Upcoming SlideShare
×

# Statistics Module 2 & 3

23,913 views

Published on

statistics

Published in: Education, Technology
10 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• brilliant reference

Are you sure you want to  Yes  No
• An Excellent Collection of Statistical Excercises

Are you sure you want to  Yes  No
Views
Total views
23,913
On SlideShare
0
From Embeds
0
Number of Embeds
17
Actions
Shares
0
1,526
2
Likes
10
Embeds 0
No embeds

No notes for slide

### Statistics Module 2 & 3

1. 2. MASTER OF ARTS IN NURSING MAN602 Statistical Methods in Nursing
2. 3. INTRODUCTION Undoubtedly, statistics is a very useful tool in the various 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 STATISTICS AS A SCIENCE 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 field of studies, professional endeavors, and even for personal profit. You will then understand the nature and meaning of statistics, its brief historical development, the difference between sample and population, the meaning and kinds of variables, and the importance of statistics especially in the field of research.
3. 4. <ul><li>GENERAL OBJECTIVES </li></ul><ul><li>At the end of this module, you are expected to: </li></ul><ul><li>State the nature and scientific definition of statistics; </li></ul><ul><li>Trace the brief historical development of statistics; </li></ul><ul><li>Distinguish sample from a population; </li></ul><ul><li>Enumerate and differentiate the kinds of variables; and </li></ul><ul><li>Explain the uses of statistics. </li></ul><ul><li>TIME FRAME 3 Hours </li></ul>
4. 5. PRE-TEST 1 Test your knowledge if the basic ideas in statistics. As much as possible, avoid guessing. At any rate, this test in 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
5. 6. ______ 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 statistics B. inferential statistics C. correlational statistics D. statistics
6. 7. _____ 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
7. 8. _____ 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 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
8. 9. B. Ratio C. Ordinal D. Interval _____ 9. The number of make students in a class is referred to a/an __________ variable. B. Nominal C. Ordinal D. Ratio ______ 10. Which of the following statements is not true about the uses of Statistics? A. Interval A. It can predict the behavior of individuals like students, workers, school administrations, etc. B. It can give precise description of data.
9. 10. C. It can be used to test a hypothesis in research. D. It can be used to solve emotional problems. LESSON 1.1 THE NATURE AND SCIENTIFIC DEFINITION OF STATISTICS The Nature of Statistics The employment of statistics in man’s 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
10. 11. 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 or relation and correlation 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 the methods of collecting, gathering, presenting, analyzing and interpreting data. The data gathering includes the collection of information through questionnaires, observations, interviews, experiments, test, etc. The information are usually converted into numerical or quantitative data. The data collected can be displayed
11. 12. through the use of graphs, tables, figures and other ways of exhibiting the data. There are two ways of presenting data in tabular form. 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 the information takes place by application of statistical 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.
12. 13. Activity 1.1 Consider the following research situations then specify the appropriate or the best manner of gathering data whether interview, questionnaire, experiment, observation, test, etc. ________ 1. Job Satisfaction of Public School Teachers ________ 2. Emotionally Disturbed Grade School Children ________ 3. Sexually Harassed 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
13. 14. _____ 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 XII
14. 15. LESSON 1. 2 BRIEF HISTORICAL VELOPMENTS OF STATISTICS In the ancient times, statistics was utilized to provide information that pertains to activities 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
15. 16. by some mathematics among them were Pascal, Leibnitz, and James Bernoulli. It is very interesting to note along this line that according to some winners of the Lotto game, the chances of winning is attributed to the application of their knowledge of probability and statistics . In relation the historical development of statistics, De Moivre (1773) discovered the equation for the normal distribution . The discovery of the said equation became the basis of the 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 of Laplace became so popular because of the application of statistics to astronomy.
16. 17. Another significant event in the development of statistics occurred when a Belgian statistician 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 of percentiles . Along with Galton was Karl Pearson (1857-1936) who exerted 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.
