative frequency along each ordered data.
Example 2.2 A demographer is interested in the number of children a family may have, he/she took
sample of 30 families and obtained the following observations.
Number of children in a sample of 30 families
4 2 4 3 2 8
3 4 4 2 2 8
5 3 4 5 4 5
4 3 5 2 7 3
3 6 7 3 8 4
Construct a frequency distribution for this data.
Solution
These individual observations can be arranged in ascending or descending order of magnitude in which
case the series is called array. Array of the number of children in 30 families is:
2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8

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And the variable” number of children in a family” can assume only the values 0, 1, 2, 3, . . . which are
discrete; therefore, we construct ungrouped frequency distribution consists of individual data,
frequencies, and cumulative frequencies.
Frequency distribution of children in a 30 families is as follow:
No of children
(Class)
No of family
(Frequency) LCF MCF
2 5 5 30
3 7 12 25
4 8 20 18
5 4 24 10
6 1 25 6
7 2 27 5
8 3 30 3
Each individual value is presented separately, that is why it is named ungrouped frequency distribution
C. Grouped Frequency Distribution
It is a frequency distribution when several numbers are grouped in one class; the data must be grouped
in which each class has more than one unit in width. We use this type of frequency distribution when the
range of the data is large, and for data from continuous variable. But, sometimes for large volume of
discrete we use grouped frequency distribution.
Some of basic terms that are most frequently used while we deal with grouped frequency distribution are
Lower Class Limits are the smallest number that can belong to the different class.
Upper Class Limits are the largest number that can belong to the different classes.
Class Boundaries (true class limits) are the number used to separate classes, but without the gaps
created by class limits.
Class midpoints are the midpoints of the classes. Each class midpoint can be found by adding the
lower class limit/boundary to the upper class limit boundary and dividing the sum by 2.
Class width is the difference between two consecutive lower class limits or two consecutive
lower class boundaries.

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Steps in constructing grouped frequency distribution
1. Find the highest and the lowest values
2. Find the range; Minimum
Maximum
Range 
 or L
H
R 

3. Select the number of classes desired. Here, we have two choices to get the desired number of
classes:
i. Use Struge’s rule. That is, n
K log
32
.
3
1
 where K is the number of class and n is the
number of observations. Round the decimal to the upper nearest integer.
ii. Select the number of classes arbitrarily between 5 and 20 conventionally. If you fail to
calculate K by Struge’s rule, this method is more appropriate.
When we choose the number of classes, we have to think about the following criteria
The classes must be mutually exclusive. Mutually exclusive classes have non overlapping class
limits so that values can’t be placed in to two classes.
The classes must be continuous. Even if there are no values in a class, the class must be included
in the frequency distribution. There should be no gaps in a frequency distribution. The only
exception occurs when the class with a zero frequency is the first or last. A class width with a
zero frequency at either end can be omitted without affecting the distribution.
The classes must be equal in width. The reason for having classes with equal width is so that
there is not a distorted view of the data. One exception occurs when a distribution is open-ended.
i.e., it has no specific beginning or end values.
4. Find the class width (W) by dividing the range by the number of classes
W =
R
K
or W =
Range
Number of Classes
Note that: Round the value of W up to the nearest whole number if there is a reminder. For
instance, 4.7 ≈ 5 and 4.12 ≈ 5.
5. Select the starting point as the lowest class limit. This is usually the lowest score (observation). Add
the width to that score to get the lower class limit of the next class. Keep adding until you achieve
the number of desired classes (K) calculated in step 3.

20
6. Find the upper class limit; subtract unit of measurement(U) from the lower class limit of the second
class in order to get the upper class limit of the first class. Then add the width to each upper class
limit to get all upper class limits. Take care of the last class to include the maximum value of data.
Unit of measurement: Is the smallest value of difference between consecutive observations or
sometimes it is next value. For instance, 28, 23, 52, and then the unit of measurement of this data
set is one. Because take one datum arbitrarily, say 23, then the next value will be 24.
Therefore, U = 24 − 23 = 1. If the data set is 24.12, 30, 21.2, then give priority to the datum
with more decimal place. Take 24.12 and guess the next possible value. It is 24.13. Therefore,
0.01
=
24.13
-
24.12
=
U
Note that: U=1 is the maximum value of unit of measurement and is the value when we
don’t have a clue about the data.
7. Find the class boundaries.
2
U
Limit
Class
Lower
Boundary
Class
Lower 
 and
2
U
Limit
Class
Upper
Boundary
Class
Upper 
 .
In short,
2
U
-
LCL
=
LCB and
2
U
UCL
=
UCB  .
8. Tally the data and write the numerical values for tallies in the frequency column.
9. Find cumulative frequency. We have two type of cumulative frequency namely less than cumulative
frequency and more than cumulative frequency. Less than cumulative frequency is obtained by
adding successively the frequencies of all the previous classes including the class against which it is
written. The cumulate is started from the lowest to the highest size. More than cumulative frequency
is obtained by finding the cumulate total of frequencies starting from the highest to the lowest class.
For example, the following frequency distribution table gives the marks obtained by 40 students:

21
The above table shows how to find less than cumulative frequency and the table shown below
shows how to find more than cumulative frequency.
Example 2.3: Consider the following set of data and construct the frequency distribution.
11 29 6 33 14 21 18 17 22 38
31 22 27 19 22 23 26 39 34 27
Solution:
Using steps to construct grouped frequency distribution
1. Highest value=39, Lowest value=6
2. 33
=
6
-
39
=
R
3. 6
5.32
=
log20
3.32
+
1
=
K 
4. 6
5
.
5
6
33




K
R
W

22
5. Select starting point. Take the minimum which is 6 then add width 6 on it to get the next class
LCL.
6. Upper class limit. Since unit of measurement is one. 11
1
12 
 . So, 11 is the UCL of the first
class. Therefore, 11
6  is the first class limit.
7. Find the class boundaries. Take the formula in step 7. 0.5
-
LCL
=
LCB i
i and
5
.
0

 i
i UCL
UCB
With respective the class boundaries, the class marks each class limit are:-
Class marks 8.5 14.5 20.5 26.5 32.5 38.5
8. 9 and 10

23
Example 2.4 The following data are percentage coverage of forest in countries in Africa. Construct
frequency distribution by using sturge’s rule.
30, 25, 23, 41, 39, 27, 41, 24, 32, 29, 35, 31, 36, 33, 36, 42, 35, 37, 41, and 29
Solution
1. Given no. of observation( n ) = 20,then no. of classes
K = 1 + 3.32 5
log20
10  , where k is number of classes.
2. Class width( W) = 4
5
23
42




k
e
lowestvalu
ue
highestval
Classes Class boundary Class mark Frequency
23 - 26 22.5 –26.5 24.5 3
27 - 30 26.5 – 30.5 28.5 4
31 - 34 30.5 – 34.5 32.5 3
35 - 38 34.5 – 38.5 36.5 5
39 - 42 38.5 – 42.5 40.5 5
Total 20
Relative Frequency Distribution
An important variation of the basic frequency distribution uses relative frequencies, which are easily
found by dividing each class frequency by the total of all frequencies. A relative frequency distribution
includes the same class limits as a frequency distribution, but relative frequencies are used instead of
actual frequencies. The relative frequencies are sometimes expressed as percent.
s
frequencie
all
of
Sum
Frequency
Class
=
Frequency
Relative
Relative frequency distribution enables us to understand the distribution of the data and to compare
different sets of data.

24
Example 2.5: Relative Frequency and Percentage Table
Class interval Frequency Relative Frequency Percentage
10 but less than 20 3 0.055 5.5
20 but less than 30 5 0.091 9.1
30 but less than 40 9 0.164 16.4
40 but less than 50 18 0.32 7 32.7
50 but less than 60 10 0.182 18.2
60 but less than 70 8 0.145 14.5
70 but less than 80 2 0.036 3.6
Total 55 1.000 100
It may be noted that at times the use of relative frequencies is more appropriate than absolute
frequencies. Whatever two or more sets of data contain different number of observation, a comparison
with absolute frequencies will be incorrect. In such cases, it is necessary to use the relative frequency
2.3.2. Diagrammatic Presentation of the Data
We have discussed the techniques of classification and tabulation that help us in organizing the collected
data in a meaningful fashion. However, this way of presentation of statistical data does not always prove
to be interesting to a layman. Too many figures are often confusing and fail to convey the massage
effectively.One of the most effective and interesting alternative way in which a statistical data may be
presented is through diagrams and graphs. There are several ways in which statistical data may be
displayed pictorially such as different types of graphs and diagrams.
General steps in constructing diagram or graphs
1. Draw and label the x and y axes
2. Choose a suitable scale for the frequencies or cumulative frequencies and label it on the y axis.

25
3. Represent the class boundaries for the histogram or Ogive or the midpoint for the frequency polygon
on the x axis.
4. Plot the points, and draw the bars or lines
Most of time we use diagram form of presentation for categorical and numerical data, some of the
diagrams are as follow:
A. Pie Chart
Pie chart can used to compare the relation between the whole and its components. Pie chart is a circular
diagram and the area of the sector of a circle is used in pie chart. Circles are drawn with radii
proportional to the square root of the quantities because the area of a circle is 2
r
 .
To construct a pie chart (sector diagram), we draw a circle with radius (square root of the total). The
total angle of the circle is 0
360 .
The angles of each component are calculated by the formula
0
360
x
Total
Part
Component
Sector
of
Angle  These angles are made in the circle by mean of a protractor to
show different components. The arrangement of the sectors is usually anti-clock wise.
Example2.6: The following table gives the details of monthly budget of a family. Represent
these figures by a suitable diagram.
Solution: The necessary computations are given below:

26
Figure 2.1 pie chart of monthly budget of a family
B. Bar Charts
The bar charts (simple bar chart, multiple bar charts, and stratified or stacked bar chart) use vertical or
horizontal bins to represent the frequencies of a distribution. While we draw bar chart, we have to
consider the following two points. These are
Make the bars the same width
Make the units on the axis that are used for the frequency equal in size
i. Simple Bar Chart is used to represents data involving only one variable classified on spatial,
quantitative or temporal basis.In simple bar chart, we make bars of equal width but variable length, i.e.
the magnitude of a quantity is represented by the height or length of the bars.The following steps are
undertaken in drawing a simple bar diagram:
Draw two perpendicular lines one horizontally and the other vertically at an appropriate place of
the paper.
food
40%
clothing
6%
House Rent
27%
Fuel and
Light
7%
misclaneou
s
20%
Monthly budget of family

27
Take the basis of classification along horizontal line (X-axis) and the observed variable along
vertical line (Y-axis) or vice versa.
Marks signs of equal width for each class and leave equal or not less than half width in between
two classes.
Finally, marks the values of the given variable to prepare required bars.
Example 2.7: Draw simple bar diagram to represent the profits of a bank for 5 years.
ii. Multiple Bars
When two or more interrelated series of data are depicted by a bar diagram, then such a diagram is
known as a multiple-bar diagram. Suppose we have export and import figures for a few years. We can
display by two bars close to each other, one representing exports while the other representing imports
figure shows such a diagram based on hypothetical data.

28
Multiple bar chart should be noted that multiple bar diagrams are particularly suitable where some
comparison is involved.
iii. Stratified (Stacked) Bar Chart is used to represent data in which the total magnitude is divided into
different or components. In this diagram, first we make simple bars for each class taking total
magnitude in that class and then divide these simple bars into parts in the ratio of various components.
This type of diagram shows the variation in different components within each class as well as between
different classes. Stratified bar diagram is also known as component bar chart.
Example 2.8: The table below shows the quantity in hundred Kgs of Wheat, Barley and Oats
produced on a certain farm during the years 1991 to 1994. Draw stratified bar
chart.

29
Solution: to make the component bar chart, first of all we have to take year wise total
production. The required diagram is given below:
iv. Deviation Bar Diagram:
When the data contains both positive and negative values such as data on net profit, net expense, percent
change etc.
Example 2.9:Suppose we have the following data relating to net profit (percent) of commodity.
Solution:
-150
-100
-50
0
50
100
150
Soap Sugar Coffee
Net profit
Commodity Net profit
Soap
Sugar
Coffee
80
-95
125

30
2.3.3. Graphical Presentation of Data
Often we use graphical presentation form for continuous data type; results from the grouped frequency
distribution and continuous variables distributed over time.
A. Histogram
Histogram is a special type of bar graph in which the horizontal scale represents classes of data values
and the vertical scale represents frequencies. The height of the bars correspond to the frequency values,
and the drawn adjacent to each other (without gaps).
We can construct a histogram after we have first completed a frequency distribution table for a data set.
TheX axis is reserved for the class boundaries.
Example2.9: The histogram for the data in example 2.4 is
41.5
35.5
17.5
11.5
5.5
Class boundaries
7.0
6.0
5.0
4. 0
3.0
2.0
1.0
0.0
23.5 29.5
Frequency

31
Relative frequency histogram has the same shape and horizontal (x axis) scale as a histogram, but the
vertical (y axis) scale is marked with relative frequencies instead of actual frequencies.
B. Frequency Polygon
A frequency polygon uses line segment connected to points located directly above class midpoint values.
The heights of the points correspond to the class frequencies, and the line segments are extended to the
left and right so that the graph begins and ends on the horizontal axis with the same distance that the
previous and next midpoint would be located.
Example 2.10:The frequency polygon for the data in example 2.4 is given as follow.
C. Ogive Graph
An ogive (pronounced as “oh-jive”) is a line that depicts cumulative frequencies, just as the cumulative
frequency distribution lists cumulative frequencies. Note that the Ogive uses class boundaries along the
horizontal scale, and graph begins with the lower boundary of the first class and ends with the upper
boundary of the last class. Ogive is useful for determining the number of values below some particular
value. There are two type of Ogive namely less than Ogive and more than Ogive. The difference is that
less than Ogive uses less than cumulative frequency and more than Ogive uses more than cumulative
frequency on y axis.
Example 2.10: Take the data in example 2.4 and draw less than and more than Ogive curve.
38.5
32.5
26.5
20.5
14.5
8.5
Midpoints
7.0
6.0
5.0
4.0
3.0
2.0
2.5 44.5
Frequency
polygon

32
Chapter Three
3. Measures of Central Tendency (MCT)
3.1. Measure of Central Tendency
Suppose the students from two or more classes appeared in the examination and we wish to compare the
performance of the classes in the examination or wish to compare the performance of the same class
after some coaching over a period of time. When making such comparisons, it is not practicable to
compare the full frequency distributions of marks. For such statistical analysis, we need a single
representative value that describes the entire mass of data given in the frequency distribution. This single
representative value is called the central value, measure of location or an average around which
individual values of a series cluster. This central value or an average enables us to get a gist of the entire
mass of data, and its value lies somewhere in the middle of the two extremes of the given observations.
It should be clear to you that the concept of a measure of central tendency is concerned only with
quantitative variables and is undefined for qualitative variables as these are immeasurable on a scale.
The three major objectives of measures of central tendency are
35.5
29.5
23.5
17.5
11.5
5.5
Class Boundaries
20
15
10
5
0
41.5
Less than Ogive
More than Ogive
F
r
e
q
u
e
n
c
y

33
To summarize a set of data by single value
To facilitate comparison among different data sets
To use for further statistical analysis or manipulation
3.2. Summation Notation
1. )
(sigma is used to facilitate the writing of sum
2. 





n
i
n
i x
x
x
x
x
1
3
2
1 .
.
.
.
3. 





n
i
n
n
i
i y
x
y
x
y
x
y
x
1
2
2
1
1 .
.
4.        










n
i
n
n
i
i y
x
y
x
y
x
y
x
1
2
2
1
1 .
.
. =  
 

n
i
n
i
i
i y
x
1 1
5. 






n
i
n
CX
CX
CX
CX
CXi
1
3
2
1 .
.
. =C (X1 +X2 + X3 + . . .+ Xn) = C 

n
i
Xi
1
6. 






n
i
nc
C
C
C
C
C
1
.
.
.
.
7.        









n
i
n
i c
x
c
x
c
x
c
x
1
2
1 ...
= x1 + x2 +. . . + xn + c + . . . + c
=  
 

n
i
n
i
C
xi
1 1
= 


n
i
i nc
x
1
N.B  
 









n
i
n
i
i
i x
x
1
2
1
2
and      


  i
i
i
i y
x
y
x
3.3. Desirable Properties of Good Measures of Central Tendency

34
1. It should be easy to calculate and understand.
2. It should be based on all the observations during computation.
3. It should be rigidly defined. The definition should be clear and unambiguous so that it leads to
one and only one interpretation by different persons.
4. It should be representative of the data, if it’s from sample. Then the sample should be random
enough to be accurate representative of the population.
5. It should have sampling stability. It shouldn’t be affected by sampling fluctuations. This means
that if we pick (take) two independent random samples of the same size from a given
population and compute the average for each of these samples then the value obtained from
different samples should not vary much from one another.
6. It shouldn’t be affected by the extreme value. If a few very small and very large items are
presented in the data, they will influence the value of the average by shifting it to one side or of
other side and hence the average chosen should be such that is not influenced by the extreme
values.
3.4. Types of Measures of Central Tendency
In statistics, we have various types of measures of central tendencies. The most commonly used types of
MCT includes:-
-Mean - Mode
- Median -Quantiles (Quartiles, Decilies, Percentiles)
3.4.1. Mean
Mean is obtained for sample and population data. There are four type of mean which is suitable for a
particular type of data.
i. Arithmetic mean (A.M)
ii. Geometric mean (G.M)
iii. Harmonic mean (H.M)
iv. Weighted mean (W.M)
i. Arithmetic Mean ( )
In classification and presentation of data, we observed that the values of the variable or observations
could be put in the form of raw data (individual data), simple (ungrouped) frequency distribution, and
grouped frequency distribution.

