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LEARNING MODULES

IN

STATISTICS(With Computer Application)

By

Prof. Felipe V. de Guzman

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MODULE IN STATISTICS

MODULE ONE

STATISTICS AS A SCIENCE

INTRODUCTION

Undoubtedly, statistics is a very useful tool in the various activities of man. During the

primitive period, people were not conscious that they were already using statistics in counting

events, activities, things, etc. They were not also aware that they were using statistics in

determining the birth rate, crop yield, occurrence of events at a certain period of time, etc.

The use of statistics in modern time is of course different from those of primitive past.

The modern man utilizes statistics, as a science, in the various fields of studies, professional

endeavors, and even for personal profit.

This module will enable you to understand the nature and meaning of statistics, its brief

historical development, the difference between sample and population, the meaning and kinds ofvariables, and the importance of statistics especially in the field of research.

GENERAL OBJECTIVES

At the end of this module, you are expected to:

1. State the nature and scientific definition of statistics

2. Trace the brief historical development of statistics3. Distinguish sample from a population

4. Enumerate and differentiate the kinds of variables

5. Explain the uses of statistics

TIME FRAME

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P R E - T E S T 1

Test your knowledge of the basic ideas in statistics. As much as possible, avoid

guessing. At any rate, this test is not graded. Choose the letter of the best answer from the

given four choices. Write your answer on the blank before the number.

1. From the research point of view, statistics as a science deals with the following

activities:

A. collection and gathering of data

B. presentation and analysis of data

C. interpretation of data

D. all of the above

2. In counting events, objects, people, etc., the measurements that are collected from

the original information are called __________.

A. data

B. scores

C. raw data

D. none of the above

3. In making generalizations about the population from which the sample has been

drawn, the measure to use is called _______.

A. descriptive statisticsB. inferential statistics

C. correlational statistics

D. statistics

4. It refers to the aggregates of people, objects, materials, etc. of any form.

A. population

B. sample

C. estimate

D. statistic

5. If you are interested with just a few members of the population to represent their

traits and properties, then these selected few members constitute a/an _____.

A. sample

B. aggregate

C. estimate

D. statistic

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6. This term refers to a property, trait or characteristic whereby the members of the

group vary or differ from one another.

A. variable

B. constant

C. measurement

D. none of the above

7. A variable which allows making of statements only of equality or difference among

the members of a group.

A. nominal variable

B. ratio variable

C. interval variable

D. ordinal variable

8. If you judge individuals according to their level of job satisfaction by ranking

them, the resulting variable is a/an ______.

A. nominal

B. ratio

C. ordinal

D. interval

9. The number of male students in a class is referred to a/an ________ variable.

A. interval

B. nominal

C. ordinal

D. ratio

10. Which of the following statements is not true about the uses of statistics?

A. It can predict the behavior of individuals like students, workers, schooladministrators, etc.

B. It can give a precise description of data.

C. It can be used to test a hypothesis in research.

D. It can be used to solve emotional problems.

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THE NATURE AND SCIENTIFIC DEFINITION

OF STATISTICS

The Nature of Statistics

The employment of statistics in mans various activities during the past several

centuries is said to be in a limited sense. Its usefulness was trapped basically in counting or

determining the number of events that have occurred at a certain period of time, birth rate,

mortality rate, etc.

In counting activities, events, things, etc., the measurements that are gathered are

referred to raw data. These data may be treated by statistical tools in order to relate,associate, or describe the data. In the method of description, the statistical tool to apply is

called descriptive statistics. In the method of relating and correlating two variables,

correlational statistics is utilized. Finally, in drawing generalizations regarding the

population from which the sample has been gathered, the tool to utilize is inferential

statistics.

Scientific Definition of Statistics

Statistics can be defined operationally. From the point of view of a researcher,

statistics is a science which deals with methods of collecting, gathering, presenting,

analyzing and interpreting data. The data gathering includes the collection of information

through questionnaires, observations, interviews, experiments, tests, etc. The information

are usually converted into numerical or quantitative data. The data collected can be

displayed through the use of graphs, tables, figures, and other ways of exhibiting the data.