17. 18. Finally, at the rise of the twentieth century, William S. Gosset developed method 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 science of statistics for experimental designs.
18. 19. <ul><li>Activity 1.2 </li></ul><ul><li>Fill in the blanks with the correct answer. </li></ul><ul><li>The inclination of man to gabling led to the early development of _______________. </li></ul><ul><li>________________ discovered the equation for normal distribution upon which man of the theories of inferential statistics have been based. </li></ul><ul><li>The normal distribution or the bell-shaped distribution was referred to ______________. </li></ul><ul><li>The work of Laplace gained popularity for it was about the application of statistics to _______________. </li></ul><ul><li>______________ made an application of statistics in the field of psychology and education. </li></ul><ul><li>The greatest contribution of Sir Francis Galton to the development of statistics to _______________. </li></ul>
19. 20. <ul><li>Pearson was probably responsible for evolving the present theories of ________________. </li></ul><ul><li>_____________ developed methods for decision-making derived from smaller sets. </li></ul><ul><li>_____________ developed statistics for experimental designs. </li></ul><ul><li>LESSON 1.3 SOME BASIC CONCEPTS USED </li></ul><ul><li>IN STATISTICS </li></ul><ul><li>What is a sample? </li></ul><ul><li>Suppose you are interested to study the behavior of handicapped students in a classroom 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 student is referred to as sample. </li></ul>
20. 21. 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 the research, the concern is to look at the properties of the aggregate or group rather than the characteristic of each member. What is a constant ? The word constant refers to a property whereby the members of a particular sample or 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 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.
21. 22. <ul><li>What are the levels of measurement of a variable? </li></ul><ul><li>There are four levels of measurement. </li></ul><ul><li>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 compare 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. </li></ul><ul><li>Ordinal variable: This variable refers to a property or characteristic wherein the members of a group are compared say, one is greater that the other or one is less than the other member. Ranking students based on the results of the midterm examinations, will always have the first, second, third and so on. In this case, the first in rank is higher than those who obtained other ranks. </li></ul>
22. 23. 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 that 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 or equality and 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.
23. 24. 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 representative 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 of 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 characteristics or property of members in a particular aggregate say individuals are ranked.
24. 25. B. Write the level of measurement that corresponds to the variable in each item. Write your answer on the blank before each number. ______ 7. Second born and fifth born child in a family. ______ 8. Frequencies pf 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. LESSON 1.4 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 statement of judgment 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.
25. 26. 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 instructor’s work performance can also be predicted through the results of teacher inventory test. In this light, 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.
26. 27. <ul><li>Activity 1.4 </li></ul><ul><li>What do you think are other uses of statistics? Enumerate at least 5. Explain your answer. </li></ul><ul><li>_________________________________________________________________________________________________________________________________________________________ </li></ul><ul><li>_________________________________________________________________________________________________________________________________________________________ </li></ul><ul><li>_________________________________________________________________________________________________________________________________________________________ </li></ul><ul><li>_________________________________________________________________________________________________________________________________________________________ </li></ul>
27. 28. <ul><li>5. ___________________________________________________ </li></ul><ul><li>___________________________________________________ </li></ul><ul><li>___________________________________________________ </li></ul><ul><li>POST TEST 1 </li></ul><ul><li>1. Fill in the blanks with the correct answer: </li></ul><ul><li>In the method to relate or associate two variables, the measure to apply is known as ___________________. </li></ul><ul><li>To make generalizations about the population from which the sample has been drawn, the measure to use is known as __________________. </li></ul><ul><li>_______________ involves getting information with the employment of interviews, questionnaires, observations, psychological tests, etc. </li></ul>
28. 29. 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 any form. 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.