35
a) Arithmetic Mean from raw(individual) data
Let X be a variable which takes valuesx1 ,x2 ,x3 ,…………….,xn, in a sample size of n from a
population of size N for n < N then A.M of a set of observations is the sum of all values in a series
divided by the number of items in the series.





 i
n
X
n
n
X
X
X
X
1
.
.
.
2
1
for sample data
Example:3.1Suppose he scores of a student on six examinations were 10 ,15,30,7,42 , 79 and 83 ,find
the arithmetic mean of scores of students.
Solution: there are seven observations (n=7). Symbolically, the arithmetic mean is:
n
x
X
n
i
i


 1
= (10 + 15 + 30+ 7 + 42 + 79 + 83) / 7 = 266 / 7 = 38
Example 3.2: The following data are on age of 20 household heads which are obtained during water
distribution survey in certain town. Find the average age of household heads.
30, 25, 23, 41, 39, 27, 41, 24, 32, 29, 35, 31, 36, 33, 36, 42, 35, 37, 41, and 29
3
.
33
20
666
20
29
...
25
30
1
.
.
.
1
2
1










 

n
i
i
n
X
n
n
X
X
X
X
Interpretation: the average age of 20 household heads is 33.3 years
NB: If mean is mentioned, it implies arithmetic mean, but other means are identified by their name.
It may be noted that the Greek letter  is used to denote the mean of the population and N to
denote the total number of observations in a population. Thus, the population mean







N
i
i
N
X
N
N
X
X
X
1
2
1 1
.
.
.

b) Arithmetic mean from discrete (ungrouped) frequency distribution

36
In discrete frequency distribution we multiply the values of the variable (X) by their respective
frequencies (f) and get the sum of the products ( i
fX
 ). The sum of the products is then divided by the
total of the frequencies (  fi = n).
Values (Xi) Frequency(fi) Xi*fi
X1 f1 x1 f1
X2 f2 x2 f2
.
Xi
.
.
fi
.
Xk fk xk fk
 i
f  i
i f
x
Thus, according to this method, the formula for calculating arithmetic mean for simple /discrete/
frequency distributions becomes:
n
fXi
fi
fXi
X


 
 where, 
 fi
n
Example 3.3: Following table gives the wages paid to 125 workers in a factory. Calculate the arithmetic
mean of the wages.
Wages (in birr): 200 210 220 230 240 250 260
No. of workers: 5 15 32 42 15 12 4
Wages(x) No. of Workers(f) fx
200
210
220
230
240
250
260
5
15
32
42
15
12
4
1000
3150
7040
9660
3600
3000
1040
Total

 f
N =125   28490
fX

37
Solution:using above results in the table we substitute into formula and obtain
birr
N
fx
X 92
.
227
125
28490




Interpretation: the average wage paid to 125 workers in the factory is 227.92 birr
c) Arithmetic mean for continuous (grouped) frequency distribution
Arithmetic mean for continuous frequency distribution is given by:








 
 

n
i
i
i
k
i
i
m
f
f
X
1
1
1
, where mi = midpoint of each ith
class interval, k is no. of class
Example 3.4:The following table gives the marks of 58 students in probability and Statistics.
Calculate the average marks of these students.
Marks Number of students
0-10 4
10-20 8
20-30 11
30-40 15
40-50 12
50-60 6
60-70 2
Solution

38
Marks Mid-point (mi) No. of Students (fi) fi*mi
0-10 5 4 20
10-20 15 8 120
20-30 25 11 275
30-40 35 15 525
40-50 45 12 540
50-60 55 6 330
60-70 65 2 130 ______________
58

 fi   1940
i
i m
f
So, Arithmetic mean will be 







 
 

n
i
i
i
n
i
i
m
f
f
X
1
1
1
= 1940/58 = 33.45.
Properties of arithmetic mean
1. It is easy to calculate and understand.
2. All observation involved in its calculation.
3. It cannot be computed for open end classes
4. It may not be the values which the variable actually takes and termed as a fictitious (unreal)
average. E.g. The figure like on average 2.21 children per family, 3.4 accidents per day.
5. It is affected by extreme values.
6. It is Unique: - a set of data has only one mean.
7. If a constant k is added or subtracted from each value of a distribution, then the new mean for
the new distribution will be the original mean plus or minus k, respectively.
8. The sum of the deviation of various values from their mean is zero i.e.   0


 x
xi

39
9. The sum of the squares of deviation of the given set of observations is minimum when taken
from the arithmetic mean i.e.  
 

2
A
xi is minimum when taken from mean than any
arbitrary value A from a set of observation.
10. It can be used for further statistical treatment, comparison of means, test of means.
11. When a set of observations is divided into k groups and 1
x is the mean of 1
n observations of
group 1, 2
x is the mean of 2
n observations of group2, …, k
x is the mean of k
n observations of
group k , then the combined mean ,denoted by c
x , of all observations taken together is given by
=
̅ + ̅ + ⋯ + ̅
+ + ⋯ +
=
∑ ̅
∑
12. If a wrong figure has been used in calculating the mean, we can correct if we know the correct figure
that should have been used. Let
 denote the wrong figure used in calculating the mean
 be the correct figure that should have been used
 be the wrong mean calculated using , then the correct mean,then , is given by
=
+ −
Example 3.5: Last year there were three sections taking Stat 1044 course in Arba Minch University. At
the end of the semester, the three sections got average marks of 80, 83 and 76. If there were 28, 32 and
35 students in each section respectively. Find the mean mark for the entire students.
Solution: here we need to calculate combined mean












95
7556
35
32
28
)
76
(
35
)
83
(
32
)
80
(
28
3
2
1
3
3
2
2
1
1
n
n
n
x
n
x
n
x
n
xc 79.54
Example 3.6: An average weight of 10 students was calculated to be 65 kg, but latter, it was discovered
that one measurement was misread as 40 kg instead of 80 kg. Calculate the corrected average weight.
Solution: =

=
( )
= 69

40
Exercise: The average score on the mid-term examination of 25 students was 75.8 out of 100. After the
mid-term exam, however, a student whose score was 41 out of 100 dropped the course. What is the
average/mean score among the 24 students?
ii. Geometric Mean (G.M)
The geometric mean is usually used in average rates of change, ratios, percentage distribution, and
logarithmical distribution.
The geometric mean is the nth
root of the product of n positive values.
a) Geometric mean for raw (individual) data series
If X1, X2,…,Xn are n positive values, then their geometric mean is G.M =(X1X2…Xn)1/n
.
b) Geometric mean for simple (discrete) frequency distribution
When the observed values x1,x2,……….xn have the corresponding frequencies f1.f2………fn
respectively then geometric mean is obtained by
G. M = n
n
f
f
f n
x
x
x .
.
.
.
, 2
1
2
1
= 

n
i
i
i x
f
n 1
log
1
where, n= 

n
i
i
f
1
c) Geometric mean for continuous (grouped) frequency distribution
Whenever the frequency distributions are grouped (continuous), class marks of the class interval are
considered as Xi, and the above formula can be used as:
G. M = n f
n
f
f n
m
m
m .
.
.
.
, 2
1
2
1
= 

n
i
i
i m
f
n 1
log
1
where n= 

n
i
i
f
1
and mi is class mark of ith
class.
Note: in case of number of observation is more than two it may be tedious taking out from square root,
in that case calculation can be simplified by taking natural logarithm with base 10.
G . M = n
n
x
x
x .
.
.
.
, 2
1 G . M =  n
n
x
x
1
1 .
.
.
. take log in both sides.

41
log ( G . M) =  
n
x
x
n
.
.
.
,
log
1
1 =  
n
x
x
x
n
log
.
.
.
log
log
1
2
1 

 = 

n
i
i
x
n 1
log
1
G. M = Antilog 







n
i
i
x
n 1
log
1
This shows that the logarithm of G. M is the mean of the logarithms of individual observations. In
similar manner we compute for ungrouped and grouped frequency distributions.
Example3.7: The ratios of prices in 1999 to those in 2000 for 4 commodities were 0.9, 1.25, 1.75 and
0.85. Find the average price ratio.
Solution G.M = antilog
n
X i
log
= antilog
4
)
85
.
0
log
75
.
1
log
25
.
1
log
92
.
0
(log 


= antilog
4
)
1
9294
.
0
2430
.
0
0969
.
0
1
963
.
0
( 




= antilog0.5829 = 1.14
What is the arithmetic mean of the above values? X = 19
.
1
4
85
.
0
75
.
1
25
.
1
92
.
0




This is quite different from the geometric mean in case of ratio data.
Properties of geometric mean
a. Its calculations are not as such easy.
b. It involves all observations during computation
c. It may not be defined even it a single observation is negative.
d. If the value of one observation is zero its values becomes zero.
e. It is affected by sample fluctuations.
iii. Harmonic mean (H.M)
The harmonic mean is used to average rates rather than simple values. It is usually appropriate in
averaging kilometers per hour, speed, and rates at time. The Harmonic mean is the reciprocal of the
arithmetic mean of the reciprocal of the single values
a) Harmonic mean for raw data

42
If X1,X2, X3,…,Xn are n values, then their harmonic mean is






i
n X
n
X
X
X
n
M
H
1
1
...
1
1
.
2
1
b) Harmonic mean for ungrouped frequency distribution

 






k
i
i
i
i
k
k
f
n
X
f
n
X
f
X
f
X
f
n
M
H
1
2
2
1
1
where,
...
.
c) Harmonic mean for grouped frequency distribution
In calculating harmonic mean for grouped frequency distribution data mi is replace in place of Xi
in above equation under b

 






k
i
i
i
i
k
k
f
n
m
f
n
m
f
m
f
m
f
n
M
H
1
2
2
1
1
where,
...
.
Example 3.8: Find the harmonic mean of the values 2, 3 &6.
H.M =
6
/
1
3
/
1
2
/
1
3


=
6
1
2
3
3


=
6
6
3
= 3
Example 3.9: A driver covers the 300km distance at an average speed of 60 km/hr makes the return trip
at an average speed of 50km/hr. What is his average speed for total distance?
Solution
Average speed for the whole distance=
taken
time
Total
ce
dis
Total tan
=600km/11hrs=54.55km/hr.
Trip Distance Average speed Time taken
1st
300km 60km/hr 5hrs
2nd
300km 50km/hr 6hrs
Total 600km --------- 11hrs

43
Using harmonic formula it is H.M=
50
/
1
60
/
1
2

=600/110=54.55km/hr.
Note that A.M=
2
50
60 
=55km/hr G.M= 50
60 =54.7km/hr.
In general: A.M ≥G.M≥H.M
Properties of harmonic mean
i. It is based on all observation in a distribution.
ii. Used when a situations where small weight is given for larger observation and larger weight for
smaller observation
iii. Difficult to calculate and understand
iv. It is not affected by extreme values and sample fluctuation.
v. Appropriate measure of central tendency in situations where data is in ratio, speed or rate.
iv. Weighted Arithmetic Mean
In finding arithmetic mean and others type of mean for series of data, all items were assumed to be of
equally importance (each value in the data set has equal weight). But, when the observations have
different weight, we use weighted average. Weights are assigned to each item in proportion to its
relative importance.
If k
x
x
x .,
.
.
,
, 2
1 represent values of the items and k
w
w
w ,
.
.
.
,
, 2
1 are the corresponding weights, then the
weighted arithmetic mean, )
( w
x is given by
=
+ + ⋯ +
+ + ⋯ +
=
∑
∑
Example 3.10: A student’s final mark in Mathematics, Physics, Chemistry and Biology are respectively
82, 80, 90 and 70.If the respective credits received for these courses are 3, 5, 3 and 1, determine the
approximate average mark the student has got for one course.
Solution: We use a weighted arithmetic mean, weight associated with each course being taken as the
number of credits received for the corresponding course.

44
Therefore, 17
.
82
1
3
5
3
)
70
1
(
)
90
3
(
)
80
5
(
)
82
3
(















i
i
i
w
w
x
w
x
3.4.2 Median ( )
Median is defined as the value of the middle item (or the mean of the values of the two middle items)
when the data are arranged in an ascending or descending order of magnitude.
a) Median for raw data
If the n values are arranged in ascending or descending order of magnitude, the median is the middle
value if n is odd. When n is even, the median is the mean of the two middle values.
Median =
th
n





 
2
1
position element if n is odd.
=
2
1
2
2
th
th
n
n














element if n is even.
Example 3.11: Suppose we have the following series: 15, 19, 21, 7, 33, 25, 18 and 5. We have to first
arrange it in either ascending or descending order. These figures are arranged in an ascending order as
follows: 5, 7, 10, 15, 18, 19, 21, 25, and 33.
Now as the series consists of odd number of items, to find out the value of the middle item, we use the
formula Median =
th
n





 
2
1
element if n is odd.
That is the size of the 5th
item is the median which is 18.
Suppose the series consists of one more item, 23. We may, therefore, have to include 23 in the above
series at an appropriate place, that is, between 21 and 25. Thus, the series is now 5, 7, 10, 15, 18, 19, 21,
23, 25, and 33. Applying the above formula, the median is the size of 5.5th
item. Here, we have to take
i
x 82 80 90 70
i
w 3 5 3 1

45
the average of the values of 5th
and 6th
item. This means an average of 18 and 19, which gives the
median as 18.5.
b) Median for ungrouped frequency distribution
Here median is obtained by using less than cumulative frequency distribution; the median is the value of
the variable for which the cumulative frequency is just greater than or equal to 


k
i
i
f
N
1
N
where
,
2
1
Example 3.12: find median for the following frequency distribution
X Frequency Less than cumulative
frequency (LCF)
10 3 3
15 5 8
20 10 18
25 7 25
30 5 30
N=30, N/2=15, The LCF greater than or equal 15 is 18 which is corresponds to 20, thus median is 20.
c) Median for continuous frequency distribution
In the case of a continuous frequency distribution, we first locate the median class by procedures under
b. In such a way thatclass for which the cumulative frequency is just greater than or equal to



k
i
i
f
N
1
N
where
,
2
1
Then, the median is calculated by:
m
m
o
f
w
fc
n
LCL
X
Median









2
)
~
(
Where, fco is less than cumulative frequency of the class preceding (one before) the median class, fm is
frequency of the median class, LCL is lower class limit of median class and wm width of the median
class and 


k
i
i
f
n
1
Note: We may use Lower class boundary (LCb) of the median class instead of LCL.

46
Example 3.13:consider age data of 20 household heads which are obtained during water distribution
survey in certain town:30, 25, 23, 41, 39, 27, 41, 24, 32, 29, 35, 31, 36, 33, 36, 42, 35, 37, 41, and
29.Find median from grouped data.
Classes Class boundary Frequency LCF
23 - 26 22.5-26.5 3 3
27 - 30 26.5-30.5 4 7
31 - 34 30.5-34.5 3 10
35 - 38 34.5-38.5 5 15
39 - 42 38.5 – 42.5 5 20
Total 20
The median class is (31 - 34) which corresponds to n
LCF
2
1
 i.e 10 ≥10
  35
3
4
7
2
31
2
)
~
( 












n
f
w
fc
n
LCL
X
Median
m
m
o
Properties of median
- Unlike mode it is unique that is like mean there is only one median for a given set of data.
- Easy to calculate and understand.
- It is not affected by extreme value.
- It’s especially used for open ended frequency distribution when median is not found in that class.
- It is not based on all observations.
- It is not suitable for further mathematical treatment.
3.4.3 Mode
The mode is another measure of central tendency. It is the value at the point around which the items are
most heavily concentrated or the mode is the measurement that occurs in greatest frequency in data set.
A given set of data may have
 One mode – uni model e.g. A=3 ,3,7,6,2,1 =3
 Two modes – Bimodal e.g. 10,10,9,9,6,3,2,1 = 10 and 9

47
 More than two modes- multimodal. eg. 5,5,5,6,6,6,8,8,8,2,3,2 =5,6,8
 May not exist at all e.g. 1,3,2,4,5,6,7,8 no modal value
a) Mode for raw data
For simple series of data mode is the most frequent value in a certain data set.
Example 3.14: Consider the following series: 8, 9, 11, 15, 16, 12, 15, 3, 7, 15
There are ten observations in the series where 15 occurs maximum number of times. Thus, mode is 15.
b) Mode for ungrouped frequency distribution
In case of discrete frequency distribution, mode is the value of the variable corresponding to the
maximum frequency. This method can be used conveniently if there is only one value with the highest
concentration of observation.
Example 3.15: Consider the following distribution, and then determine modal value of the distribution.
X 1 2 3 4 5 6 7 8 9
f 3 1 18 25 40 30 22 10 6
The maximum frequency is 40 and corresponding value is 5, therefore, the mode is 5.
c) Mode for grouped frequency distribution
First modal class (a class with maximum frequency) could be found, and then mode for grouped
frequency distribution is determined by the following formula:
   
w
f
f
f
f
f
f
lo 













2
1
0
1
0
1
)
X̂
(
Mode
Where: o
l = is the lower value of the class in which the mode lies.
f1 = is the frequency of the class in which the mode lie.
f0 = is the frequency of the class preceding the modal class.
f2 = is the frequency of the class succeeding the modal class.

