STT 620: GRADUATE RESEARCH STATISTICS
SELF INSTRUCTION MANUAL FOR STUDENTS
TABLE OF CONTENTS
Introduction ..................................................................................................................................... 1
Chapter One:Statistics, Measurement And Analysis ...................................................................... 6
Chapter Two:Measures Of Central Tendency And Dispersion..................................................... 31
Chapter Three:The Normal Distribution ....................................................................................... 44
Chapter Four:Statistical Inference Using The Mean And Proportion ........................................... 67
Chapter Five:The Chi-Square Statistic ......................................................................................... 87
Chapter Six:Analysis Of Variance .............................................................................................. 104
Chapter Seven:Linear Regression And Correlation .................................................................... 127
References ................................................................................................................................... 151
GENERAL INTRODUCTION OF THE UNIT
INTRODUCTION
After working for sometimes in adult life, we are required to use statistics in our efforts of
analyzing data measured from real-life phenomena. The immediate question which is asked by
most of us is, why statistics has not been necessary all this time, and now it is required! The
answer to this puzzle is that all that time we have not been asked to use statistics, our lives may
have been much simpler than now, and we did not require rigorous data analysis, rigorous proofs,
and rigorous exhibition of data accuracy and veracity. Now that we are interested in all these
things, I welcome you to one of the most powerful tools available to mankind for data
collection, data analysis, and data reporting - or presentation. This tool is largely a product of the
technological developments of the 19th and 20th centuries. That is when it became necessary to be
accurate with data (of all kinds) because of the nature of production mechanisms involved in all
the industries that sprang up as a result of the industrial revolution, all current human endeavor,
and the current information age (using computers and allied equipment) at the dawn of the 21st
Page 1 of 152
,Century. For example, in this regard, we are never satisfied by reports from anybody that
something is bigger or better than another one. The current technical minded man or woman
requires to know how much bigger the object is, and how better it is. This means that there must
be reliable data (always), upon which to base our decision making. This is the nature of today’s
life. Think of the time your child comes home from school at the end of the term, with a report
form. He or she tells you that he/she was number ten in this term’s examination. Unlike our
forefathers, we are not satisfied by the mere number (or position) that our child scored (obtained)
in school. We need to know the average mark that the child obtained, so that we can know the
actual academic strength of our child. We then may consider the average or the mean grade. This
is when we are able to preliminarily rate our child’s ability. There are very few of us who have
not done this, to ascertain whether our child is good material for admission to secondary schools,
or to universities or other institutions of higher learning. This is one of the commonest and
simplest uses of Statistics, which we shall be considering soon in this module. The average mark
is called the mean grade or mean mark! Do you see how close Statistics is to our daily lives?
Then it pays to learn statistics! These days, statistical calculations are required in almost all
human activities and all academic disciplines. Advances in science and technology in all fields of
study, all production areas, and all areas of human endeavor have necessitated the use of
statistics. Some phenomena are easily quantified and can be easily measured. Other natural
phenomena are not that easy to measure. In all these cases, we resort to the use of statistics so
that we can know how to handle each case we come across. In this module we shall learn how to
analyze facts and figures, and to summarize many diffuse data types using the tools that are
available within the discipline of statistics. We may therefore define statistics as a body of
knowledge in the realm of applied mathematics with its own ethics, terminology content,
theorems and techniques. When we master this discipline through our rigorous study we seek to
master all these theorems and analytical techniques and to make them our toolkit which we can
use as our second nature in order to understand the universe around us and to make sure that we
are not duped by con-people who push facts and figures to us so that we can believe them in
order they can gain emotionally, politically, spiritually or materially from our data indigestion.
In this way we view the discipline of statistics as a tool for helping us in our daily needs of data
analysis and understanding. It may not be necessary to use the tool always, but if it is available
Page 2 of 152
,we can use it any time we need it. It is better to have it on stand-by, than to miss it altogether.
This way we become flexible in the understanding of our universe.
Origins of Statistics
Statistics has many origins. However, there are two chief sources of this discipline. One of the
earliest sources of statistics is in the book of Numbers within the Holy Bible, when it was
necessary to know how many of the children of Israel were proceeding to the Promised Land,
which flowed with milk and honey. As a learner you are requested to appreciate that the numbers
given in the Book of Numbers are facts and figures of governing any community, or any state.