29. 30. <ul><li>II. Write the level of measurement that corresponds to the variable in each item. Write your answer on the blank before the number. </li></ul><ul><li>___________ 1. Third born and fifth born child. </li></ul><ul><li>___________ 2. High and low scores in Statistics test. </li></ul><ul><li>___________ 3. Performance of boys and girls in an aptitude test. </li></ul><ul><li>____________ 4. Color preference of adults in Cebu. </li></ul><ul><li>____________ 5. Failing and passing in a qualifying </li></ul><ul><li>examination. </li></ul><ul><li>____________ 6. Frequencies of strongly agree and strongly disagree responses to the creation of E-vat. </li></ul>
30. 31. <ul><li>7. __________Performance of 100 examinees in PBET. </li></ul><ul><li>8. __________Positions or ranks of graduate students on a social adjustment scale. </li></ul><ul><li>9. __________The valedictorian and salutatorian in a graduating class. </li></ul><ul><li>10. _________Number of students who are in favor of the creation of the law on sexual harassment. </li></ul>
31. 32. ORGANIZING THE DATA Introduction: The collection of data entails a serious effort on the researcher. In doing so, the researcher must have good foresight, careful planning, and systematic organization of activities. The completion of data collection is not the end of the researcher’s task. The data must be analyzed using appropriate statistical tool or treatment. From the analysis of data, results will be obtained and test of hypotheses will be done. This module introduces the concepts of frequency, frequency, distribution, midpoint, class interval, proportion and percentage, cumulative percentage, raw data, etc. A clearer understanding of these concepts will enable researchers to construct frequency distribution for the different levels of measurement.
32. 33. <ul><li>GENERAL OBJECTIVES </li></ul><ul><li>At the end of this module, you are expected to: </li></ul><ul><li>Define or describe the following terms: </li></ul><ul><li>a. frequency </li></ul><ul><li>b. frequency distribution </li></ul><ul><li>c. cumulative frequency </li></ul><ul><li>d. cumulative percentage </li></ul><ul><li>e. midpoint </li></ul><ul><li>f. class interval </li></ul><ul><li>g. proportion and percentage </li></ul><ul><li>h. raw data </li></ul><ul><li>2. Construct at frequency distribution of the different levels of measurement. </li></ul>
33. 34. <ul><li>Pre-Test </li></ul><ul><li>Direction: Choose the letter of the correct answer. Write your answer on the blank before each number. </li></ul><ul><li>____1. The variants which have not been organized or classified in any way, and which are often recorded in the order observed. </li></ul><ul><li>a. raw data </li></ul><ul><li>b.transmuted data </li></ul><ul><li>c. raw scores </li></ul><ul><li>d. none of these </li></ul><ul><li>____2. The distribution obtained by a simple process of successive addition of the entries in the frequency column. </li></ul><ul><li>a. relative frequency b. cumulative frequency </li></ul><ul><li>c. frequency d. cumulative percentage </li></ul>
34. 35. <ul><li>____3. It is obtained by dividing each entry in the cumulative frequency column by N and multiplying by 100% </li></ul><ul><li>a. cumulative frequency c. relative frequency </li></ul><ul><li>b. cumulative percentage d. frequency </li></ul><ul><li>____4. It is obtained by adding the upper class limit and the lower class limit and dividing the sum by 2. </li></ul><ul><li>a. class interval c. class boundary </li></ul><ul><li>b. midpoint d. none of these </li></ul><ul><li>____5. A frequency distribution is defined as ______ </li></ul><ul><li>a. an orderly arrangement of data using arbitrarily defined classes or groupings and associated frequencies. </li></ul><ul><li>b. an arrangement of numbers using a standard set of groupings. </li></ul><ul><li>c. a classification of data showing the frequency of occurrence of the variants. </li></ul><ul><li>d. an arrangement of a set of raw data from highest to lowest or vice versa. </li></ul>