48
w = is the class width of the modal class.
While applying the above formula, we should ensure that the class-intervals are uniform throughout. If
the class-intervals are not uniform, then they should be made uniform on the assumption that the
frequencies are evenly distributed throughout the class. In the case of unequal class-intervals, the
application of the above formula will give misleading results.
Example 3.15:Consider Example 3.13, find mode for the frequency distribution.
Here we have two modal classes, the last two class with frequencies of 5. Therefore, we called the
distribution is bimodal.
Case 1; = 34.5 + 4 = 38.5
Case 2; = 38.5 + 4 = 38.5
3.5 Measure of Location (Quintiles)
Median is the value of the middle item which divide the data in to two equal parts and found by
arranging the data in an increasing or decreasing order of magnitude, where quintiles are measures
which divides a given set of data in to equal subdivision and are obtained by the same procedure to that
of median but data must be arranged only in an increasing order. The most commonly used ones include
quartiles, deciles and percentiles.
i. Quartiles
Quartiles are measure which divided the ordered data in to four equal parts and usually denoted by
Q1,Q2, Q3, and are obtained after arranging the data in an increasing order. Q1: is first(lower) quartile or
value for which 25% of the observation lies below it, Q2: is second quartile or value for which 50%of
the observation lies below or above it, and Q3: is third (upper) quartile or value for which 75% of the
arranged item lies below it or 25% lies above it.
a) Quartiles for raw data
For series of data the ith
quartiles is the value of the items which is at the
item
position
4
1
*
th
n
i 




 
i.e Qi = position
n
i
th





 
4
1
* item where, i=1, 2, 3

49
1
Q - is value corresponding to
th
n





 
4
1
ordered observation.
2
Q is the value of 2
th
n





 
4
1
ordered observation
3
Q is the value of the 3
th
n





 
4
1
ordered observation
Example 3.16: Find the value for which 25% of the wages of 7 workers lies below it, second and upper
quartiles for the data on the wage distribution of 7 workers.
Wage: 1100, 1150, 1080, 1120, 1200, 1160 and 1400
Arranging the data on ascending order, we have 1080, 1100, 1120, 1150, 1160, 1200, and 1400
Q1 is the value of position
th





 
4
1
7
item in the ordered observation,   position
nd
2 item =1100. Q2 is the
value of 2* 





 
item
position
th
4
1
7
  position
th
4 item=1150, Q3 is the value of 3* position
th





 
4
1
7
item =
  position
th
6 item=1200.
b) Quartiles for ungrouped frequency distribution
The ith
quartile (Qi) for the discrete frequency distribution data is the value corresponding to cumulative
frequency greater or equal to 


k
i
i
f
n
n
i
1
where,
4
* . Or the value whose cumulative frequency is greater
than or equal to 


k
i
i
f
n
n
i
1
where,
4
*
Example 3.17: Consider example 3.12, then compute the first, second and third quartile of the fd.
Solution: we have n=30. Then, 3
,
2
,
1
4
30
*
4
* 
 i
for
i
n
i , Q1= the value corresponding to LCF greater
or equal to 7.5 which is 15, Q2= the value corresponding to LCF greater or equal to 15 which is 20, Q3 =
the value corresponding to LCF greater or equal to 22.5 which is 25. i.e. (Q1=15,Q1=20, and Q1=25)
c) Quartiles for grouped frequency distribution

50
In case of grouped (continuous) frequency distribution quartiles obtained first by obtaining quartile class
by the procedure we have applied under ungrouped frequency distribution above,To find ith
quartile class
compute 





4
* n
i
and search for the minimum less than cumulative frequency greater than or equal to
this value, then the class corresponding to this cumulative frequency is ith
quartile class. Then to
determine unique value of ith
quartile applies the formula given by:
 
f
w
cf
n
i
L
Q o
i


 4
*
Where, n = the sum of the frequencies of all classes = i
f
Lo = the lower class boundary of the ith
quartile class.
cf = the cumulative frequencies of class before ith
quartile class
f = the frequency of ith
quartile class, and w is class width
Example 3.18: Consider example 3.13, find the value for which 75% of the age lies above and/or below
it from raw data and grouped dada, and second quartiles.
Solution:To find 1st
quartile class, compute 5
4
/
20
4
*
1 

n
Then Q1 lies in the class 27-30, then Lo = 26.5, cf = 3, f = 4 and w = 4. The unique value of Q1 is
computed by
 
f
w
cf
n
L
Q o


 4
1 =
  5
.
28
4
4
3
4
/
20
5
.
26 


The value of Q1 shows that only 25 % of 20 household headsare with age 28.5 or less.
Similarly, to find 2nd
quartile class compute 





4
*
2 n
=
  10
4
20
2  , The class for Q2 is 31--34, then
Lo= 30.5, cf = 7, f = 3 and w=4, the unique value of Q2 is computed by:
f
w
cf
n
L
Q o









4
2
2 X
~
5
.
34
4
3
7
20
4
2
5
.
30 










50 of 20 household heads arewith age 34.5 or less/or more.
ii. Deciles
Deciles are measures which divide a given ordered data in to ten equal parts and each part contains equal
no of elements. It has nine points known as 1st
, 2nd
,…, 9th
deciles, denoted by D1, D2,… D9respectively.

51
a) Deciles for individual data
For series of data set the ith
deciles is the value of the item which is at the position
n
i
th





 
10
1
* item
Di = position
n
i
th





 
10
1
* item where i=1,2,3,3…….,9.
b) Deciles for ungrouped frequency distribution
The ith
decile (Di) for the discrete frequency distribution data is the value corresponding to cumulative
frequency greater or equal to 


k
i
i
f
n
n
i
1
where,
10
* . Or the value whose cumulative frequency is greater
than or equal to 9
...,
3,
2,
1,
i
and
where,
10
*
1

 

k
i
i
f
n
n
i
c) Deciles for grouped frequency distribution
For grouped (continuous) data deciles can be obtained first by obtaining the ith
deciles classbysearch for
the minimum less than cumulative frequency greater than or equal to
10
*
n
i , then the class
corresponding to this cumulative frequency is ith
deciles class. Then, determine the unique decile by the
following formula:
 
f
w
cf
n
i
L
D o
i


 10 , for i=1, 2, 3…, 9.
f
deciles class
deciles class
deciles class, and w is class width
iii. Percentiles

52
Percentiles are measures having 99 points which divide a given ordered data in to 100 equal parts and
each part consists of equal number of elements. It is denoted by P1, P2... P99 and known as 1st
, 2nd
…99th
percentiles respectively.
a) Percentile for individual data
For a series of data the ith
percentiles is the value of item at the position
n
i
th





 
100
1
*
item
position
100
1
*
i.e.
th
i
n
P 




 
 where, i=1, 2, 3, 3…, 99.
b) Percentiles for ungrouped frequency distribution
The ith
percentile (Pi) for the discrete frequency distribution data is the value corresponding to
cumulative frequency greater or equal to 


k
i
i
f
n
n
i
1
where,
100
* . Or the value whose cumulative
frequency is greater than or equal to 99
...,
3,
2,
1,
i
and
where,
100
*
1

 

k
i
i
f
n
n
i
c) Percentile for grouped frequency distribution
First percentile class is obtained; to find ith
percentile class compute 





100
* n
i
and search less than
cumulative frequency less than or equal to this value, then the class corresponding to this cumulative
frequency is ith
percentile class. Then, for grouped (continuous) frequency distribution data percentile
can be obtained by:
 
f
w
cf
n
i
L
P o
i


 100 , for i=1, 2, 3…, 99.
f
percentile class
percentile class
percentile class, and w is class width
Example 3.19: Consider example 3.11 and compute D4, D8, P25, and P80

53
Chapter Four
Measures of Dispersion
4.1 Introduction
In this unit we shall discus the most commonly used measure of dispersion like Range, Quartile
Deviation, Mean Deviation, Standard Deviation, coefficient of variation. And measure shapes such as
skewness and kurtosis. We have seen that averages are representatives of a frequency distribution. They
do not tell anything about the scatterness of observations within the distribution. Suppose that we have
the distribution of the yields (kg per plot) of two paddy varieties from 5 plots each.
Variety 1: 45 42 42 41 40
Variety 2: 54 48 42 33 30
The mean yield for both varieties is 42kg. But we cannot say that the yield of the two varieties are the
same. The first variety may be preferred since it is more consistent in yield performance. From the
above example, it is obvious that a measure of central tendency alone is not sufficient to describe a
frequency distribution. In addition to it we should have a measure of scatterness of observations. The
scatterness or variation of observations from their average is called dispersion.

54
Some of the main objectives of dispersion are stated under:
1. To determine the reliability of an average: The measures of dispersion help in determining the
reliability of an average. It points out how far an average is representative of a statistical series.
If the dispersion or variation is small, the average will closely represent the individual values
and it is highly representative. On the other hand, if the dispersion or variation is large, the
average will be quite unreliable.
2. To compare the variability of two or more series: The measures of dispersion help in
comparing the variability of two or more series. It is also useful to determine the uniformity or
consistency of two or more series. A high degree of variation would mean less consistency or
less uniformity as compared to the data having less variation.
3. For facilitating the use of other statistical measures: Measures of dispersion serve the basis of
many other statistical measures such as correlation, regression, testing of hypothesis etc.
4. Basis of statistical quality control: The measure of dispersion is the basis of statistical quality
control. The extent of the dispersion gives indication to the management as to whether the
variation in the quality of the product is due to random factors or there is some defect in the
manufacturing process.
Desirable Properties of Measures of Dispersion
1. It should be based on all observations.
2. It should be easy to compute and to understand.
3. It should not be affected much by extreme values.
4. It should not be affected by sampling fluctuation
4.2 Absolute Measures of Dispersion
Absolute measure is expressed in the same statistical unit in which the original data are given such as
kilograms, tones etc. These measures are suitable for comparing the variability in two distributions
having variables expressed in the same units and of the same averaging size. These measures are not
suitable for comparing the variability in two distributions having variables expressed in different units.
Range: The simplest measure of dispersion is the range. The range is the difference between the two
extreme values (highest and lowest value) of data. Range takes only maximum and minimum values into
account and not all the values. Hence it is a very unstable or unreliable indicator of the amount of
deviation.

55
-The major area in which range is applied is statistical quality control.
-It is also applicable in the cases where extreme values are important like maximum rainfall,
temperature, etc Range =R = Xmax - Xmin
Example 4.1 Consider the following data on weight of 7 individuals.
24, 25, 30, 15, 47, 43 and 35. R = L-S = 47 - 15 = 32kg.
The relative measures of range, also called coefficient of range, is defined as
=
S
L
S
L


For a grouped data, range is the difference between the upper class boundary of the last class interval
and lower class boundary of the first class interval.
Find out range and coefficient of range of the following series
Size 5-10 11-15 16-20 21-25 26-30
Frequency 4 9 15 30 40
Here, = ℎ
= ℎ = 30 = 5
= 30 − 5 = 25
= 7143
.
0
5
30
5
30



Properties of range
 It’s easy to calculate and to understand
 It can be affected by extreme values
 It can’t be computed when the distribution has open ended classes.
 It cannot take the entire data in to account.
 It does not tell anything about the distribution of values in the series.
Inter Quartile Range: Is the difference between 3rd and 1st
quartile and it is a good indicator of the
absolute variability than range. IQR = 1
3 Q
Q  .

56
Quartile Deviation (semi – inter quartile Range) is a half of inter quartile range
QD =
   
2
1
2
2
3 Q
Q
Q
Q 


=
2
1
3 Q
Q 
Coefficient of quartile Deviation The relative measure of quartile deviation also called the coefficient
of quartile deviation is defined as: =
Properties of Quartile Deviations
i) The size of quartile deviation gives an indication about the uniformity. If QD is small, it
denotes large uniformity. Thus, a coefficient of quartile deviation is used for comparing
uniformity or variation in different distribution.
ii) Quartile deviation is not a measure of dispersion in the sense that it doesn’t show the
scatter around an average but only a distance on scale. As result it is regarded as a measure
of partition.
iii) It can be computed when the distribution has an open-ended class. it is quite suitable in the
case of open – ended distribution
iv) As compared to range, it is considered a superior measure of dispersion.
v) Since it not influenced by the extreme values in a distribution. It is particularly suitable in
highly skewed or irregular distribution.
.
Examples 4.2 For the following frequency distribution find
a) Inter– quartile range.
b) Quartile deviation
c) CQD
Class limit Frequency
21 – 22 10
23 – 24 22
25 – 26 20

57
27 – 28 14
29 – 30 14___
Total 80 n/4 = 80/4 = 20, 20th ordered
observation
The 1st
quartile class is 23 -24
    4
.
23
22
2
10
20
5
.
22
4
1 






f
w
cf
n
b
LC
Q
on
obeservati
is
Q
n
Q th
40
,
40
4
80
2
4
2 2
2 














The class interval containing 2
Q is 25 – 26.
 
 
f
w
cf
n
b
C
L
Q Q


 4
2
2
2 =
 
20
2
3
40
5
.
24
x

 = 3
.
25
And ,
60
4
3
3 







n
Q 3
Q is 60th
position observation.
The class limits containing 3
Q is 27 – 28
 
    84
.
27
14
52
60
5
.
26
4
3
3
3 






f
w
cf
n
b
C
L
Q Q
a) Inter quartile range = 1
3 Q
Q  = 27.64 - 23.44 = 4.23
b)   115
.
2
2
/
23
.
4
2
1
. 1
3 


 Q
Q
D
Q
c) CQD =4.23/51.24
The quartile deviation is more stable than the range as it defenses on two intermediate values. This is
not affected by extreme values since the extreme values are already removed. However, quartile
deviation also fails to take the values of all deviations.

58
Mean Deviation: Mean deviation is the mean of the deviations of individual values from their average.
The average may be either mean or median.
M.D=
n
A
X
 
for raw data. M.D =

 
f
A
X
f
for grouped data where A is either mean or
median.
Example 4.3 Consider the following data and compute mean deviation from mean 53, 56, 57, 59, 63,
and 66 59
6
6
1

 

i
Xi
X
Xi 53 56 57 59 63 66
deviation from mean 6 3 2 0 4 7
Mean deviation =
n
X
Xi
 
= 67
.
3
6
22

 The data deviates on average 3.67 from the arithmetic mean
Example 4.4 Calculate the mean deviation for the following data using both mean & median.
Xi :- 14 , 15 , 26 , 20 , 10 , Median 15 , mean = 17
Mean deviation = 5
/
22
15
26
15
20
15
15
15
14
15
10 









Xi 10 14 15 20 26 Total
/di / = /xi - / 5 1 0 5 11 22
/di / = /xi – Mean / 7 3 2 3 9 24

59
M. D from median = 4
.
4
5
22




n
median
xi
Mean Deviation from median is 4.4. M.
D from mean 8
.
4
5
24





n
mean
xi
Example 4.5 Calculate the mean deviation from mean and median
Xi 6 7 8 9 10 11 12
fi 3 6 9 13 8 5 4
Xi fi 18 42 72 117 80 55 48
Mean= 9
48
/
432 



i
i
i
f
x
f
Median= 9
2
9
9
2
25
24
2
1
2
2



















th
th
th
th
n
n
Xi 6 7 8 9 10 11 12 Total
fi 3 6 9 13 8 5 4 48
| | 3 2 1 0 1 2 3
| | 9 12 9 0 8 10 12 60
Where di = (Xi – median (or mean)) M. D from median = 25
.
1
48
60




i
i
i
f
d
f
Property of Mean Deviation
- The mean deviation takes all values into consideration.

60
- It is fairly stable compared to range or quartile deviation. But it is not stable as standard
deviation. Since, it mathematically ignores signs of deviations.
- It is not possible to use for further statistical investigation.
Variance (S2
or 2
)
Variance is the arithmetic mean of square deviation about the mean. When our data constitute a sample
, the variance averaging done by dividing the sum of squared deviation from mean by n-1 and it is
denoted by s2
.when our data constitute an entire population variance averaging done by dividing by N
and denoted by 2
. It is commonly used absolute measure of dispersion
- s2
=
 




n
i
i
n
x
x
1
2
1
an unbiased estimator for population variance
- 2
=   iance
puplation
X
N
N
var
;
1
1
2
  
.
The computing formula for variance is can be
simplified as given S2
=
 
1
2
1
2








 

n
n
xi
xi
n
i
,
Since    

 



2
2
2 x
x
x
x
x
xi i
i =
2
2
.
2 

 
 x
x
x
X i
i
=
2
2
.
2 x
n
x
n
xi 

 =
2
2
2
2 x
n
x
n
xi 

 =
2
2
x
n
xi 

=
2
2










 n
x
n
x
i
i =
 2
2
n
x
x
i
i

 
 S2
=
 
1
2
2


 
n
n
x
x i
i
 Variance for simple frequency distribution
xi x1 x2 . .
.
xk

61
S2
=
 
  





i
i
n
i
i
f
n
where
n
x
x
f
1
2
1
 Determination of variance from grouped frequency distributions.
 