This is where the name Statistics came from. Please read the Book of Numbers in the Old
Testament of the Holy Bible to be able to appreciate this. Also one of the other ancient events
recorded in the bible is when Jesus Christ was born. There were data needs within the ancient
Roman Empire which caused Caesar Augustus to have an enumeration of all people in the
Roman Empire. When it came to Palestine, it affected the Holy Family, and they had to travel to
Bethlehem so that the population count can be accurate for the purposes of the emperor. They
had to be counted in their district of origin, not where they worked and lived. Upon this event,
Jesus was born in Bethlehem. All these facts should make us appreciate the value of statistics for
the sake of planning and decision making purposes.
Purpose of the course
This course is designed to introduce the learner to the purpose and meaning of Statistics; and to
the use of statistics in research design, data collection, data analysis, and research reporting,
especially in educational Research.
Expected learning outcomes:
By the end of the course the learner is expected to:-
1. Collect, organize, analyze and present educational data using statistical tools
2. Explain and use sampling and sampling methods
3. Explain the nature and use of Statistics
4. Work out with simple statistical data to calculate descriptive statistics (measures of
central tendency and dispersion)
Page 3 of 152
, 5. Use various test statistics such as Z-test, t-test and F distribution to carry out
hypothesis testing
6. Use Statistics in data analysis and report writing within the environment of education
research.
7. Interpret statistical information both descriptive and inferential
This manual is organized into chapters as follows:
Chapter one: statistics, Measurements, and analysis:
The chapter deals with the basic terms used in statistics, types of measurements and
organization of data for statistical analysis including frequency tables
Chapter two: measures of central tendency and dispersion
This chapter will enable the learner calculate mean, median and mode of a set of data and
also work with measures of dispersion such as range standard deviation and variance.
Chapter three: normal distribution
In this chapter, the learner will be introduced to the normal probability distribution, its
properties and use in interpreting statistical information. Use of normal distribution tables
will be explored
Chapter four: statistical inferences using mean and proportion
The chapter deals with introduction to inferential statistics. The following concepts are
discussed in details under this chapter: estimation, sampling distribution, and the central
limit theorem, hypothesis testing of sample means and proportion, hypothesis testing for
small and large samples i.e Z –test and t- test distributions
Chapter five: chi square distribution
This chapter introduces one of the powerful non-parametric tests for testing hypothesis.
Test for goodness of fit and use of contingency tables are explored.
Chapter six: analysis of variance
The chapter discusses hypothesis testing of differences of means of more than two
samples by invoking the F distribution. One way and two way analysis of variances are
discussed in detail
Chapter seven: regression and correlation analysis:
Page 4 of 152
SELF INSTRUCTION MANUAL FOR STUDENTS
TABLE OF CONTENTS
Introduction ..................................................................................................................................... 1
Chapter One:Statistics, Measurement And Analysis ...................................................................... 6
Chapter Two:Measures Of Central Tendency And Dispersion..................................................... 31
Chapter Three:The Normal Distribution ....................................................................................... 44
Chapter Four:Statistical Inference Using The Mean And Proportion ........................................... 67
Chapter Five:The Chi-Square Statistic ......................................................................................... 87
Chapter Six:Analysis Of Variance .............................................................................................. 104
Chapter Seven:Linear Regression And Correlation .................................................................... 127
References ................................................................................................................................... 151
GENERAL INTRODUCTION OF THE UNIT
INTRODUCTION
After working for sometimes in adult life, we are required to use statistics in our efforts of
analyzing data measured from real-life phenomena. The immediate question which is asked by
most of us is, why statistics has not been necessary all this time, and now it is required! The
answer to this puzzle is that all that time we have not been asked to use statistics, our lives may
have been much simpler than now, and we did not require rigorous data analysis, rigorous proofs,
and rigorous exhibition of data accuracy and veracity. Now that we are interested in all these
things, I welcome you to one of the most powerful tools available to mankind for data
collection, data analysis, and data reporting - or presentation. This tool is largely a product of the
technological developments of the 19th and 20th centuries. That is when it became necessary to be
accurate with data (of all kinds) because of the nature of production mechanisms involved in all
the industries that sprang up as a result of the industrial revolution, all current human endeavor,
and the current information age (using computers and allied equipment) at the dawn of the 21st
Page 1 of 152
,Century. For example, in this regard, we are never satisfied by reports from anybody that
something is bigger or better than another one. The current technical minded man or woman
requires to know how much bigger the object is, and how better it is. This means that there must
be reliable data (always), upon which to base our decision making. This is the nature of today’s
life. Think of the time your child comes home from school at the end of the term, with a report
form. He or she tells you that he/she was number ten in this term’s examination. Unlike our
forefathers, we are not satisfied by the mere number (or position) that our child scored (obtained)
in school. We need to know the average mark that the child obtained, so that we can know the
actual academic strength of our child. We then may consider the average or the mean grade. This
is when we are able to preliminarily rate our child’s ability. There are very few of us who have
not done this, to ascertain whether our child is good material for admission to secondary schools,
or to universities or other institutions of higher learning. This is one of the commonest and
simplest uses of Statistics, which we shall be considering soon in this module. The average mark
is called the mean grade or mean mark! Do you see how close Statistics is to our daily lives?