35. 36. <ul><li>_____6. The number of values or cases which is found in a class interval or category is called ______. </li></ul><ul><li>a. class interval c. probability </li></ul><ul><li>b. frequency d. data/scores </li></ul><ul><li>_____7. The midpoint of class interval 15-19 is _____ </li></ul><ul><li>a. 16 c. 18 </li></ul><ul><li>b. 17 d. 19 </li></ul><ul><li>_____8. What is the cumulative percentage of the frequency in the class interval 30-34 from the table below? </li></ul><ul><li> Class interval f </li></ul><ul><li>15-19 3 </li></ul><ul><li>20-24 15 </li></ul><ul><li>25-29 2 </li></ul><ul><li>30-34 10 </li></ul>
36. 37. <ul><li>a. 33.3% c. 23% </li></ul><ul><li>b. 30% d. 20% </li></ul><ul><li>______9. The table below shows an example of the frequency distribution under ______ </li></ul><ul><li>Religion Frequency </li></ul><ul><li>Protestant 15 </li></ul><ul><li>INC 12 </li></ul><ul><li>Catholic 10 </li></ul><ul><li>a. nominal c. interval </li></ul><ul><li>ratio d. ordinal </li></ul><ul><li>______10. The table below shows an example of frequency distribution under ________data. </li></ul>
37. 38. <ul><li>Class interval f. % </li></ul><ul><li> 15-19 10 22.2% </li></ul><ul><li> 20-24 12 26.6% </li></ul><ul><li> 25-29 8 17.7% </li></ul><ul><li> 30-34 5 11.1% </li></ul><ul><li> 35-39 10 22.2% </li></ul><ul><li>a. nominal c. interval </li></ul><ul><li>b. ordinal d. ratio </li></ul>
38. 39. <ul><li>FREQUENCY DISTRIBUTION OF NOMINAL DATA </li></ul><ul><li>A tailor transforms raw cloth material into suits; a chef converts raw food material into more palatable versions, served at the dinner table. On the same manner, the researcher aided by formulas and techniques, can transform raw data into an organized set of measures which can be used to test hypotheses. </li></ul><ul><li>What can a researcher do to organize the gathered data? How to transform the raw data into an easy-to-understand summary form? Perhaps, the first step will be the construction of a frequency distribution in a form of table. </li></ul><ul><li>Frequency distribution of nominal data has the following characteristics or properties. These characteristics are also held true to other levels of measurement of data. </li></ul>
39. 40. <ul><li>Table number – it facilitates the reader to locate the table easily </li></ul><ul><li>Title – it gives the reader an idea as to the nature of the data being presented. </li></ul><ul><li>Example: Frequency Distribution of Nominal Data </li></ul><ul><li>Table I Sex of Students Majoring in Psychology </li></ul><ul><li>Sex of Students Frequency (f) </li></ul><ul><li>Male 74 </li></ul><ul><li>Female 26 </li></ul><ul><li>Total 100 </li></ul><ul><li>Table 1, the left-hand column indicates what characteristic is being presented ( sex of students ) and contains the categories of analysis ( male and female ). An adjacent column headed </li></ul>
40. 41. <ul><li>“ Frequency” or “f” and indicates the number of cases in each category (74 and 26 , respectively) as well as the total number of cases (N=100). A glance in the frequency distribution in Table I clearly shows that more males are majoring in Psychology. </li></ul><ul><li>Comparing Distributions </li></ul><ul><li>Comparing the number of students majoring in Psychology at PWU Taft and those who are majoring in the same course at PWU Q.C. requires statistical knowledge and comparison. Thus, making comparisons between frequency distributions is a procedure often used to clarify results and add information. </li></ul><ul><li>Recalling the hypothetical example above mentioned, you might ask: Are majors in Psychology at PWU Taft more likely to be male than at PWU QC? To provide an answer, you might compare the students majoring at PWU Taft and at PWU QC in the said major course. </li></ul>