)
1
(
1
2
2





n
x
m
f
s
n
i
i
i
Where mi is mid value of class
Activity Compute variance for the following frequency distribution
Class interval 1-5 6-10 11-15 16-20
frequency 4 1 2 3
Properties of Variance
1. The variance is always non – negative (S2
 0 )
2. If every element in the distributions are multiplied by a constant C the new variance is
2
2
2
old
new S
C
S 
Old  
 

 1
/
.,
.
.
.
,
,
2
2
2
1 n
x
x
S
x
x
x i
old
n
New
 
1
,
.
.
.
.
,
,
2
2
2
1




n
x
c
cx
S
cx
cx
cx i
new
n
=
 
  
1
2



n
x
x
c i
=
 
1
2
2



n
x
x
c i
=
 
1
2
2



n
x
x
c i
= 2
2
old
S
C
3. When a constant c is added to all measurement of the distribution, the variance doesn’t change
xi (old) = xi , x2 , . . . . , xn
fi f1 f2 . .
.
fk

62
xi (new) = x1 + c1 , x2 + c , . . . . , xn + c
 
n
c
x
n
c
x
X
i
i
new
 
 


 =
n
nc
n
xi

 = c
X 
 
 
 
 
old
i
i
new S
n
x
x
n
c
x
c
x
S 2
2
2
1
1











4. The variance of constant measured n times is zero. .
0
, 2

 S
c
x
Example 4.6 If the mean & variance of x are 10 & 5 , respectively. Find the mean and variance of y,
where y = 10x - 5
 
 
n
x
n
x
n
y
y
n
i
i



 




 5
10
5
10
1
= 5
10 
x = 10 (10) – 5 = 100 – 5 = 95
var (y) =
 
1
2



n
y
yi
=
 
1
95
5
10
2




n
x
=
 
1
10
10
2
2



n
x
=
 
 

1
10
10
2
2
n
xi
= 100 (5) = 500
Standard Deviation (S.D)
The standard deviation is defined as the square root of the mean of the squared deviations of individual
values from their mean.
S.D =
 
1
2



n
X
X
- Its advantage over variance is that it is in the same unit as the variable under consideration.
- It is a measure of average variation in the set of data.

63
Example 4.7 Compute the variance & S.D. for the data given below.
xi 32 36 40 44 48 Total
frequency 2 5 8 4 1 20
Xi : 32 36 40 44 48 Total
Fi: 2 5 8 4 1 20
Xi fi: 64 180 320 176 48 788
Xi
2
fi: 2048 6480 12,800 7,744 2,304 31,376
 

 



1
2
2
2
i
i
i
i
i
f
n
f
x
x
f
S =
 
19
20
2
788
31376 
= 328.8/19 = 17.31
2
S
S 
 = 31
.
17 = 4.16
- If the s .d of set of data is small then the values are scattered widely about the mean.
316
.
3
11
,
11 2
2



 S
S
S
Example 4.8 Calculate the S.D for the following grouped frequency distribution.
Class intervals Frequency(fi)
1 – 3 1
3 – 5 9
5 – 7 25
7 – 9 35
9 – 11 17
11 – 13 10
13 – 15 3
Totale 100

64
Class intervals Frequency (fi )mi mi
2
fi mi fi
1 – 3 1 2 4 2
3 – 5 9 4 144 36
5 – 7 25 6 900 150
7 – 9 35 8 2240 280
9 – 11 17 10 1700 170
11 – 13 10 12 1440 120
13 – 15 3 14 588 42
Total 100 7016 800
 

 



1
2
2
2
i
i
i
i
i
f
n
m
f
m
f
S =
 
99
100
2
800
7016 
=6.2 49
.
2
22
.
6
2


 S
S
4. 3 Relative Measure of Dispersion
Suppose that the two distributions to be compared are expressed in the same units and their means are
equal or nearly equal. Then their variability can be compared directly by using their standard deviations.
However, if their means are widely different or if they expressed indifferent units of measurement, we
cannot use the standard deviation as such for comparing their variability. We have to use the relative
measures of dispersion in such situation.
Coefficient of variation (CV): The CV is a unit free measure. It is always expressed as percentage.
CV =
Mean
SD
100%
The CV will be small if the variation is small. Of the two groups, the one with less CV is said to be more
consistent. The coefficient of variation is unreliable if the mean is near zero. Also it is unstable if the
measurement scale used is not ratio scale. The CV is informative if it is given along with the mean and
standard deviation. Otherwise, it may be misleading.

65
Example 4.9 Consider the distribution of the yields (per plot) of two paddy varieties. For the first
variety, the mean and standard deviation are 60kg & 10kg, respectively. For the second variety, the
mean and standard deviation are 50kg & 9kg, respectively. Then we have,
CV = (10/60)100%=16.7%, for first variety.
CV = (9/50)100%=18.0%, for second variety.
It is apparent that the variability in first variety is less as compared to that in the second variety.
Coefficient of Mean Deviation: The coefficient of mean deviation is founded by dividing the mean
deviation by the measure of central tendency about which the deviation is computed.
- It is a relative measure of dispersion, coefficient of mean deviation can be computed as
CMD =
Median
deviation
Mean
CMD
or
Mean
ion
Meandeviat

Example 4.10 Coefficient of the mean deviation from mean & median for above example 4.4 is
Mean Deviation from median is 4.4, M. D from mean = 8
.
4
5
24

C. M. D from median =
median
median
from
deviatin
mean
=
15
4
.
4
= 0.293
C. M. D from mean =
mean
mean
from
deviation
mean
=
17
8
.
4
= 0.283
The standard Score: The standard score is denoted by Z and defined as
 
S
x
x
Z i 
 Where
S – Standard deviation of the distribution, Xi each observation value. This measures the deviation of
individual observation from the mean of the total observation in the unit of standard deviation and
termed as Z – Score. The Z – scores of individuals in different groups are then added to give a true
Measure of relative performance.

66
Example 4.11 Compare the performance of the following two students.
Candidate Marks in economics Marks in Acct. Total
A 84 75 159
B 74 85 159
Average mark for Accounting is 50 with standard deviation of 11 and average marks for Economics is
60 with standard deviation 13.Whose performance is better A or B?
Z score for A
⎩
⎪
⎨
⎪
⎧ 846
.
1
13
60
84


Economics
273
.
2
11
50
75


Accounting
Total Z score for A = 1.846 + 2.27 = 4.119
Z score for B
⎩
⎪
⎨
⎪
⎧ 077
.
1
13
60
74


Economics
182
.
3
11
50
85


g
Accounitin
Total Z – Score for B = 1.077 + 3.182 = 4.25
Since B’s Z – score is higher; student B had good performance than student A.
4.4 Measure of Shapes
We have seen that averages and measure of dispersion can help in describing the frequency distribution.
However, they are not sufficient to describe the nature of the distribution. For this purpose, we use the
other concepts known as Skewness and Kurtosis.
Skewness: Skewness means lack of symmetry. When the values are uniformly distributed around the
mean a distribution is said to be symmetrical. For example, the following distribution is symmetrical
about its mean 3.

67
Xi : 1 2 3 4 5
fi : 5 9 12 9 5
In a symmetrical distribution the mean, median and mode coincide, that is, X = X
~
= X̂ .
X = X
X ˆ
~
 Symmetrical distribution
When a distribution is skewed to the right; mean > median > mode. If we take income distribution for
different number of families; Income distribution is skewed to the right mean that a large number of
families have relatively have low income and a small number of families have extremely high income.
In such a case, the mean is pulled up by the extreme high incomes and the relation among these three
measures is as shown in figure. Here, we find that mean > median > mode. When a distribution is
skewed to the left, then mode > median > mean. This is because here mean is pulled down below the
median by extremely low values.
Right(+) skewed Distribution
Left(-)skewedDistribution

68
Karl person’s Measure of skewness: In case
the distribution is symmetric we will have
Arithmetic mean. = Median = Mode; unless they
will not be equal if the distribution is skewed.
Therefore the distance between the A.M. and the
Mode (A.M – Mode) can also be used as a
measure of skewness. However since the
measure of skewness should be a pure number
we define as

Mode
M
A
Sk


.
,
Where
is the standard deviation of the distribution.
For distribution which are bell shaped and are moderately skewed, we have an approximate relationship
between the A.M, Median and mode.
A. M – Mode = 3 (A. M – Median)
Accordingly we may define skewness as follows
 

Median
M
A
Sk


.
3
For a symmetrical distribution Sk= 0. If the distribution negatively skewed, then the value of Sk is
negative, and if it is positively skewed then Sk is positive. The range for values of Sk is from -3 to 3. The
other measure uses the β (beta) coefficient which is given by, β1 = µ3
2
/µ2
3
. Where µ2 & µ3 are the
second and the third central moments.
The second central moment is nothing but the variance. The sample estimate of this coefficient is b1 =
m3
2
/m2
3
where m2& m3 are sample central moments given by,
m2 =
 
1
2



n
X
X
or
 
1
2



n
X
X
f
, m3 =
 
1
3



n
X
X
or
 
1
3



n
X
X
f
For a symmetrical distribution b1 is zero. And also Skewness is positive or negative depending upon
whether m3 is positive or negative.


69
Example 4.12 The first four moments about mean of the distribution are 0, 2.5, 0.7, and 18.75. Test the
Skewness of distribution
Kurtosis: A measure of the peakedness or
convexity of a curve is known as Kurtosis.
All the three curves are symmetrical about
the mean. Still they are not of the same
type. One has different peak as compared to
that of others. Curve (1) is known as meso-
kurtic (normal curve); curve (2) is known as
leptokurtic (leaping curve) and curve (3) is
known as platy-kurtic (flat curve). Kurtosis
is measured by Pearson’s coefficient, β2. It
is given by β2 = µ4/µ2
2
4
4



The sample estimate of this coefficient is b2 = m4/m2
2
, where m4 is the 4th
central moment given by
m4 =
 
1
4



n
X
X
. The distribution is called meso-kurtic if the value of b2 = 3. When b2 is more than
3 the distribution is said to be leptokurtic. And also, if b2 is less than 3 the distribution is said to be
platykurtic.
Example 4.13 The measure of skewness and kurtosis are given below for data in table.
Value(xi) 3 4 5 6 7 8 9 10
Frequency(f) 4 6 10 26 24 15 10 5

70
Value(xi) Frequency(f) d=X- X f*d2
f*d3
3 4 -3.7 54.76 -202.612
4 6 -2.7 43.74 -118.098
5 10 -1.7 28.90 -49.130
6 26 -0.7 12.74 -8.918
7 24 0.3 2.16 0.648
8 15 1.3 25.35 32.955
9 10 2.3 52.90 121.670
10 5 3.3 54.45 179.685
m2=s2
=
 
1
2



n
X
X
fi
=
99
275
=2.7777
m4 =
 





99
13
.
2074
1
4
n
X
X
fi
20.9508, m3=
 
1
3



n
X
X
fi
=
99
8
.
43

= -0.4424
b1 =
 
 
0091
.
0
7777
.
2
4424
.
0
3
2
3
2
2
3



m
m
,
b2 =
 
7153
.
2
7777
.
2
9508
.
20
2
2
2
4


m
m
It is negatively skewed since m3 is negative. The value of b2 is 2.7153 which is less than 3. Hence the
distribution is platykurtic.

71
CHAPTER FIVE
PROBABILITY AND PROBABILITY DISTRIBUTIONS
5.1. Basic Concepts of Elementary Probability
Introduction
An experiment is any activity that generates outcome(s). If an experiment is conducted under homogeneous
and similar conditions, we come across two types of situations or phenomena. The results or the
outcome is unique or certain which are called as deterministic phenomena by which the result can be
predicted certainly from an experiment performed. For instance, in case of ohm’s law voltage, V =
I*R, Force, F = m *a , the distance (S) covered by a particle after a time t is given by S= 2
2
1
at
Vt 
,…, etc. are deterministic phenomena because if we know the right hand side, we certainly determine
the left hand side.
The results or the outcomes of an experiment are not unique, but can be one of the several possible
outcomes; this type of model is called probabilistic or non-deterministic (random) phenomena.
Example -In random tossing of a coin, we are not sure of getting head or tail.
- A sex of new born baby may be male or female
- A manufacturer cannot be ascertained (sure) of the future demand of his product with certainty.
As everybody knows our world is a full of uncertainty ,even ,no one knows exactly what comes after a minute,
an hour…etc. but we can guess the chance that something will happen. The word probability or chance is very
commonly used in day-to-day conversation, and generally, people have some idea what it means. Terms like
possible, probable, or likely and so on, have all similar meanings.
Probability: can be defined as a measure of the likelihood that a particular event will occur or it is a
science of decision making with calculated risk in face of uncertainty. It is a numerical measure with a
value between 0 and 1 of such likelihood. Where the probability of zero indicates that the given event
cannot occur and the Probability of one assures certainty of such an occurrence.

72
Definitions of some Probability Terms
1. Random Experiment: is statistical process of measurement or observation that can be repeated in
any single trial of which the outcome is unpredictable. For example, tossing of a fair coin, rolling a
die etc is considered as a statistical experiment.
2. Outcome: is the result of a random experiment.
Example
Experiment Outcomes
Tossing of a fair coin Head, tail
Rolling a die 1, 2, 3, 4, 5, 6
Selecting an item from a production lot defective (faulty), non-defective (good)
Introducing a new product Success, failure
3. Sample space: A sample space is the collection of all possible outcomes of an experiment. For
example, there are two possible outcomes of a toss of a fair coin, which are a head and a tail. Then
the sample space, for this experiment denoted by S would be: S = {H, T}. Each possible outcome in
the sample space is called sample point.
4. Event is a subset of the sample space or it is asset containing sample points of a certain sample
space under consideration. For examples, getting two heads in the trial of tossing three fair coins
simultaneously would be an event.
5. Elementary event (simple event) is a single possible outcome of an experiment.
6. Composite (compound) event is an event having two or more elementary events in it. For example,
rolling a die sample space = {1,2,3,4,5,6) an event having { 5} is simple event where as having
even number={2,4,6} is compound (composite)event.
7. Mutually exclusive events: Two events are said to be mutually exclusive, if both events cannot
occur at the same time as outcome of a single experiment. In other word two events E1 and E 2 said
to be mutually exclusive events if there is no sample point in common to both events E1 and E 2 .
For example, if we roll a fair dice, then the experiment is rolling the die and sample space (S) is
S = 
 6
,
5
,
4
,
3
,
2
,
1 If we are interested the outcome of event E1 getting even numbers and E 2 odd
numbersE1 = {2, 4, 6} E 2 = {1, 3, 5}. Clearly E1  E 2 = Ø. Thus E1 and E 2 are mutually
exclusive events.

73
8. Independent Events: Two events A and B are said to be independent events if the occurrence of
event A has no influence (bearing) on the occurrence of event B. For example, if two fair coins are
tossed, then the result of one toss is totally independent of the result of the other toss. The
probability that a head will be the outcome of any one toss will always be ½, irrespective of
whatever the outcome is of the other toss. Hence, these two events are independent. On the other
hand, consider drawing two cards from a pack of 52 playing cards. The probability that the second
card will be an ace would depend up on whether the first card was an ace or not. Hence these two
events are not independent events.
Another example a bag contains balls of two different colours say yellow and white. Two balls are
drawn successively .First ball is drawn from a bag and replaced after notes its colour. Let us assume that
it is yellow and denote this event by A. Another ball is drawn from the same bag and its colour is noted
let this event denoted by B. Clearly, the result of first draw has no effect on the result of the second
draw. Hence, the events A and B are independent events.
9. Equally likely outcomes: In a certain experiment, if each outcome in the sample space has the same
chance to occur, then we say that the outcomes are equally likely outcomes.
Fundamental Principles of Counting Techniques
If the number of possible outcomes in an experiment is small, it is relatively easy to list and count all
possible events. When there are large numbers of possible outcomes an enumeration of cases is often
difficult, tedious, or both. Therefore, to overcome such problems one can use various counting
techniques or rules.
i. Addition rule: Suppose that a procedure designated by 1, can be performed in n1 ways. Assume that
second procedure designated by 2 can be performed in n 2 ways. Suppose furthermore that it is not
possible both procedures 1 and 2 are performed together. The number of ways in which we can
perform 1 or 2 procedures is n1 + n 2 ways. This can be generalized as follows if there are k
procedures and ith
procedure may be performed in ni ways, i=1, 2, …, k , then the number of ways in
which we perform procedure 1 or 2 or … or k is given by n1 +n 2 +…+ n k =

k
i
i
n
1
, assuming that no
two procedures performed together.