There are two ways of presenting data in tabular form: In text or summary table or in

reference table. The text or summary table is usually found in the body of the research

work. The reference table is usually found in the appendices of the research work. The data

analysis is a procedure wherein the resolution of information takes place by the application ofstatistical principles. It involves the employment of any statistical method and the choice of

which depends largely upon the objectives of the research problem. After the analysis of

data has been undertaken, the results can be explained and interpreted. The findings of the

study will then be compared to the existing theories and earlier researches or studies in a

particular field.

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Activity 1.1

Consider the following research situations then specify the appropriate or the bestmanner of gathering data whetherinterview, questionnaire, experiment, observation, test,

etc.

1. Job Satisfaction of Public School Teachers

2. Emotionally Disturbed Grade School Children

3. Sexually Harrassed Adolescents

4. Effect of Modularized Instruction to Graduate Students Academic

Performance

5. Profile of the Faculty in Catholic Schools

6. Factors Affecting the Performance of Staff Nurses in the Rural

Areas

7. Comparative Study on the Various Instructional Strategies

Applied to Handicapped Learners

8. Diagnosing the Needs of Adult Learners in Tertiary Level

9. Development of Instructional Materials in Hydraulics

10. The Management Practices of Private School Principals in

Region I

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BRIEF HISTORICAL DEVELOPMENT OF STATISTICS

In the ancient times, statistics was utilized to provide information that pertains toactivities that include farming, collection of taxes, number of soldiers in a particular nation,

number of events that occurred in a particular period of time, agricultural crops and even in

athletic endeavors of man. The employment of statistics was later developed into an inferential

science sometimes in the sixteenth century. As an inferential science, it largely depended on the

theory of probability. The development continued through the researches made by the people in

various fields during the past 400 years.

The inclination of man into gambling led to the development of the probability theory.

During those times, the gamblers asked help from the mathematicians to teach them the

techniques on how to win the games. The requests for such techniques were considered by some

mathematicians among them were Pascal, Leibnitz, and James Bernoulli. It is very interesting tonote along this line that according to some winners of the Lotto game, the chances of winning is

attributed to the application of their knowledge ofprobability and statistics.

In relation to the historical development of statistics, De Moivre (1733) discovered the

equation for the normal distribution. The discovery of the said equation became the basis of

development in many theories of inferential statistics. The normal distribution which is a bell-

shaped distribution was also referred to as the Gaussian distribution. It was during this time

that the work ofLaplace became so popular because of the application of statistics to astronomy.

Another significant event in the development of statistics occurred when a Belgian

statistics named Adolph Quetelet (1796-1874) made an application of statistics in the field of

psychology and education. He was considered to be the first statistician to demonstrate the

statistical techniques derived in one area of research and applied to other areas.

Another statistician who contributed his knowledge of statistics in the field of social

sciences was Sir Francis Galton (1822-1911). The application of statistics to heredity and

eugenics was probably the most notable contribution of Galton to the development of statistics.

He also discovered the computation ofpercentiles. Along with Galton was Karl Pearson (1857-

1936) who exerted his efforts and cooperated with Galton in developing the theory of

correlation and regression. While Pearson was probably responsible for evolving the

theories of sampling at present.

Finally, at the rise of the twentieth century, William S. Gosset developed methods

for decision-making derived from smaller sets of data. Gosset worked in a brewery. He made

a study and published its results under the name student. He disguised his real name because

the brewery company which is owned by an Irish prohibited research since results of the study

might prove useful to its competitors. The idea of Gosset was continued by another

statistician named Sir Ronald Fisher (1890-1962) who was responsible for developing the

science of statistics for experimental designs.

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Activity 1.2

Fill in the blanks with the correct answer.

1. The inclination of man to gambling led to the early development of _________________.

2. ________________ discovered the equation for normal distribution upon which many of

the theories of inferential statistics have been based.

3. The normal distribution or the bell-shaped distribution was referred to _______________.

4. The work of Laplace gained popularity for it was about the application of statistics to

_____________________.

5. __________________ made an application of statistics in the field of psychology and

education.

6. The greatest contribution of Sir Francis Galton to the development of statistics was his

application of statistics in the field of _____________________.

7. Pearson was probably responsible for evolving the present theories of ______________.

8. _________________ developed methods for decision-making derived from smaller sets

of data.

9. __________________ developed statistics for experimental designs.

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SOME BASIC CONCEPTS USED IN STATISTICS

What is a sample?