41. 42. <ul><li>Take a look at this example: </li></ul><ul><li>Table 2 SEX OF STUDENTS MAJORING IN PSYCHOLOGY AT PWU TAFT AND PWU QC </li></ul><ul><li> Psychology Majors </li></ul><ul><li> PWU Taft PWU QC </li></ul><ul><li>Sex of Students f f </li></ul><ul><li>Male 80 70 </li></ul><ul><li>Female 20 30 </li></ul><ul><li>Total 100 100 </li></ul>
42. 43. <ul><li>As shown in the table, 80 out of 100 who are majoring in Psychology at PWU Taft, and 70 out of 100 who are also majoring in Psychology at PWU QC, are males. Thus, males predominate among Psychology majors in both schools, this tendency is more pronounced at PWU Taft. </li></ul><ul><li>One must be careful in making comparisons which require data that are not presented. </li></ul><ul><li>Proportions and Percentages </li></ul><ul><li>When the researcher studies distributions of equal total size, the frequency data can be used to make comparisons between the groups. Thus, the number of males majoring in Psychology at PWU Taft and QC can be directly compared, because there are exactly 100 students majoring on each campus. It is generally not possible, however, to study distributions having number of cases. For instance, how to make sure that precisely 100 students at both universities will decide to major in Psychology? </li></ul>
43. 44. <ul><li>For more general use, there is a method of standardizing frequency distributions for size. A way to compare groups despite differences in total frequencies. There are two popular and useful methods of standardizing for size and comparing distributions. These are the proportion and the percentage. </li></ul><ul><li>The proportion compares the number of cases in a given category with the total size of the distribution. Any frequency can be converted to a proportion P by dividing the number of cases in any category f by the total number of cases in the distribution N . The formula is: </li></ul><ul><li>f </li></ul><ul><li>P = N </li></ul>
44. 45. <ul><li>Despite the usefulness of the proportion, may people prefer to indicate the relative size of a series of numbers in terms of the percentage, the frequency of occurrence of a category per 100 cases . To calculate a percentage, the formula is. </li></ul><ul><ul><ul><ul><ul><li>f </li></ul></ul></ul></ul></ul><ul><li>% = (100) </li></ul><ul><li>N </li></ul><ul><li>Activity 2.1. </li></ul><ul><li>Given the following data, construct a frequency distribution in a form of table. Always have a table number and title for your frequency distribution. Use the space provided for your answer. </li></ul><ul><li>1. In the School Year 1995-1996, there were 250 Education students majoring in Mathematics at Philippine Normal University. Out of that total number of Education students, 105 were males. </li></ul>
45. 46. <ul><li>2. A researcher tried to compare the number of enrollees in two universities for the School Year 1993-1994. There were 350 male and 650 females who enrolled at University Y. At University A, there were 250 male enrollees and 750 were females. </li></ul><ul><li>3. The proportions of males who were majors of Mathematics at PNU. </li></ul><ul><li>4. The proportion of females who were majors of Mathematics at PNU. </li></ul><ul><li>5. The percentage of male enrollees in University Y for the School Year 1993-1994. </li></ul><ul><li>6. The percentage of females who were of majors of Mathematics at PNU. </li></ul><ul><li>7. The percentage of female enrollees in University A for the School Year 1993-1994. </li></ul>