74
Example 5.1: Suppose that we are planning a trip and deciding between bus and train transportation. If
there are 3 bus routes and 2 train routes to go from A to B, find the available routes for the trip.There are
3+2 = 5 possible routes for someone to go from A to B.
ii. Multiplication Rule: Suppose that procedure 1 can be performed in n1 ways. Let us assume
procedure 2 can be performed in n 2 ways. Suppose also that each way of doing procedure 2 may be
followed by any way of doing procedure 1, then the procedure consisting of n1 followed by n2 may
be performed by n1 * n2 ways. We also generalize to k procedures.
Example 5.2: An airline has 6 flights from A to B, and 7 flights from B to C per day. If the flights are to
be made on separate days, in how many different ways can the airline offer from A to C?In operation 1
there are 6 flights from A to B, 7 flights are available to make flight from B to C. Altogether there are
6*7 = 42 possible flights from A to C.
Example 5.3: Suppose that in a medical study patients are classified according to their blood type as
A, B , AB, and O; according to their RH factors as (+) or (-) and according to their blood pressure as
high, normal or low, then in how many different ways can a patient be classified ?
The1st
classification has done in 4 ways; the 2nd
in 2 ways, and the 3rd
in 3 ways. Thus patient can be
classified in 4*2*3 = 24 different ways.
iii. Permutations: Permutation is an arrangement of all or parts of a set of objects with regard to order.
Rule 1: The number of permutations of n distinct objects taken all together is n! Or
In particular, the number of permutations of n objects taken n at a time is
nPn =
  !
0
!
!
! n
n
n
n


= n!. In definition 0! = 1! = 1.
Example 5.4:In how many ways 4 people are lined up to get on a bus (or to sit for photo graph)?
In 4! = 4*3*2*1 = 24 Ways.

75
Rule-2: A permutation of n different objects taken r at a time is an arrangement of r out of the n objects,
with attention given to the order of arrangement. The number of permutations of n objects taken r at a
time is denoted by nPr, or P (n,r) and is given by
nPr = n(n-1)(n-2)…(n-r+1) =
 !
!
r
n
n

Example 5.5: The number of permutations of letters a, b & c taken two at a time is
3
P2 = 6. These are ab, ba, ac, ca, bc & cb.
Rule-3: The number of permutation of n objects taken all at a time, when n1 objects are alike of one
kind, n2 objects are alike of second kind, …, nk objects are alike of kth
kind is given by:
 










 k
i
i
k
i
i
k
n
n
n
n
n
n
n
1
1
3
2
1
!
!
!
!...
!
!
!
Example 5.6 The total number of arrangement of the letters of the word STATISTICS taken all at a time
is given by
!
1
!
2
!
1
!
3
!
3
!
10
= 50,400 since there are 3s’s, 3t’s, 1a, 2i’s and 1c.
Note:-The number of arrangements of n distinct objects around circular object (table) is (n-1)! and when
the method of selection or arrangement of r objects from n objects with repetition the
possible numbers of arrangements are nr
.
Example5.7: Arba Minch University Registrar Office wants to give identity number for students by
using 4 digits. The number should be considered by the following numbers only: {0, 1, 2, 3, 4, 5, and 6}.
Hence, how many different ID Numbers could be preferred by the Registrar
a. Without repeating the number?
b. With repetition of numbers?

76
We have 7 possible numbers for 4 digits. But the required number of digits for ID number is 4.Hence n
= 7 & r = 4.The possible number of ID numbers given for students without repeating the digit is nPr, =
 !
!
r
n
n
  !
4
7
!
7

 = 7*6*5*4 = 840.
 The possible number of ID numbers given for students with repeating the digit is
nr
= 74
= 7*7*7*7 = 2401
iv. Combinations: Combination is the selection of objects without regarding order of arrangement.A
combination of n different objects taken r at a time is a selection of r out of n objects, with no
attention given to the order of arrangement. The number of combinations of n objects taken r at a
time is denoted by the symbol or nCr is given by
=
    
  !
!
!
!
!
1
...
2
1
r
P
r
n
r
n
r
r
n
n
n
n r
n







Example 5.8: The number of combinations of letters a, b& c taken two at a time is 3C2=
!
1
!
2
!
3
= 3.
These are ab, ac and bc. Note that ab is the same combination as ba, but not the same in permutation.
Example 5.9: Suppose in the box 3 red, 3 white and 5 black equal sized balls are there. We want to
draw 3 balls at a time. How many ways do we have from each type?
→ Solution 























1
5
1
3
1
3
= 3(3)5 = 45 ways.
Different Approaches to Probability
There are four different conceptual approaches to study of probability theory
i. Classical or Mathematical Approach
If a random experiment results in N exhaustive, mutually exclusive and equally likely outcomes; out of
which n are favorable to the happening of an event A, then the probability of occurrence of A, usually
denoted by P (A) is given by: N
n
cases
of
No
exhaustive
A
to
cases
favorable
A
P /
.
)
( 


77
Example 5.10: In a given basket there is 3 yellow, 4 black and 3 white balls. What is the probability of
selection of one black ball?
Let event A drawing of black ball, P (A) =
cases
of
No
exhaustive
A
to
cases
favorable
.
=
10
4
= 0.4
ii. Empirical or Frequency Approach
The classic definition of probability has a disadvantage in that of the words equally likely is vague. In
fact, since these words seem to be synonymous with “equally probable”, the definition is circular
because we are essentially defining probability in terms of itself. For this reason, a statistical definition
of probability has been advocated by some people. According to this the estimated probability, or
empirical probability, of an event is taken to be the relative frequency of occurrence of the event when
the number of observations is very large. The probability itself is the limit of the relative frequency as
the number of observations increases indefinitely.
Example 5.11: If 1000 tosses of a coin result in 529 heads, the relative frequency of heads is 529/1000
= 0.529. If another 1000 tosses results in 493 heads, the relative frequency in the total of 2000 tosses is
2000
493
529 
=0.511.
According to the statistical definition, by counting in this manner we should ultimately get closer and
closer to a number that represents the probability of a head in a single toss of the coin. From the results
so far presented, this should be 0.5 to one significant figure.
iii. Axiomatic Approach
Both the classical and frequency approaches have serious drawbacks, the first because the words equally
likely are vague and the second because the large number involved is vague. Because of these
difficulties, statisticians have been led to an axiomatic approach of probability.
Given a sample space of a random experiment S, the probability of the occurrence of any event A is
defined as a set function P (A) satisfying the following axioms:
1: P (A) is defined, is real and non-negative i.e. 0 ≤ P (A)
2: P(S) = 1 where S is the sample space (sure event)

78
3: If A1, A2 … An is any finite or infinite sequence of disjoint events of S, then










k
i
i
i
k
i
A
P
A
U
P
1
1
)
(
Results of Axioms
1. P 0
)
( 
 , for impossible event (null set)
2. P(Ac
) = 1-P(A)
3. If A and B are two independent events then the chance of occurrence of both two events is
P (A B) = P (A) P (B)
iv. Subjective Approach
A probability derived from an individual's personal judgment about whether a specific outcome is likely
to occur. Subjective probabilities contain no formal calculations and only reflect the subject's opinions
and past experience. Subjective probabilities differ from person to person. Because the probability is
subjective, it contains a high degree of personal bias. An example of subjective probability could be
asking Arsenal fan, before the football season starts, the chances of Arsenal winning the world
champions. While there is no absolute mathematical proof behind the answer to the example, fans might
still reply in actual percentage terms, such as the Arsenal having the 95% chance of winning the world
champions.
Events as a Set
If A and B are two events then
 A B: the happening of at least event A or B.
 A B: the simultaneously happening of both events A and B.
 Ac
: A does not happen (complement of event A).
 Ac
 Bc
: neither A nor B happens
 Ac
 B: B occurs alone or exactly B occurs or only B occurs.
 (A Bc
)  (Ac
 B): exactly one of the two events A and B happens

79
The probability of occurrence of at least one of the two events A and B is given by:
P (A )
(
)
(
)
(
) B
A
P
B
P
A
P
B 
 


If A and B are mutually exclusive events, then
P (A )
(
)
(
) B
P
A
P
B 


Conditional Probability and Independence
Let there be two events A and B. Then the probability of event A given that the outcome of event B is
given by: P [A|B] =
]
[
]
[
B
P
AnB
P
Where: P [A |B] is interpreted as the probability of event A on the
condition that event B has occurred. In this case P [A n B] is the joint probability of event A and B, and
P [B] is not equal to zero.
And 0
P(A)
where
,
)
(
)
(
)
/
( 


A
P
A
B
P
A
B
P
Example 5.12:120 employees of a certain factory are given a performance test and are divided in to two
groups as those with good performance (G) and those with poor performance (P), then the result is
given below
Good performance (G) Poor performance(P) Total
Male (M) 60 25 80
Female (F) 25 15 40
Total 85 35 120
The probability of a person to be male given that it has a good performance is
P (M|G) =
 
)
(G
P
G
M
P 
=
120
/
85
120
/
60
=
17
12
The probability of a person to be female given that it has a poor performance is
P (F|P) =
 
)
(P
P
P
M
P 
=
120
/
35
120
/
15
=
7
3

80
Example 5.13: A jar contains black and white marbles. Two marbles are chosen without replacement.
The probability of selecting a black marble and a white marble is 0.34, and the probability of selecting a
black marble on the first draw is 0.47. What is the probability of selecting white marble on the second
draw, given that the first marble drawn is black?
  72
.
0
47
.
0
34
.
0
)
(
)
(
| 


Black
P
White
and
Black
P
Black
White
P
Example 5.14:The probability that it is Friday and that a student is absent is 0.03. Since there are 5
schooldays in a week, the probability that it is Friday is 0.2. What is the probability that a student is
absent given that today is Friday?
  15
.
0
2
.
0
03
.
0
)
(
)
(
| 


Firday
P
Absent
and
Friday
P
Friday
Absent
P
It often happens that the knowledge that a certain event A has occurred has no effect on the probability
that some other event B has occurred, that is, )
(
)
|
( A
P
B
A
P  . One would expect that in this case, the
equation )
(
)
|
( B
P
A
B
P  would also be true. If these equations are true, we may say B is independent
of A. In another words two events A and B are said to be independent if )
(
)
|
( A
P
B
A
P 
)
(
)
|
( B
P
A
B
P  and P (A B) = P (A) P (B)
5.2. Random Variable and Probability Distributions
Random Variables
A variable whose value is determined by random experiment is called random variable. Mathematically
a random variable is defined as a function from the sample space S to the line of real numbers R. In
other words random variable is a function that assigns a real number to each outcome in the sample
space of a random experiment. i.e. X: SR
A random variable takes a possible outcome and assigns a number to it.
Example: Flip a coin three times, let X be the number of heads in three tosses.
S= {(HHH),(HHT), (HTH), (HTT), (THH), (THT), (TTH), (TTT)}
X (HHH) =3,
X (HHT) =X (HTH) =X (THH) =2

81
X (HTT) =X (THT) =X (TTH) =1, X (TTT) =0
X= {0, 1, 2, 3}. X assumes a specific number of values with some probabilities.
Random variables are of two types:
1. Discrete Random Variables: are variables which can assume only a specific number of values.
They have values that can be counted. Examples:
 Tossing a coin n times and count the number of heads. 

 Number of children in a family. 

 Number of car accidents per week. 
 Number of defective items in a given company. 
 Number of bacteria per two cubic centimeter of water.
2. Continuous Random Variable: are variables that can assume all values between any two
given values. Examples:
 Height of students at certain college. 

 Mark of a student. 

 Life time of light bulbs. 
 Length of time required to complete a given training. 
Probability Distributions
Probability distribution of a random variable is the formula, table or graph that gives the possible value
of a random variable X with their associated probability P(X). A probability distribution consists of
possible values of the random and the corresponding probabilities of the values. It can be classified as
discrete or continuous depending on the nature of the random variable. Discrete probability distributions
are commonly called probability mass function (pmf) and denoted by P(X) = P(X=x). Similarly,
continuous probability distributions are known as probability density function (pdf) denoted by




 dx
x
f
X
f )
(
)
(
Example: Consider the experiment of tossing a coin three times. Let X be the number of heads.
Construct the probability distribution of X.
Solution:First identify the possible value that X can assume then calculate the probability of each
possible distinct value of X and express X in the form of frequency distribution.
X=x 0 1 2 3
P(X=x) 1/8 3/8 3/8 1/8

82
Properties of Probability Distribution
The probability distribution of a random variable should satisfy the following conditions.
1. ( ) ≥ 0,
( ) ≥ 0,
2. ∑ P(X = x) = 1 ,
∫ f (x)dx 1 , if X is continuous.
Note:
1. If X is a continuous random variable then
( < < ) = ( )
2. Probability of a fixed value of a continuous random variable is zero.
⇒P (a  X  b) P (a ≤ X  b) P (a  X ≤ b) P (a ≤ X ≤ b)
3. Probability means area for continuous random variable.
The probability distribution can be represented graphically by plotting P(X) against X, just as relative
frequency distributions. By cumulating probabilities, we obtain cumulative probability distributions,
which are analogous to cumulative relative frequency distributions. The cumulative distribution is:






 








x
x
X
dt
t
f
x
X
P
x
X
P
X
F
continous
is
X
if
)
(
discrete
is
X
if
)
(
)
(
)
(
Expectation and Variance of a Random Variable
Expected Value
1. Let a discrete random variable X assume the values X1, X2, ….,Xn with the probabilities P(X1),
P(X2), ….,P(Xn) respectively. Then the expected value of X , denoted as E(X) is defined as:
E( X ) X1P( X1) X 2 P( X 2).... X n P( X n) = ∑ ( = )
2. Let X be a continuous random variable assuming the values in the interval (a, b)
such that ∫ ( ) = 1, ℎ
( ) = ( )

83
Variance of a Random Variable
Let X be given random variable with expected value (mean) of E(x), then the variance of X is given by:



















continous
is
X
if
)
(
)
)
(
(
discrete
is
X
if
)
(
))
(
(
))
(
(
)
(
2
2
dx
x
f
X
E
x
x
X
P
X
E
x
X
E
x
E
X
Var
x
In the form of expectation variance of X is Var(X ) E(X 2
)−[E(X )]2
or
Var(X) = E(X(X-1)) + E(X) – (E(X))2
Where:
( ) = ( = ) ,
= ( = ),
Example 1: Compute the mean and variance of the random variable X, which denotes the number
showing up when a single die is rolled.
Solution: First we have to find the frequency distribution,
X=xi 1 2 3 4 5 6
P(X=xi) 1/6 1/6 1/6 1/6 1/6 1/6
( ) = 1
1
6
+ 2
1
6
+ 3
1
6
+ 4
1
6
+ 5
1
6
+ 6
1
6
= 3.5
( ) = (1 − 3.5)
1
6
+ (2 − 3.5)
1
6
+ (3 − 3.5)
1
6
+ (4 − 3.5)
1
6
+ (5 − 3.5)
1
6
+ (6 − 3.5)
1
6
= 2.9167
Example 2: Compute the mean and variance of the following probability distribution.
( ) =
[0,4],
0 ℎ .
( ) = ( )


=
1
4
=
1
4
1
2
4
0
=
1
4
1
2
4 −
1
2
0 = 2

84
( ) = ( − ) ( )


= ( − 2)
1
4
=
1
4
( − 4 + 4) =
1
4
1
3
−
4
2
+ 4
4
0
=
1
4
1
3
4 −
4
2
4 + 4 ∙ 4 − 0 =
4
3
5.3. Common Probability Distribution
Simple probabilities can be computed from elementary consideration. However, in dealing with
probabilities of whole classes of events, we have to consider more efficient ways of analysis of
probability. For this purpose we should know the concept of a probability distribution. In this section we
shall examine in detail some probability distribution s that have proved themselves empirically to be
useful probability models for any type of practical problems. The probability functions are derived
mathematically based on certain assumptions that are assumed to be hold for random phenomenon. We
have many probability distributions; discrete probability distributions are Bernoulli distribution,
binomial distribution, Poisson distribution, multinomial distribution, geometric distribution, hyper
geometric distribution, etc. Continuous probability distributions are uniform distribution, normal
distribution, expontial distribution, t-distribution, chi-square distribution, F-distribution, gamma
distribution, Weibul distribution, etc.
In practice we have use some of both discrete and continuous probability models.
Common Discrete Probability Distributions
i. Binomial Distribution
It is used to represent the probability distribution of discrete random variables. Binomial means two
categories. The successive repetition of an observation (trial) may result in an outcome which possesses
or which does not possess a specified character. Our primary interest will be either of these possibilities.
Conventionally, the outcome of primary interest is termed as success. The alternative outcome is termed
as failure. These terminologies are used irrespective of the nature of the outcome. For example, non-
germination of a seed may be termed as success.
In binomial distribution the experiment consisting the following criteria/assumptions
 The experiment consists of n identical trials
 There is only two outcomes in Bernoulli trials (success or failure)

85
 The number of trials n is fixed and independent
 At each trial the probability of success (p) remains the same
The variable X which represents the count of the number of successes in Bernoulli trials will be a
discrete random variable. The probability distribution of such discrete random variable X is called the
binomial distribution. The binomial distribution is given by the probability mass function (pmf):
P(X=x) = x
n
x
q
p
x
n 








for all possible values of X where,
n= number of trials, x= number of successes in a trial, n-x = number of failures in a trial, p =
probability of success (= x/n), q = 1 - p = probability of failure, and 







x
n
= the possible number of
ways in which x successes can occur.
The binomial distribution is determined by two parameters n and p.
The expected value of the binomial distribution is E(x) = np and the variance is Var(x) = npq.
Example 1: A given mid-exam contains 10 multiple choice questions, and each question has four
alternatives with one exact answer. Find the probability that the student exactly answered
i. 3 questions iii. At least 3 questions
ii. 8 questions
Using binomial distribution we can get the probability value easily. That is n = 10,
p = ¼ (the chance of getting answer from 4 alternatives) q = 1- p = 1- ¼ = ¾
The possible marks for a student from 10 questions are X = 0, 1, 2, 3. . . 10.
P(X = x) = x
n
x
q
p
x
n 








i. P(X = 3) = 







3
10
(0.25)3
(0.75)7
= 0.250
ii. P(X = 8) = 







8
10
(0.25)8
(0.75)2
= 0.00386
iii. P(X >= x) = 1 - P(X < x). Hence P(X >= 3) = 1 – P(X < 3)