Suppose you are interested to study the behavior of handicapped students in a class-

room situation. It will be very tedious if you will consider to select the thousands of this type

of students in a semester. Instead you will only consider some of them to be selected using an

appropriate sampling technique. The portion of the totality of the handicapped students is

referred to as sample.

What is a population?

The term population refers to the aggregates of things, objects, people, events, etc.

This could be population of students, engineers, accountants, school administrators, etc. In

research, the concern is to look at the properties of the aggregate or group rather than the

characteristics of each member.

What is a constant?

The word constant refers to a property whereby the members of a particular sample or

an aggregate do not differ from one another. For instance, a particular sex, say male, is a

constant because the members do not differ.

What is a variable?

The term variable refers to a property whereby the members of an aggregate differ

from one another. Thus, members of the group may vary or differ in the color of eyes, height,

weight, civil status, etc.

What are the levels of measurement of a variable?

There are four levels of measurement.

1. Nominal variable: This variable refers to a characteristic or property of the members of the

group or aggregate defined by an operation which allows making of statements only of

equality or difference. We can say that a member is different or the same compared to another

member of the group. For instance, two male students are the same in sex while another two

males may be different in height and weight.

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2. Ordinal variable: This variable refers to a property or characteristic wherein the members of

a group are compared say, one is greater than the other or one is less than the other member.

Ranking students based on the results of their midterm examinations, will always have the

first, the second, third and so on. In this case, the first in rank is higher than those who

obtained other ranks.

3. Interval variable: This refers to a property or characteristic defined by an operation which

allows making of statements of equality rather than statements of greater than or less than and

sameness or difference. An interval variable does not have a true zero point.

4. Ratio variable: This refers to a property defined by an operation which allows making of

statement of equality and of ratios. This means that one value may be thought of as five times

another, triple of a certain number, and so on. The measurements in the ratio variable are made

from an arbitrary zero point.

Activity 1.3

A. Identify the concept: write your answer on the blank before the number.

1. The aggregates of objects, events, people, etc.

2. The representatives of an aggregate of handicapped learners in the

tertiary level.

3. It refers to a property or trait whereby the members of the group do

not differ from one another.

4. It refers to a characteristic or property whereby the members of a

group vary or differ from one another.

5. The level of measurement wherein the property of members in a

group are considered in terms of sameness or difference.

6. The scale of measurement of a variable wherein the characteristic

or property of members in a particular aggregate say individuals

are ranked.

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B. Write the level of measurement that corresponds to the variable in each item. Write your

answer on the blank before the number.

7. Second born and fifth born child in a family.

8. Frequencies of passing and failing the course in Research.

9. Performance of 50 students in Statistics test.

10. Socio-economic status of 30 subjects in a class.

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THE USES OF STATISTICS

Statistics has an indespensable role particularly in the field of research. It enables a

researcher to make a flawless and accurate statements of judgements about a relationship of

two or more variables. For instance, describing the academic performance of the students in

terms of the computed mean, standard deviation, correlation in relation with another factor of

academic performance results. Thus, statistics can be utilized to give a precise description of

data.

In an educational research, the academic performance can be predicted through the

result of an entrance tests such as aptitude test, personality test, etc. An instructors work

performance can also be predicted through the results of teacher inventory test. In thislight, statistics is useful in predicting the behavior of individuals.

In order to determine the relationship between two or more variables, an appropriate

statistical measure must be utilized. For instance, a correlational study may employ

statistical measures such as t-test, chi-square test, F-test, and others. With this purpose,

statistics can be used to test a hypothesis.

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Activity 1.4

What do you think are other uses of statistics? Enumerate at least 5. Explain youranswer.

1.

2.

3.

4.

5.

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POST TEST 1

I. Fill in the blanks with the correct answer:

1. In the method to relate or associate two variables, the measure to apply is known as

_________________________.

2. To make generalizations about the population from which the sample has been drawn, the

measure to use is known as _________________________.

3. ___________________ involves getting information with the employment of interviews,

questionnaires, observations, psychological tests, etc.

4. ______________ is the resolution of information into simpler elements by the application of

statistical principles.

5. The most notable contribution of Sir Francis Galton to the development of statistics was the

application of statistics to heredity and eugenics and his discoveries by ____________.