46. 47. <ul><li>8. The proportion of male enrollees in University A for the School Year 1993-1994. </li></ul><ul><li>9. The proportion of female enrollees in University Y for the School Year 1993-1994. </li></ul><ul><li>10. The percentage of males who were majors of Mathematics at PNU. </li></ul><ul><li>PREQUENCY DISTRIBUTIONS OF NOMINAL, ORDINAL AND INTERVAL DATA </li></ul><ul><li>Nominal data are labeled rather than scaled. The categories of nominal-level distributions do not have to be listed in any particular order. </li></ul><ul><li>The following table shows an example of ordinal data presented in three different, but equally acceptable arrangements. </li></ul>
47. 48. <ul><li>Table 3 THE DISTRIBUTION OF RELIGIOUS PREFERENCES </li></ul><ul><li>(SHOWN IN THREE WAYS) </li></ul><ul><li>Religion f Religion f Religion f </li></ul><ul><li>Protestant 35 Catholic 50 Iglesia 20 </li></ul><ul><li>Catholic 50 Iglesia 20 Catholic 50 </li></ul><ul><li>Iglesia 20 Protestant 35 Protestant 35 </li></ul><ul><li>Total 105 Total 105 Total 105 </li></ul>
48. 49. <ul><li>In contrast, the score values in ordinal or interval distributions represent the degree to which a particular characteristic is present. The ordinal and interval categories are always arranged in order. The arrangement is usually from highest to lowest values. </li></ul><ul><li>The table below shows an example of interval data arranged correctly and incorrectly. </li></ul>
49. 50. <ul><li>Table 4 FREQUENCY DISTRIBUTION OF ATTITUDES TOWARD E-VAT ON A COLLEGE CAMPUS: INCORRECT AND CORRECT PRESENTATIONS </li></ul><ul><li>Attitude Toward E-Vat f Attitude Toward E-Vat f </li></ul><ul><li>Slightly favorable 2 Strongly favorable 0 </li></ul><ul><li>Somewhat unfavorable 11 Somewhat favorable 1 </li></ul><ul><li>Strongly favorable 0 Slightly favorable 2 </li></ul><ul><li>Slightly unfavorable 6 Slightly unfavorable 6 </li></ul><ul><li>Strongly unfavorable 20 Somewhat unfavorable 11 </li></ul><ul><li>Somewhat favorable 1 Strongly unfavorable 20 </li></ul><ul><li> Total 40 Total 40 </li></ul><ul><li>INCORRECT CORRECT </li></ul><ul><li>Which version do you find easier to read? </li></ul>
50. 51. <ul><li>Grouped Frequency Distribution of Interval Data </li></ul><ul><li>Interval-level scores are sometimes spread over a wide range, making the resultant simple frequency distribution long and difficult to read, but it can be presented clearly by considering the separate scores into a number of smaller groups. Each group or category in a grouped frequency distribution is known as a class interval , whose size is determined by the number of score values it contains. </li></ul><ul><li>Class Limits </li></ul><ul><li>In accordance with its size, each class interval has an upper limit and a lower limit. The highest and lowest scores in any category seem to be the limits. Thus, we might say that the upper and lower limits of 50-54 are 54 and 50, respectively. Class boundaries are located at the point halfway between adjacent class intervals and, therefore, serve to close the gap between them. Thus, the upper boundary of the class interval 40-44 is 44.5 and the lower boundary of the class interval 45-49 is also 44.5. </li></ul>
51. 52. <ul><li>The Class Midpoint </li></ul><ul><li>Another characteristic of a class interval is its midpoint (m). It is the middlemost score value in the class interval. As an example, the midpoint of the class interval 55-59 is 57 as illustrated below: </li></ul><ul><li>lowest + highest score </li></ul><ul><li>m = </li></ul><ul><li>2 </li></ul><ul><li>55 + 59 </li></ul><ul><li>= </li></ul><ul><li> 2 </li></ul><ul><li>m = 57 </li></ul><ul><li>Therefore the midpoint of 55-59 is 57. </li></ul>