86
= 1 – {P(X = 0) + P(X = 1) + P(X = 2)}
P(X = 0) = 







0
10
(0.25)0
(0.75)10
= 0.0563
P(X = 1) = 







1
10
(0.25)1
(0.75)9
= 0.1877
P(X = 2) = 







2
10
(0.25)2
(0.75)8
= 0.2816
.’. P(X >= 3) = 1 – (0.0563 + 0.1877 + 0.2816) = 0.4744
The mean = np = 2.5. The variance = npq = 1.875
Example 2: Suppose that a population of size N = 500 consists of 300 dominants and 200 recessive. For
a sample of size n = 10, calculate the probabilities:-
a) Exactly 2 individuals will be recessive.
b) At least 2 individuals will be recessive.
c) At most 1 individual will be recessive.(Exercise)
d) At most 5 individuals will be recessive.(Exercise)
Let X = recessive, p = probability of recessives = 200/500 = 2/5.
a) P[X=2] =     1210
.
0
5
/
3
5
/
2
2
10 8
2









b) P[X ]
1
[
]
0
[
1
]
2 




 X
P
X
P , but P[X=0] = 006047
.
0
0
10 10









q
P[X=1] = 040320
.
0
1
10 9









pq .
Hence, P[X2] = 1 - (0.006047 + 0.040320) = 0.9536
Mean = np = 4 V(X) = npq = 12/5
The binomial distribution approaches normal distribution as the number of trials n tends to large (n→  )
for any fixed value of p. A rule of thumb is that for p < 0.5, the normal approximation is adequate if np

87
>5. Departures from the given conditions result in less accurate approximations. When n is very large
and p is very small (n→∞ &p→0) the binomial distribution approaches Poisson distribution.
iii. Poison Distribution
The Poisson distribution is also used to represent the probability distribution of a discrete random
variable. It is employed in describing random events that occur rarely over a continuum of time or space,
such as number of car accident in certain road corr-section, number of errors in digital communication,
number of type fill errors, etc. The Poisson distribution bears a close similarity to the binomial
distribution. Suppose that we are interested in the number of occurrences of an event E in a time period
of length t. This time period can be split into n equal intervals, each of length t/n. These n intervals can
be treated as n trials by Bernoulli process. But there is difficult. Since the event occurs at various points
of time, it can occur twice or more in one of the trials of length t/n. In case of binomial distribution the
event is dichotomous, and hence there is no possibility of such multiple occurrences within a single trial.
In order to overcome this difficulty we make n larger and larger. When n is large, the trials are shorter in
terms of length of time. As a result, the probability of occurrence of an event in a single trial would be
smaller. It is equivalent of saying that it is a rare event. The binomial distribution can still be used to
represent the distribution of such random events. However, the computations become tedious since n is
very large. This can be explained by example.
Suppose that the number of insects caught in a trap is being studied and that the data are collected on the
number of insects caught per hour. Assume that the probability that an insect will be caught in any
single minute is 0.06. Assume further that the events of insects being trapped are mutually independent
and the probability p = 0.06 remains same for all the minutes. We may use the binomial distribution to
calculate the number of insects caught per hour by considering each minute as a separate Bernoulli trial.
If x is the number of insects caught in a minute then we have: P[X=x] =     x
x
x








 60
94
.
0
06
.
0
60
Instead of dividing the hour into minutes the seconds may be used as basic units. Then the value of p
would be reduced to, p=0.06/60=0.001. Considering each second as a Bernoulli trial, we would have a
sample size 60 60=3600 for a period of one hour. The binomial distribution would now be:
P[X=x] =     x
x
x








 3600
999
.
0
001
.
0
3600

88
Thus when n becomes larger and larger the computations using binomial become tedious. Fortunately, it
has been shown by Poisson that the value of x
n
x
q
p
x
n 








approaches the value of
 
!
x
e
np p
n
X 
, when n
becomes large and p becomes small in such a way that the equality, np =  is maintained.
The probability mass function of Poisson distribution is given by:
P[X=x] =
!
x
e x



. Where,  = np = mean number of times an event occurs.
x = the number of times an event occur. e = Naperian base = 2.7182…
The value of e 

can be obtained directly from mathematical tables. In case of Poisson distribution the
counts of alternative events, i.e., failures are not of interest. This is a contrast between binomial and
Poisson distributions. For Poisson distribution all that we need is np, the mean number of successes. We
need not know about n and p individually. Thus, the Poisson distribution is determined by the parameter
.
 The special property of Poisson distribution is that its mean and variance are same to  .
i.e. In magnitude; mean = variance = .

Example 3:In Black Lion Hospital, the average new born female baby in every 24 hour is 7. What is the
probability that
i. No female babies are born in a day?
ii. Only three female babies are born per day?
iii. Two female babies are born in 12 hours?
In this case  = 7 per day
No female baby born in a day  P(X = 0) =
!
0
70
7

e
= e-7
= 0.0138189
Only three female babies are born  P(X = 3) =
!
3
73
7

e
= 0.78998
Two female babies are born in 12 hours → in this case  = 7⁄2 = 3.5
P(X = 2) =
!
2
)
5
.
3
( 2
5
.
3

e
= 0.184959

89
Example 4: In some experiments it was observed that the incidence of stem fly in black gram was 6
percent. Suppose we examine 50 black gram plants in a field at random. What is probability that at most
3 plants will be found to be affected by stem fly?
The probability that a plant is affected by stem fly is given as 0.06. The number of plants observed (n =
50). Hence,  = np = 3. The required probability is
P[X  3] = P[X = 0] + P[X = 1] + P[X = 2] + P[X = 3]
P[X = x] =
!
x
e x



P[X = 0] =
!
0
30
3

e
= e-3
P[X = 1] =
!
1
31
3

e
= 3e-3
P[X = 2] =
!
2
32
3

e
= 4.5e-3
P[X = 3] = 3
3
3
3
5
.
4
6
27
!
3
3 



 e
e
e
 P[X 3
 ] = 13e-3
. From mathematical table it can found that e-3 =
0.0498.
Therefore P[X3] = 13 0498
.
0
 = 0.6474.
Common Continuous Probability Distributions
i. Normal Distribution
The most important and widely used probability distribution is normal distribution. It is also known as
Gaussian distribution. Most of the distributions occurring in practice, for instance, binomial, Poisson,
etc., can be approximated by normal distribution. Further, many of the sampling distributions like
Student’s t, F, & χ2
distributions tend to normality for large samples. Therefore, the normal distribution
finds an important place in statistical inference.
The normal distribution is used to represent the probability distribution of a continuous random variable
like life expectancies of some product, the volume of shipping container, etc.

90
Its probability density function is expressed by the relation,

 2
1
)
( 
X
f e 2
1

2





 


X
Where:  = a constant equaling 22/7, e = Naperian base equaling 2.718,  = population mean,  =
population standard deviation, x = a given value of the random variable in the range - .



 x
For a normal distribution the frequency curve will be symmetrical or bell shaped. However, not all
symmetrical curves are normal. The shape of the normal curve is completely determined by two
parameters 
 & . For any given , there can be a number of normal curves each with a different  .
Likewise, for any given  , there can be a number of normal curves each with a different . In order to
make such all distributions readily comparable with each other, their individuality as expressed by their
mean and standard deviation has to be suppressed. This is done by transforming the normal variable into
standard normal variable.
The standard normal variable is denoted by Z and is given by Z =



X
.
The distribution of the standard normal variable is known as standard normal distribution and given by:
f(x) =

2
1
e 2
2
1
z

, in the range - 


 z .
For standard normal distribution, .
1
0 
 
 and Tables are readily available for different values
of Z. Because of the symmetrical nature of the normal distribution the tables are presented only for the
positive values of Z.
Note: Area under curve is equal to one. But area above or below z = 0 is 0.5
Example 5: On a final examination in mathematics, the mean was 72 and the standard deviation was 15.
i) Determine the standard score of the students receiving the grades:
a) 60 b) 93 c) 72
ii) Determine students’ grade that have standard score
a) -1 b) 1.6

91
iii) Find the probability that any student score between 60 & 93. i.e. P[ 60< X < 93]
Solution: Where X is mark of student
ia) Z = 8
.
0
15
72
60





S
X
X
ib) Z = 4
.
1
15
72
93


ic) Z = 0
iia) X = X + ZS = 72 + -1(15) = 57
iib) X = X + ZS = 72 + 1.6(15) = 96
iii) P [60  X  93] = P [
S
X
S
X
X
S
X 



 93
60
] = P [-0.8 4
.
1

 Z ] =
P [-0.8 4192
.
0
2881
.
0
]
4
.
1
0
[
]
8
.
0
0
[
]
4
.
1
0
[
]
0 











 Z
P
Z
P
Z
P
Z = 0.7073 (This
is from standard normal table).
Example. P(0<Z<1.24)= 0.3925
From the table of Normal curves it can be seen that 68.26% of the area lies within the range of 
  ,
95.46% within the range of 
 2
 , and 99.74% within the range of 
 3 . This is an important
property of normal distribution which is frequently used in statistical inference.

92
The normality of a distribution can be tested using several methods. Three important methods can be
mentioned. They are graphic methods, Pearson’s test and goodness of fit. On the basis of Pearson’s
coefficients, 2
1 & 
 we can say whether a distribution is normal or not. If 2
1 &
0 
  = 3, the
distribution is said to be normal. (See chapter four about skewness and kurtosis)
ii. Chi-square Distribution ( on
Distributi

2
 )
The square of a standard normal variable is called a chi-square variate with one degree of freedom. Thus
if is a random variable following normal distribution with mean and standard deviation , then
( )
is a standard normal variate. is a chi-square variate with 1 degree of freedom. If
, , … , are independent random variables following normal distribution with means , , … ,
and standard deviations , , … , respectively then the variate
2
 = + + ⋯ +
= + + ⋯ + = ∑ this is the sum of the square of independent standard normal
variates, follows chi-square distribution with degree of freedom.
Applications of chi-square distribution
Chi-square distribution has a number of applications. Some of which are listed below
Chi-square test of goodness of fit
Chi-square test for independence of attributes
To test whether the population has a specified value of the variance.
iii. Student’s Distribution
In probability and statistics, Student's t-distribution (or simply the t-distribution) is a probability
distribution that arises in the problem of estimating the mean of a normally distributed population when
the sample size is small. It is the basis of the popular Student's t-tests for the statistical significance of
the difference between two sample means, and for confidence intervals for the difference between two
population means. The Student's t-distribution is a special case of the generalized hyperbolic
distribution.

93
Suppose X1, ..., Xn are independent random variables that are normally distributed with expected value µ
and varianceσ2
. Let the sample mean and sample variances are:
It can be shown that the random variable
The quantity has a chi-square distribution with n − 1 degrees of freedom. It is readily shown that the
quantity
Z is normally distributed with mean 0 and variance 1, since the sample mean is normally distributed
with mean µ and standard error . Moreover, it is possible to show that these two random variables
the normally distributed one and the chi-square-distributed one are independent. Consequently the
pivotal quantity,
which differs from Z in that the exact standard deviation σ is replaced by the random variable , has a
Student's t-distribution as defined above. The distribution of T is now called the t-distribution.

94
CHAPTER SIX
6. SAMPLING AND SAMPLING DISTRIBUTION
6.1. Basic Concepts of Sampling
When secondary data are not available for the problem under study, a decision may be taken to collect
primary data by using any of the methods of data collection discussed in the chapter one. The required
information may be obtained by following either the census method or the sample method.
Under the census or complete enumeration survey method, data are collected for each and every unit
(person, household, field, shop, factory etc.), as the case may be of the population or universe, which is
the complete set of items, which are of interest in any particular situation. If it is not essential to conduct
the complete enumeration, then a sample of some suitable size is selected from the population and the
study is carried out on the sample. This study is called sample survey. Most of the research work is done
through sample surveys.
Sampling is the process or method of selecting sample from the population (universe)
To easily follow sampling remind the terms such as population, sample, statistic, parameter, target
population, sampling unit, sampling frame, sample size, etc.
Essentials of Sampling
1. Saves Time, Energy and Cost: As the size of the sample is small as compared to the population, the
time and cost involved on sample study are much less than the complete counts. For complete count
huge funds are required. There is always the problem of finances. A small sample can be studied in a
limited time and total cost of sample study is very small. Thus the sample study requires less time
and less cost.
2. Reliability: If we collect the information about all the units of population, the collected information
may be true. But we are never sure about it. We do not know whether the information is true or
completely false. Thus we cannot say anything with confidence about the quality of information. We
say that the reliability is not possible. This is a very important advantage of sampling. The inference
about the population parameters is possible only when the sample data is collected from the selected
sample. Sample data is also used to check the accuracy of the census data.

95
3. Representativeness: A sample should be so selected that it truly represents the universe otherwise
the results obtained may be misleading. To ensure representativeness the random method of
selection should be used.
4. Adequacy: The size of sample should be adequate; otherwise it may not represent the characteristics
of the universe.
5. Independence: All items of the sample should be selected independently of one another and all
items of the universe should have the same chance of being selected in the sample. By independence
of selection we mean that the selection of a particular item in one draw has no influence on the
probabilities of selection in any other draw.
6. Homogeneity: When we talk of homogeneity we mean that there is no basic difference in the nature
of units of the universe and that of the sample. If two samples from the same universe are taken, they
should give more or less the same unit.
6.2.Types of Sampling Techniques
The various methods of sampling can be grouped under two broad heads:
1. Probability sampling (also known as random sampling) and
2. Non-probability (or non-random) sampling.
1. Random Sampling Techniques (probability sampling techniques)
Probability sampling methods are those in which every item in the population has a known chance, or
probability, of being chosen for sample. This implies that the selection of sample items is independent of
the person making the study-that is, the sampling operation is controlled so objectively that the items
will be chosen strictly at random. It may be noted that the term random sample is not used to describe
the data in the sample but the process employed to select the sample. Randomness is thus a property of
the sampling procedure instead of an individual sample. As such, randomness can enter processed
sampling in a number of ways and hence random samples may be of kinds.In other words, there is no
personal bias regarding the selection. Five common random sampling techniques are:
Simple Random sampling
Systematic Random sampling
Stratified Random sampling
Cluster Random sampling
Multi-stage sampling

96
2. Non-Random Sampling Technique
Non-random sampling is a process of sample selection without the use of randomization. In other words,
a non-random sample is selected basis other than the probability consideration such as convenience,
judgment, voluntary, quota, etc. In non-probability sampling techniques all elements of a population
have no known chance of inclusion or some outcomes have a zero chance of being selected as a sample.
The most familiar examples of non-random sampling techniques are:
Judgment (Purposive) sampling
Quota sampling
Convenience sampling
Volunteer sampling
Note: The most important difference between random and non-random sampling is that whereas the
pattern of sampling variability can be ascertained in case of random sampling. In non-random sampling,
there is no way of knowing the pattern of variability in the process.
Simple Random Sampling
Simple random sampling refers to that sampling technique in which each and every unit of the
population has an equal opportunity of being selected in the sample. In simple random sampling which
items get selected in the sample is just a matter of chance-personal bias of the investigator does not
influence the selection. It should be noted that the word random does not mean ‘haphazard’ or ‘hit-or-
miss’-it rather means that the selection process is such that chance only determines which items shall be
included in the sample. As pointed out by Chou, when a sample of size n is drawn from a population
with N elements, the sample is a simple random sample if any of the following is true. And, if any, the
following are true:
 All n items of the sample are selected independently of one another and all N items in the
population have the same chance of being included in the sample. By independence of selection
we mean that the selection of a particular item in one draw has no influence on the probabilities
of selection in any other draw.
 At each selection, all-remaining, items in the population have the same chance of being drawn. If
sampling is made with replacement, i.e. when each unit drawn from the population is returned
prior to drawing the next unit each item has a probability of 1/ N of being drawn at each
selection.