6. The equation for the normal distribution was discovered by ____________________.

7. ______________ refers to the groups or aggregates of people, events, materials, etc. of anyform.

8. ______________ refers to a property whereby the members of a group or aggregate do not

differ from one another.

9. The measures of the population are called _______________________.

10. ______________ refers to the properties or characteristics whereby the members of the group

or aggregate vary or differ from one another.

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II. Write the level of measurement that corresponds to the variable in each item. Write your

answer on the blank before the number.

1. Third born and fifth born child.

2. High and low scores in Statistics test

3. Performance of boys and girls in an aptitude test.

4. Color preferences of adults in Cebu.

5. Failing and passing in a qualifying examination.

6. Frequencies of strongly agree and strongly disagree responses tothe creation of E-vat.

7. Performance of 100 examinees in PBET.

8. Positions or ranks of graduate students on a social adjustment

scale

9. The valedictorian and salutatorian in a graduating class.

10. Number of students who are in favor of the creation of the law on

sexual harrassment.

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MODULE 3 GRAPHIC PRESENTATION OF DATA ........................ 29

Lesson 3.1 - Presenting Data in a Pie Chart & Bar Graph ......................... 33

Lesson 3.2 - Presenting Data in a Frequency Polygon ................................ 36

MODULE 4 MEASURES OF CENTRAL TENDENCY .................... 44

Lesson 4.1 - The Mean for the Ungrouped Data ........................................ 47

Lesson 4.2 - The Mean of the Grouped Data .............................................. 51

Lesson 4.3 - The Median of Ungrouped and Grouped Data ....................... 53Lesson 4.4 - The Mode of the Ungrouped and Grouped Data .................... 58

MODULE 5 MEASURES OF VARIATION ......................................... 64

Lesson 5.1 - The Range and the Mean Deviation ....................................... 66

Lesson 5.2 - The Variance and Standard Deviation .................................... 69

TITLE

MODULE 1

TABLE OF CONTENTS

page

MODULES IN STATISTICS

STATISTICS AS A SCIENCE ....................................

Lesson 1.1 - The Nature and Science Definition of Statistics .......................4

Lesson 1.2 - Brief Historical Development of Statistics ..............................6

Lesson 1.3 - Some Basic Concepts Used in Statistics ................................. 8

Lesson 1.4 - The Uses of Statistics ............................................................. 11

Lesson 2.1 - Frequency Distribution of Nominal Data ..............................

Lesson 2.2 - Frequency Distributions of Nominal,

Ordinal and Interval Data ....................................................

18

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Lesson 8.1 - Understanding Analysis of Variance ............................ 126

Lesson 8.2 - The Anova Using The Raw Score Method ........................... 134

MODULE 9 MEASURES OF CORRELATION ................................. 143

Lesson 9.1 - The Concept of Correlation, Coefficients of Correlation ........ 145

Lesson 9.2 - Simple Linear Regression In A Scatter Diagram ................... 147

Lesson 9.3 - Correlation Between Interval Data ......................................... 149

Lesson 9.4 - Correlation Between Ordinal Data ......................................... 154

Lesson 9.5 - Correlation Between Nominal Data ....................................... 159

MODULE 10 THE CHI-SQUARE TEST ............................................... 163

Lesson 10. - Characteristics of the Chi-Square Test ................................... 165

Lesson 10.2- Test of Goodness of Fit .......................................................... 167

Lesson 10.3- Chi-Square Test of Independence .......................................... 172

BIBLIOGRAPHY

page

MODULE 6 NORMAL PROBABILITY DISTRIBUTION ................ 78

Lesson 6.1 - Characteristics of a Normal Probability Distribution .............. 81

Lesson 6.2 - Areas Under the Normal Curve .............................................. 84

Lesson 6.3 - The Applications of the Standard Normal Distribution .......... 89

MODULE 7 TESTING DIFFERENCES BETWEEN MEANS ...........

Lesson 7.1 - The Null Hypothesis: No Difference Between Means

The Research Hypothesis: A Difference Between Means ..

Lesson 7.2 - Directional and Non-Directional Hypothesis ........................

Lesson 7.3 - The Steps in Hypothesis Testing ................................

Lesson 7.4 - The T-Tests of Significance ....................................................

MODULE 8 ONE FACTOR ANALYSIS OF VARIANCE .................

98

101

104

108

111

123