Then it pays to learn statistics! These days, statistical calculations are required in almost all
human activities and all academic disciplines. Advances in science and technology in all fields of
study, all production areas, and all areas of human endeavor have necessitated the use of
statistics. Some phenomena are easily quantified and can be easily measured. Other natural
phenomena are not that easy to measure. In all these cases, we resort to the use of statistics so
that we can know how to handle each case we come across. In this module we shall learn how to
analyze facts and figures, and to summarize many diffuse data types using the tools that are
available within the discipline of statistics. We may therefore define statistics as a body of
knowledge in the realm of applied mathematics with its own ethics, terminology content,
theorems and techniques. When we master this discipline through our rigorous study we seek to
master all these theorems and analytical techniques and to make them our toolkit which we can
use as our second nature in order to understand the universe around us and to make sure that we
are not duped by con-people who push facts and figures to us so that we can believe them in
order they can gain emotionally, politically, spiritually or materially from our data indigestion.
In this way we view the discipline of statistics as a tool for helping us in our daily needs of data
analysis and understanding. It may not be necessary to use the tool always, but if it is available
Page 2 of 152
,we can use it any time we need it. It is better to have it on stand-by, than to miss it altogether.
This way we become flexible in the understanding of our universe.
Origins of Statistics
Statistics has many origins. However, there are two chief sources of this discipline. One of the
earliest sources of statistics is in the book of Numbers within the Holy Bible, when it was
necessary to know how many of the children of Israel were proceeding to the Promised Land,
which flowed with milk and honey. As a learner you are requested to appreciate that the numbers
given in the Book of Numbers are facts and figures of governing any community, or any state.
This is where the name Statistics came from. Please read the Book of Numbers in the Old
Testament of the Holy Bible to be able to appreciate this. Also one of the other ancient events
recorded in the bible is when Jesus Christ was born. There were data needs within the ancient
Roman Empire which caused Caesar Augustus to have an enumeration of all people in the
Roman Empire. When it came to Palestine, it affected the Holy Family, and they had to travel to
Bethlehem so that the population count can be accurate for the purposes of the emperor. They
had to be counted in their district of origin, not where they worked and lived. Upon this event,
Jesus was born in Bethlehem. All these facts should make us appreciate the value of statistics for
the sake of planning and decision making purposes.
Purpose of the course
This course is designed to introduce the learner to the purpose and meaning of Statistics; and to
the use of statistics in research design, data collection, data analysis, and research reporting,
especially in educational Research.
Expected learning outcomes:
By the end of the course the learner is expected to:-
1. Collect, organize, analyze and present educational data using statistical tools
2. Explain and use sampling and sampling methods
3. Explain the nature and use of Statistics
4. Work out with simple statistical data to calculate descriptive statistics (measures of
central tendency and dispersion)
Page 3 of 152
, 5. Use various test statistics such as Z-test, t-test and F distribution to carry out
hypothesis testing
6. Use Statistics in data analysis and report writing within the environment of education
research.
7. Interpret statistical information both descriptive and inferential
This manual is organized into chapters as follows:
Chapter one: statistics, Measurements, and analysis:
The chapter deals with the basic terms used in statistics, types of measurements and
organization of data for statistical analysis including frequency tables
Chapter two: measures of central tendency and dispersion
This chapter will enable the learner calculate mean, median and mode of a set of data and
also work with measures of dispersion such as range standard deviation and variance.
Chapter three: normal distribution
In this chapter, the learner will be introduced to the normal probability distribution, its
properties and use in interpreting statistical information. Use of normal distribution tables
will be explored
Chapter four: statistical inferences using mean and proportion
The chapter deals with introduction to inferential statistics. The following concepts are
discussed in details under this chapter: estimation, sampling distribution, and the central
limit theorem, hypothesis testing of sample means and proportion, hypothesis testing for
small and large samples i.e Z –test and t- test distributions
Chapter five: chi square distribution
This chapter introduces one of the powerful non-parametric tests for testing hypothesis.
Test for goodness of fit and use of contingency tables are explored.
Chapter six: analysis of variance
The chapter discusses hypothesis testing of differences of means of more than two
samples by invoking the F distribution. One way and two way analysis of variances are
discussed in detail
Chapter seven: regression and correlation analysis:
Page 4 of 152