97
If sampling is without replacement, i.e. when each unit drawn from the population is not returned
prior to drawing the next unit, the probability of selection of each item remaining in the population at
the first draw is 1/N, at the second draw is 1/(N-1),at the third draw is 1 / (N-2), and so on. It should
be noted that sampling with replacement has very limited and special use in statistics-we are mostly
concerned with sampling without replacement.
 All the possible samples of a given size n are equally likely to be selected.
What Methods to Ensure Randomness?
To ensure randomness of selection one may adopt eitherthe lottery method ortable of random numbers.
Lottery Method: This is a very popular method of taking a random sample. Under this method, all
items of the universe are numbered or named on separate slips of paper of identical size and shape.
These slips are then folded and mixed up in a container or drum. A blindfold selection then made of the
number of slips required to constitute the desired sample size. The selection of items thus depends
entirely on chance. This method would be quite clear with the help of an example. If we want to take a
sample of 10 persons out of a population of 100, the procedure is to write the names of the 100 persons
on separate slips of paper, fold these slips, mix them thoroughly and then make a blindfold selection of
10 slips. The above method is very popular in lottery draws where a decision about prizes is to be made.
However, while adopting lottery method it is absolutely essential to see that the slips are of identical
size, shape and color, otherwise there is a lot of possibility of personal prejudice and bias affecting the
results.
Table of Random Numbers: The lottery method discussed above becomes quite cumbersome as the
size of population increases. An alternative method of random selection is that of using the table of
random numbers.
The random numbers are generally obtained by some mechanism which, when repeated a large number
of times, ensures approximately equal frequencies for the numbers from 0 to 9 and also proper
frequencies for various combinations of numbers (such as 00, 01,… 999, etc.) that could be expected in
a random sequence of the digits 0 to 9.

98
2952 6641 3992 9792 7969 5911 3170 5624
4167 9524 1545 1396 7203 5356 1300 2693
2670 7483 3408 2762 3563 1089 6913 7991
0560 5246 1112 6107 6008 8125 4233 8776
2754 9143 1405 9025 7002 6111 8816 6446
It is important that the starting point in the table of random numbers be selected in some random fashion
so that every unit has an equal chance of being selected.
The remaining both random Sampling methods (systematic, stratified, cluster, and multi-stage sampling)
and non-random sampling methods (Judgment, Quota, Convenience, and Volunteer sampling) are left
for you as reading assignment.
6.3. Sampling Distribution
Sampling distribution is not the distribution of sample, but the distribution of the statistic. That means if
we take many samples of equal sizes and get statistic for each of those samples; the distribution of all
these statistics is sampling distribution.Often we are interested in sampling distribution of common
statistics of the interest such as sample mean, sample proportion, sample variance, their functional form.
i. Sampling Distribution of the Sample Mean
A sampling distribution of the sample mean is a distribution obtained by using the means computed
from random samples of a specific size taken from a population. To illustrate the basic idea of sampling
distribution, we will consider very small hypothetical populations of 5 farmers, who use fertilizer in
their farming. Suppose the amount of fertilizer used (in ) by each of the farmer is 70, 78, 80, 80, and
95. Then, we obtain the following characteristics of the population
A. Population Parameters: Let denotes the amount of fertilizer used by each farmer =
1, 2, …, 5 = 5,
= ∑ = 70 + 78 + 80 + 80 + 95 = 403 = = 80.6 , which is the population mean
=
∑( )
=
( . ) ( . ) ⋯ ( . )
= 65.44
This population characteristics are mostly unknown and should be estimated from the sample

99
B. Possible samples and Sampling Distribution
Estimate the population parameters from a sample of three farmers that are randomly select from
population of 5 using sampling without replacement. The total number of possible samples is 10,
and is shown in the following table with their means. Let represents the framer.
No Sample Amount in the sample Sample Mean
1 70, 78, 80 76.0
2 70, 78, 80 76.0
3 70, 78, 95 81.0
4 70, 80, 80 76.67
5 70, 80, 95 81.67
6 70, 80, 95 81.67
7 78, 80, 80 79.33
8 78, 80, 95 84.33
9 78, 80, 95 84.33
10 80, 80, 95 85.0
The sample mean ( ) is obtained by dividing the sum of the three weights of fertilizer in that sample by
3. For example, the mean of the first sample is = 76.00.Using the value of as given above,
we record the frequency distribution of . Then, we obtain the probabilities of the random variable
by dividing the frequency of by the sum of the frequencies. These are listed below
Values of Frequencies ( ) Probability ( )
76.0 2 152.00 0.2
79.67 1 76.67 0.1
79.33 1 79.33 0.1
81.00 1 81.00 0.1
81.67 2 163.34 0.2
84.33 2 168.66 0.2
85.00 1 85.00 0.1
Total 10 806.00 1

100
This table gives the sampling distribution of . If we draw just one sample of three framers from the
population of five farmers, we may draw any one of the 10 possible farmers. Hence, the sample mean
can assume any of the values listed above with the corresponding probabilities. For example, the
probability of the mean 81.67 is ( = 81.67) = 0.2 . Therefore, the sample average, , is a random
variable that depends on which sample is selected. The value varies from 76.00 to 85 which are lower or
higher than the population mean = 80.6 . The average of the estimates of all possible samples for any
sample size is the true population value. That is, the expected value of , denoted by [ ], taken over
all possible samples equals population mean, i.e., [ ] = , in which [ ] =
∑
∑
= = 80.6.
If ~ ( , ), then sample mean ~ ( , / ).
ii. Sampling Distribution of Sample Proportion
Let P represent the proportion of elements in a large population having some characteristic; that is, the
proportion of ‘successes,’ where success corresponds to having that characteristic. If simple random
samples of size n are taken from a population where the proportion of ‘successes’ is p, then the sampling
distribution of ̂ has the following properties:
1. = : The average of all the possible ̂ values is equal to the parameter p. in other words, ̂ is
an unbiased estimator of p.
2. =
( )
: The standard deviation for ̂ decreases as the sample size n increases. For a fixed
sample size, the maximum standard deviation is attained at p=0.5.
3. ̂~ ,
( )
: If n is “sufficiently” large, the distribution of ̂ eventually looks like a normal
distribution with mean p and variance
( )
. The necessary size depends on the value of the
population proportion. It must be large enough that ≥ 5 (1 − ) ≥ 5.
Example: If the population proportion of people who favor a certain issue is 0.3, the sampling
distribution of ̂, when the sample size is 400, is approximately normal with a mean of 0.3 and
standard deviation of
( )
=
. ( . )
=0.023.

101
Let’s use this information to calculate some probabilities that the sample proportion ̂ will take on
certain values. Since we have a quantity that is normally distributed, to find probabilities we need to
convert the values to standard z-scores and use the normal table.
a. What is the probability that less than 25% of the sample will favor the issue?
( ̂ < 0.25) = <
0.25 − 0.3
0.023
= ( < −2.17) = 0.0150
b. What is the probability that the sample proportion ̂ who favor the issue is between 0.25 and 0.35?
(0.25 < ̂ < 0.35) =
. .
.
<
. .
.
= (−2.17 < < 2.17) = 0.9850 − 0.0150 =
0.9700
c. Will the probability that the sample proportion ̂ who favor the issue is between 0.27 and 0.33 be
larger or smaller than the probability calculated in part b.? smaller.
d. Find the probability. (0.27 < ̂ < 0.33) =
. .
.
<
. .
.
= (−1.30 < < 1.30) =
0.9032 − 0.0968 = 0.8064
Central Limit Theorem
As the sample size increases, the shape of the distribution of the sample means taken from a
population with mean and standard deviation of will approach a normal distribution. As previously
shown, this distribution will have a mean and a standard deviation .
The central limit theorem can be used to answer question about sample means in the same manner that
the normal distribution can be used to answer questions about individual values. The only difference is
that a new formula must be used for the values. It is
=
−
√
=
√ ( − )
Note that: is the sample mean and the denominator is the standard error of the mean.
When using the central limit theorem, one must remember two things:

102
1. When the original variable is normally distributed, the distribution of the sample means will be
normally distributed, for any sample size
2. When the distribution of the original variable departs from normality, a sample size of 30 or
more is needed to use the normal distribution as an approximation for the distribution of the
sample mean. The larger the sample, the better the approximation will be
Example: The blood cholesterol levels of a population of workers have mean 202 and standard
deviation 14. If a sample of 36 workers is selected, approximate the probability that the sample mean of
their blood cholesterol levels will lie between 198 and 206.
Solution: It follows from the central limit theorem that is approximately normal with mean = 202
and standard deviation
√
= 14
√36
= 2.3333. Thus the standardize variable
=
− 202
2.3333
has an approximately standard normal distribution. To compute {198 ≤ ≤ 206} , first we
must write the inequality in terms of the standardized variable . This results in the equality
{198 ≤ ≤ 206} =
198 − 202
2.3333
≤
− 202
2.3333
≤
206 − 202
2.3333
= {−1.714 ≤ ≤ 1.714}
= 2 {0 ≤ ≤ 1.714} = 0.913
Sample Size Determination
An important decision that has to be taken in adopting a sampling technique is about the size of the
sample. Size of sample means the number of sampling units selected from the population for
investigation. Experts have expressed different opinions on this point. For example, some have
suggested that the sample size should be 5 per cent of the size of population while others are of the
opinion that sample size should be at least 10 per cent. However, these views are of little use in practice
because no hard and fast rule can be laid down that sample size should be 5 per cent, 10 per cent or 25
per cent of the universe size. It may point out that more size alone does not ensure representativeness. A
smaller sample, but well selected sample, may be superior to a larger but badly selected sample.

103
If the size of the sample is small it may not represent the universe and the inference drawn about the
population may be misleading. On the other hand, if the size of sample is very large, it may be too
burdensome financially, require a lot of time and may have serious problems of managing it. Hence the
sample size should neither be too small nor too large. It should be ‘optimum’. Optimum size, according
to parten, is one that fulfills the requirements of efficiency, representativeness, reliability and flexibility.
The following factors should be considered while deciding the sample size: the size of the universe, the
resources available, the degree of accuracy or precision desired, homogeneity or heterogeneity of the
universe, nature of study, and method of sampling adopted, nature of respondents.
To determine sample size a number of formulae have been devised for determining the sample size
depending upon the availability of information. A few formulae are given below:
n=
2






d
zs
Where: n = Sample size, z= Value at a specified level of confidence or desired degree of precision, s=
Standard deviation of the population, d= Difference between population mean and sample mean.
Example: Determine the sample size if s = 6, population mean = 25, sample mean = 23 and the desired
degree of precision is 99 percent.
n=
2






d
ZS
, s= 6, d= 25-23 = 2, z = 2.576 (at 1% level the z value is 2.576)
Substituting the values: 60
72
.
59
728
.
7
2
6
576
.
2 2
2
or
n 






 


104
CHAPTER SEVEN
7. ESTIMATION AND HYPOTHESIS TESTING
Statistical Inference is the process of making interpretations or conclusions from sample data for the
totality of the population.It is only the sample data that is ready for inference.
Definitions
 Confidence Interval: An interval estimate with a specific level of confidence
 Confidence Level: The percent of the time the true value will lie in the interval estimate given.
 Consistent Estimator: An estimator which gets closer to the value of the parameter as the sample
size increases.
 Degrees of Freedom: The number of data values which are allowed to vary once a statistic has been
determined.
 Estimator: A sample statistic which is used to estimate a population parameter. It must be unbiased,
consistent, and relatively efficient.
 Estimate: Is the different possible values which an estimator can assumes, or particular value of an
estimator.
 Interval Estimate: A range of values used to estimate a parameter.
 Point Estimate: A single value used to estimate a parameter.
 Relatively Efficient Estimator: The estimator for a parameter with the smallest variance.
 Unbiased Estimator: An estimator whose expected value is the value of the parameter being
estimated.
In statistics there are two ways through which inference can be made.
 Statistical estimation
 Statistical hypothesis testing.
7.1. Statistical Estimation
This is one way of making inference about the population parameter where the investigator does not
have any prior notion about values or characteristics of the population parameter. There are two ways
estimation.

105
1) Point Estimation: is calculating a single value from the sample data to estimate the parameter.
It is a procedure that results in a single value as an estimate for a parameter. For example calculating
sample mean from sample data to estimate population mean is point estimation. Point estimation can be
done by method of moments, maximum likelihood method, and Bayesian method.
2) Interval estimation
It is the procedure that results in the interval of values as an estimate for a parameter, which is interval
that contains the likely values of a parameter. It deals with identifying the upper and lower limits of a
parameter. The limits by themselves are random variable.
Point and Interval estimation of the population mean: µ
 Point Estimation
Another term for statistic is point estimate, since we are estimating the parameter value. A point
estimator is the mathematical way we compute the point estimate. For instance, sum of i
x over n is the
point estimator used to compute the estimate of the population means,  .That is
n
x
X
i

 is a
point estimator of the population mean.
 Confidence interval estimation of the population mean
Although X possesses nearly all the qualities of a good estimator, because of sampling error, we know
that it's not likely that our sample statistic will be equal to the population parameter, but instead will fall
into an interval of values. We will have to be satisfied knowing that the statistic is "close to" the
parameter. That leads to the obvious question, what is "close"?
We can phrase the latter question differently: How confident can we be that the value of the statistic falls
within a certain "distance" of the parameter? Or, what is the probability that the parameter's value is
within a certain range of the statistic's value? This range is the confidence interval.
The confidence level is the probability that the value of the parameter falls within the range specified by
the confidence interval surrounding the statistic.

106
There are different cases to be considered to construct confidence intervals for population mean.
Case 1:If sample size is large and if the population is normal with known variance
Recall the Central Limit Theorem, which applies to the sampling distribution of the mean of a sample.
Consider samples of size n drawn from a population, whose mean is  and standard deviation is  with
replacement and order important. The population can have any frequency distribution. The sampling
distribution of X will have a mean 
 
x and a standard deviation n
x 
  , and approaches a
normal distribution as n gets large. This allows us to use the normal distribution curve for computing
confidence intervals.
1
var
0 



 iance
and
mean
with
on
distributi
normal
a
has
n
X
Z


n
Z
error
of
measure
a
is
where
X
n
Z
X













.
,
- For the interval estimator to be good the error should be small. How it be small?
 By making n large
 Small variability
 Taking Z small
- To obtain the value of Z, we have to attach this to a theory of chance. That is, there is an area of size


1 such that 

 



 1
)
( 2
2
Z
Z
Z
P
Where: is the probability that the parameter lies outside the interval
2

Z Stands for the standard normal variable to the right of which 2
 probability lies,
i.e 2
)
( 2 
 
 Z
Z
P

























1
)
(
1
)
(
2
2
2
2
n
Z
X
n
Z
X
P
Z
n
X
Z
P
)
,
( 2
2 n
Z
X
n
Z
X 
 
 

 is a %
100
)
1
( 
 confidence interval for 

107
But usually 2
 is not known, in that case we estimate by its point estimator S2
)
,
( 2
2 n
S
Z
X
n
S
Z
X 
 

 is a %
100
)
1
( 
 confidence interval for  .
Here are the z values corresponding to the most commonly used confidence levels.
Case 2: If sample size is small and the population variance
2
 is not known.
.
deg
1 freedom
of
rees
n
with
on
distributi
t
has
n
S
X

 
A 100(1-α)% confidence interval for population mean is
)
,
( 2
2 n
S
t
X
n
S
t
X 
 


The unit of measurement of the confidence interval is the standard error. This is just the standard
deviation of the sampling distribution of the statistic.
Examples:
1. From a normal sample of size 25 a mean of 32 was found .Given that the population standard
deviation is 4.2. Find
a) A 95% confidence interval for the population mean.
b) A 99% confidence interval for the population mean.
)
1
(
100 
 %  2
 2

Z
90 0.10 0.05 1.645
95 0.05 0.025 1.96
99 0.01 0.005 2.58

108
8. Solution:
a)
)
65
.
33
,
35
.
30
(
65
.
1
32
25
2
.
4
*
96
.
1
32
int
.
96
.
1
025
.
0
2
,
05
.
0
95
.
0
1
,
2
.
4
,
32
)
2
2
















n
Z
X
be
will
erval
required
The
table
from
Z
X
a







)
17
.
34
,
83
.
29
(
17
.
2
32
25
2
.
4
*
58
.
2
32
int
.
58
.
2
005
.
0
2
,
01
.
0
99
.
0
1
,
2
.
4
,
32
)
2
2
















n
Z
X
be
will
erval
required
The
table
from
Z
X
b







Note that for the example 1 n=25 which is done by case 2 but we use Z for demonstration.
2. A drug company is testing a new drug which is supposed to reduce blood pressure. From the six
people who are used as subjects, it is found that the average drop in blood pressure is 2.28 points,
with a standard deviation of .95 points. What is the 95% confidence interval for the mean change in
pressure?
Thus, we can be 95% confident that the mean decrease in blood pressure is between 1.28 and 3.28.
)
28
.
3
,
28
.
1
(
008
.
1
28
.
2
6
95
.
0
*
571
.
2
28
.
2
int
.
5
571
.
2
025
.
0
2
,
05
.
0
95
.
0
1
,
95
.
0
,
28
.
2
2
2

















n
S
t
X
be
will
erval
required
The
table
from
df
with
t
S
X






109
7.2. Hypothesis Testing
This is also one way of making inference about population parameter, where the investigator has prior
notion about the value of the parameter.
Statistical hypothesis: is an assertion or statement about the population whose plausibility is to be
evaluated on the basis of the sample data.
Test statistic: is a statistics whose value serves to determine whether to reject or accept the hypothesis to
be tested. It is a random variable and its value depends on sample data.
Testing hypothesis: is deciding whether the claim is supported by the experiment evidence obtained
through random sample
There are two types of hypothesis:
Null hypothesis: It is the hypothesis to be tested. It is the hypothesis of equality or the hypothesis of no
difference. Usually it is denoted by H0.
Alternative hypothesis: It is the hypothesis available when the null hypothesis has to be rejected. It is
the hypothesis of difference. Usually it is denoted by H1 or Ha.
Types and size of errors: Testing hypothesis is based on sample data which may involve sampling and
non-sampling errors. The following table gives a summary of possible results of any hypothesis test:
Decision
Reject H0 Don't reject H0
Truth
H0 Type I Error Right Decision
H1 Right Decision Type II Error
- Type I error: Rejecting the null hypothesis when it is true.
- Type II error: Failing to reject the null hypothesis when it is false.

110
Note that:
 There are errors that are prevalent in any two choice decision making problems.
 There is always a possibility of committing one or the other errors.
 Type I error ( ) and type II error (  ) have inverse relationship and therefore, cannot be minimized at
the same time.
 In practice we set  at some value and design a test that minimize  . This is because a type I error is
often considered to be more serious, and therefore more important to avoid, than a type II error.
General steps in hypothesis testing:
1. The first step in hypothesis testing is to specify the null hypothesis (H0) and the alternative hypothesis
(H1).
2. The next step is to select a significance level,
3. Identify the sampling distribution of the estimator and calculate a test statistic analogous to the
parameter specified by the null hypothesis.
4. Identify the critical region.
5. Making decision and conclusion of the result.
Hypothesis testing about the population mean (  )
Suppose the assumed or hypothesized value of  is denoted by 0
 , then one can formulate two sided (1)
and one sided (2 and 3) hypothesis as follows:
1. 0
1
0
0 :
: 


 
 H
vs
H
2. 0
1
0
0 :
: 


 
 H
vs
H
3. 0
1
0
0 :
: 


 
 H
vs
H
Case 1: When sampling is from a normal distribution with
2
 known
The relevant test statistic is
n
X
Z





111
After specifying  we have the following regions (critical and acceptance) on the standard normal
distribution corresponding to the above three hypothesis.
H0 Reject H0 if Accept H0 if Inconclusive if
0

  2

Z
Zcal  2

Z
Zcal  2
2 
 Z
Z
or
Z
Z cal
cal 


0

  
Z
Zcal 
 
Z
Zcal 
 
Z
Zcal 

0

  
Z
Zcal  
Z
Zcal  
Z
Zcal 
Where:
n
X
Zcal

0


Case 2: When sampling is from a normal distribution with
2
 unknown and small sample size
The relevant test statistic is
n
S
X
t


 ~ .
freedom
of
degrees
1
n
with
t 
After specifying  we have the following regions on the student t-distribution corresponding to the
above three hypothesis.
H0 Reject H0 if Accept H0 if Inconclusive if
0

  2

t
tcal  2

t
tcal  2
2 
 t
t
or
t
t cal
cal 


0

  
t
tcal 
 
t
tcal 
 
t
tcal 

0

  
t
tcal  
t
tcal  
t
tcal 
Where:
n
S
X
tcal
0




112
Case3: When sampling is from a non- normally distributed population or a population whose
functional form is unknown.
If a sample size is large one can perform a test hypothesis about the mean by using:
.
,
.
,
2
0
2
0
unknown
is
if
n
S
X
known
is
if
n
X
Zcal









The decision rule is the same as case I.
Examples:
1. Test the hypotheses that the average height content of containers of certain lubricant is 10 liters if the
contents of a random sample of 10 containers are 10.2, 9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4, 10.3, and 9.8
liters. Use the 0.01 level of significance and assume that the distribution of contents is normal.
Solution: Let µ is population mean and µo=10
Step 1: Identify the appropriate hypothesis
0
1
µ
:
H
vs
10
µ
:
H 1
0 

Step 2: select the level of significance, α=0.01
Step 3: Select an appropriate test statistics
T- Statistic is appropriate because population variance is not known and the sample size is also small.
25
.
0
,
06
.
10 
 S
X
76
.
0
10
25
.
0
10
06
.
10
0






n
S
X
tcal

Step 4: Identify the critical region.

113
Here we have two critical regions since we have two tailed hypothesis
.
tan
)
2498
.
3
,
2498
.
3
(
2498
.
3
)
9
(
005
.
0
region
ce
accep
is
t
is
region
critical
The



Step 6: Decision and Conclusion
At 1% level of significance we accept H0, since tcal is in the acceptance region or since calculated t is not
greater than absolute value of tabulated t. Thus, we conclude that average height content of containers of
certain lubricant is 10 liters. In other words at 1% level of significance, we have no evidence to say that
the average height content of containers of the given lubricant is different from 10 litters, based on the
given sample data.
2. The mean life time of a sample of 16 fluorescent light bulbs produced by a company is computed to be
1570 hours. The population standard deviation is 120 hours. Suppose the hypothesized value for the
population mean is 1600 hours. Can we conclude that the life time of light bulbs is decreasing? (Use
05
.
0

 and assume the normality of the population)
Solution: Let µ is population mean and µo=1600 is hypothesized population mean
Step 1: Identify the appropriate hypothesis
1600
:
1600
: 1
0 
 
 H
vs
H
Step 2: select the level of significance, )
(
05
.
0 given


Step 3: Select an appropriate test statistic
Z- Statistic is appropriate because population variance is known.
But sample size is small, so let we do for demonstration
1.0
16
120
1600
1570
n
σ
µ
X
Z 0
cal 





Step 4: identify the critical region.
.
tan
)
,
645
.
1
(
645
.
1
05
.
0
region
ce
accep
is
Z
is
region
critical
The




Step 5: Computations and Decision

114
At 5% level of significance accept H0 , since Zcal is in the acceptance region and we conclude that there
is no evidence to say that that the life time of light bulbs is decreasing, based on the given sample data.
3. (Exercise)It is known in a pharmacological experiment that rats fed with a particular diet over a certain
period gain an average of 40 gms in weight. A new diet was tried on a sample of 20 rats yielding a
weight gain of 43 gms with variance 7 gms2 . Test the hypothesis that the new diet is an improvement
assuming normality.
a) State the appropriate hypothesis
b) What is the appropriate test statistic? Why?
c) Identify the critical region(s)
d) On the basis of the given information test the hypothesis and make conclusion.

115
CHAPTER EIGHT
8. REGRESSION ANDCORRELATION ANALYSIS
8.1 Introduction
Regression analysis, in the general sense, means the estimation or prediction of the unknown values of
one variable from known values of the other variable .In regression analysis there are two types of
variables. The variable whose value is influenced or to be predicted is called dependent (regressed or
explained) variable, and the variable which influences the values or is used for prediction is called
independent variable (regressor or predictor or explanatory). If the Regression curve is a straight line,
we say that there is linear relationship between the variables under study, non-linear elsewhere.
When only two variables are involved, the functional relationship is known as simple regression. If the
relationship between the two variables is a straight line, it is known as simple linear regression;
otherwise it is called as simple non-linear regression. When there are more than two variables and one of
them is assumed dependent upon the other, the functional relationship between the variables is known as
multiple regressions. Moreover, correlation analysis is concerned with mathematical measure of the
extent or degree of relationship between two variables.
Regression analysis is performed if one wants to know relationship between
a) Income –consumption
b) Sales of ice-cream –with temperature of the day
c) Industrial production and consumption of electricity
d) The yield of crops, amount of rainfall, type of fertilizer, humidity.
e) Weight and height, age and strength, blood pressure and time of exercise.
8.2. Simple Linear Regression (SLR)
Simple linear regression is refers to the linear relationship between two quantitative variables in such
way that one of them influences the other. The simple linear regression of Y on X in the population is
given by Y= 

 
 X
Where,  = y intercept (Value of Y when X is zero)
 = slope of the line or regression coefficient (change in Y for unit change of value of X)

116
ε = Error term = (y- ŷ ) is unexplained part by the X. where x
y 
 ˆ
ˆ
ˆ 

Basic Assumptions of SLR
1. There is linear relationship between dependent variable y and explanatory variable x
2. Expected value of error term is zero and its variance is constant (δ2)
3. Error term is approximately normally distributed with mean zero and constant variance (δ2
).
4. The dependent variable has normal distribution with mean x
y 
 
 and variance δ2
5. Data on independent variables are fixed numbers.
Estimation of the regression Parameters
The y-intercept  and the slope  are the population parameters. We generally obtain the estimates of

 and from the sample. The estimator of 
 and are denoted by a and b, respectively. Thus the
fitted regression line is
ŷ = a+ bx
The values of a and b are obtained using the method of least squares. According to the principle of least
squares, one should select a and b such that  2
e will be as small as possible, that is, we minimize
SSE=  2
e = S=  
 2
 
 bx
a
y
To minimize this function, first we take the partial derivatives of SSE with respect to a and b. Then the
partial derivatives are equal to zero separately. These will result in the equations known as normal
equations.
For the straight line, y= a+ bx the normal equations are
 
 bx
na
y
  

 2
x
b
x
a
xy
By solving these normal equations, we can get the values of a and b as follow.
The best estimate of β is given by:

117
   
 
  



 





 2
2
2
2
x
x
n
y
x
xy
n
n
x
x
n
y
x
xy
b
The best estimate of  is given by:
x
b
y
a 

The regression line indicates the average value of the dependent variable Y associated with a particular
value of the independent variable X. The slope b, hereafter referred to as regression coefficient which
indicates the change in Y with a unit change in X.
Example 8.1 The following table gives the ages and blood pressure of 10 women
Age(x) 56 42 36 47 49 42 60 72 63 55
Blood pressure(y) 147 125 118 128 145 140 155 160 149 150
a) Determine the least square regression equation of blood pressure on age of women
b) Estimate the blood pressure of a women whose age is 45 years.
Solution
 
x 522  y = 1417  xy =75188 x = 52.2
 2
x = 28348  2
y = 202493 7
.
141

y
The estimated value of a and b can be obtained as respectively
11
.
1
)
522
(
28348
*
10
1417
*
522
75188
*
10
)
( 2
2
2







 
  
x
x
n
y
x
xy
n
b
The best estimate of  is given by
   76
.
83
5
.
52
11
.
1
7
.
141 



 x
b
y
a
a) The least square regression equation of blood pressure on age of women is given by

118
ŷ =a + bx , where ŷ is estimated blood pressure and x is age of woman
=83.76+1.11x
b) Estimated blood pressure of awoman whoseage 45 years is given as follows.
When x= 45
ŷ =a + bx
=83.76+1.11*45=133.71
Example 8.2: From the following data obtain the regression equation of Y on X
Sales(X) : 91 97 108 121 67 124 51 73 111 57
Purchase(Y)):75 75 69 97 70 91 39 61 80 47
Solution
n= 10 ,   900
x
,   700
y ,   66900
xy ,   87360
2
x
61
.
0
)
900
(
87360
*
10
700
*
900
66900
*
10
)
( 2
2
2







 
  
x
x
n
y
x
xy
n
b
1
.
15
)
900
61
.
0
700
(
10
1
)
(
1
1







 
 

n
i
i
n
i
i x
b
y
n
x
b
y
a
x
y 61
.
0
1
.
15 


8.3. Correlation Analysis
Correlation analysis is concerned with measuring the strength (degree) of the relationship between two
or more variables. Some of correlations are simple correlation, partial correlation, and autocorrelation.
Simple correlation is used if we are interested in knowing the extent of interdependence between two
variables or to measure degree of linear relation between two variables.

119
a) Karl Pearson’s Coefficient of Simple Correlation
The Karl Pearson correlation coefficient denoted by )
,
( y
x
r or xy
r or simply r, is defined as the ratio of
the covariance between X and Y to the product of their standard deviations:
 










2
2
)
(
)
(
)
)(
(
)
,
cov(
y
y
x
x
y
y
x
x
y
x
r
i
i
i
i
y
x

The simplified formula used for computational purpose is
   





 
 
  
n
y
y
n
x
x
n
y
x
xy
r
2
2
2
2    
  



)
)
(
)(
)
(
( 2
2
2
2
y
y
n
x
x
n
y
x
xy
n
An increase in one variable may cause an increase in the other variable, or a decrease in one variable
may cause decrease in the other variable. When the variables move in the same direction like this they
are said to be positively correlated. The positive correlation may be termed as direct correlation. If a
decrease in one variable causes an increase in the other variable or visa versa, the variables are said to be
negatively correlated. The negative correlation may be termed as inverse correlation. In case the two
variables are not at all related they are said to be independent or uncorrelated.
Example
i. Amount of rainfall and yield of crop(up to a point) has positive correlation
ii. Price and demand of a commodity has negative correlation.
Properties of Simple Correlation Coefficient
 Coefficient of correlation lies between –1≤ r ≤1
 If r =0 indicate that there is no linear relationship between two variables.
 If r = -1 or +1 indicate that there is perfect negative (inverse) or positive (direct) linear
relationship between two variables respectively.

120
 A coefficient of correlation(r) that is closes to zero shows the relationship is quite weak,
whereas r is closest to +1 or -1, shows that the relationship is strong.
Note that
 The strength of correlation does not depend on the positiveness and negativeness of r.
 The slope of simple linear regression (coefficient of regression) and correlation coefficient
should be the same in sign.
The correlation between two variables is linear if a unit changes in one variable result in a constant
change in the other variable. Correlation can be studied through plotting scattered diagrams
Example 8.3 Calculate simple correlation coefficient (r) for the data on advertising and sales
expenditure and interpret it.
Advertising(x): 39 65 62 90 82 75 25 98 36 78

121
Sales (y) : 47 53 58 86 62 68 60 91 51 84
  650
x ,  660
y ,   45604
xy ,   47648
2
x ,   45784
2
y
8
.
0
78
.
0
22240
*
53980
27040
)
)
660
(
45784
*
10
)(
)
650
(
47648
*
10
(
660
*
650
45604
*
10
2
2







r
There is strong positive (direct) linear relationship between sales and advertisement since simple
correlation coefficient approaches to 1
Example 8.4:Calculate and interpret simple correlation coefficient for data on blood pressure and age of
10 women
Age(x) 56 42 36 47 49 42 60 72 63 55
Blood
pressure(y)
147 125 118 128 145 140 155 160 149 150
 
x 522  y = 1417  xy =75188 x = 52.2
 2
x = 28348  2
y = 202493 7
.
141

y
9
.
0
89
.
0
17041
*
10996
12206
)
)
1417
(
202493
*
10
)(
)
522
(
28348
*
10
(
1417
*
522
75188
*
10
2
2







r
 There is strong direct linear relationship between blood pressure and age of women, since
correlation coefficient approaches to +1.
Coefficient of Determination (R2
)
It is defined as the proportion of the variation in the dependent variable Y that is explained, or accounted
for, by the variation of the independent variable X. Its value is the square of the coefficient of
correlation, thus we denote it by r2
and it is usually expressed in the form of percentage.Example
compute and interpret coefficient of determination for above example on age and blood pressure.

122
Given that simple correlation coefficient between blood pressure and age is 0.89, hence coefficient of
determination is square of the coefficient of correlation (r2
)=(0.89)2
=79.21% which implies that 79.21%
variation in the blood pressure of women is accounted for, by the variation of the age of women.
Rank Correlation
Sometimes we come across statistical series in which the variables under consideration are not capable
of quantitative measurement, but can be arranged in serial order. This happens when we dealing with
qualitative characteristics(attributes ) such as beauty, efficient ,honest ,intelligence ,….etc., in such
case one may rank the different items and apply the spearman method of rank difference for finding
out the degree of relationship. The greatest use of this method(rank correlation) lies in the fact that
one could use it to find correlation of qualitative variables, but since the method reduces the
amount of labor of calculation ,it is sometimes used also where quantitative data is available. It is
used when statistical series are ranked according to their magnitude and the exact size of individual item
is not known. Spearman’s correlation coefficient is denoted by rs. If the ranks are given, denote the
difference R1i - R2i by di and obtain the total of di. Then the following formula is applied
 











1
6
1 2
2
n
n
d
rs
If the actual data is given, rank it in ascending or descending order and follow the above procedures.
 Note that the values of rank correlation (rs),.also lies between -1 and +1 inclusive.
Example 8.5 Ten competitors in a beauty contest are ranked by two judges in the following order.
Compute and interpret opinion of two judges with regard to beauty out looking.
1st
judge(x) 1 6 5 10 3 2 4 9 7 7
2nd
judge(y) 3 5 8 4 7 10 2 1 6 9
d=(x-y) -2 1 -3 6 -4 -8 2 8 1 -2
d2
4 1 9 36 16 64 4 64 1 4
  203
2
d

123
    2303
.
0
1
10
10
203
*
6
1
1
6
1 2
2
2









n
n
d
rs
 Hence the pair of judges has opposite (divergent) tastes for beauty since rank correlation
coefficient is negative.
Example 9.6 Calculate rank correlation coefficient between advertisement cost and sales from the
following data and interpret it.
Advertisement(x): 39 65 62 90 82 75 25 98 36 78
Sales (y) : 47 53 58 86 62 68 60 91 51 84
Solution
Rank of X : 8 6 7 2 3 5 10 1 9 4
Rank of Y : 10 8 7 2 5 4 6 1 9 3
d = x - y :-2 -2 0 0 -2 1 4 0 0 1
d2
:- 4 4 0 0 4 1 16 0 0 1
  30
2
d
    82
.
0
11
9
11
2
1
1
10
10
30
6
1
1
6
1
10 2
2
2












 x
n
n
d
r
n s rs = 0.82
It implies that there is strong positive linear relationship between advertisement cost and sales since
correlation coefficient approaches to +1.

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