52. 53. <ul><li>Determining the Number of Intervals </li></ul><ul><li>In presenting interval data in a grouped frequency distribution, the researcher considers the number of categories to employ. Textbooks generally use as few as 3 or 4 intervals to as many as 20 intervals. In this sense, it would be wise to remember that grouped frequency distributions are utilized to emphasize a group pattern. </li></ul><ul><li>Cumulative Distributions </li></ul><ul><li>It is sometimes desirable to display frequencies in a cumulative manner especially in locating a position of one case relative to overall group performance. </li></ul><ul><li>Cumulative frequencies (cf) are defined as the total number of cases having any given score or a score that is lower. Thus, the cumulative frequency (cf) of any group or category (or class interval) is obtained by adding the frequency in that category to the total frequency for all category below it. </li></ul>
53. 54. <ul><li>The table below shows an example of a frequency distribution of interval data. </li></ul><ul><li>Table 5 GROUPED FREQUENCY DISTRIBUTION OF MIDTERM EXAMINATION SCORES FOR 50 STUDENTS IN STATISTICS WITH COMPUTER APPLICATIONS </li></ul><ul><li>Class Interval f % m cf </li></ul><ul><li>85-89 5 10 87 50 </li></ul><ul><li>80-84 5 10 82 45 </li></ul><ul><li>75-79 10 20 77 35 </li></ul><ul><li>70-74 5 10 72 30 </li></ul><ul><li>65-69 8 16 67 25 </li></ul><ul><li>60-64 7 14 62 17 </li></ul><ul><li>55-59 10 20 57 10 </li></ul><ul><li>Total 50 100% </li></ul>
54. 55. <ul><li>Cumulative percentage </li></ul><ul><li>In addition to cumulative frequency, you can also construct a distribution which indicates cumulative percentage (c%), the percent of case having any score or a score that is lower. To calculate cumulative percentage, the following formula can be employed: </li></ul><ul><li>cf </li></ul><ul><li>c% = (100%) </li></ul><ul><li>N </li></ul><ul><li>where </li></ul><ul><li>cf = the cumulative frequency in any category </li></ul><ul><li>N = the total number of cases in the distribution </li></ul><ul><li>Consider the following table and study how the cumulative percentage are being obtained. </li></ul>
55. 56. <ul><li>Table 6 CUMULATIVE PERCENTAGE DISTRIBUTION OF TEST SCORES IN STATISTICS FOR 50 STUDENTS </li></ul><ul><li>Class interval f cf c% </li></ul><ul><li>35-39 8 8 6 </li></ul><ul><li>40-44 5 13 10 </li></ul><ul><li>45-49 7 20 14 </li></ul><ul><li>50-54 12 32 24 </li></ul><ul><li>55-59 10 42 20 </li></ul><ul><li>60-64 5 47 10 </li></ul><ul><li>65-69 3 50 6 </li></ul><ul><li> Total N = 50 100% </li></ul>
56. 57. <ul><li>To illustrate, the cumulative percentage for the class interval 65-69 is obtained as shown in the following solution: </li></ul><ul><li>c% = (100%) 3 </li></ul><ul><li> 50 </li></ul><ul><li> = (100%) .60 </li></ul><ul><li> = 6% </li></ul><ul><li>Activity 2.2. </li></ul><ul><li>1. Construct a table for the following distribution of club preferences by 100 students in tertiary level: </li></ul>
57. 58. <ul><li>Club Frequency </li></ul><ul><li>Sports 30 </li></ul><ul><li>Mission 10 </li></ul><ul><li>Religious 5 </li></ul><ul><li>Math & Science 15 </li></ul><ul><li>Drama 40 </li></ul><ul><li>Total 100 </li></ul>
58. 59. <ul><li>2. Construct a comprehensive tabular presentation of the following nominal data taken from the opinions of 200 students about the creation of sexual harassment. </li></ul><ul><li>Level of agreement and disagreement Frequency </li></ul><ul><li>Strongly agree 75 </li></ul><ul><li>Agree 47 </li></ul><ul><li>Uncertain 23 </li></ul><ul><li>Disagree 30 </li></ul><ul><li>Strongly disagree 25 </li></ul><ul><li>Total 200 </li></ul>
59. 60. <ul><li>3. Complete the table below to be able to construct a frequency distribution based on the following scores obtained in Statistics test by 50 students. </li></ul><ul><li>SCORES: </li></ul><ul><li>70 64 60 55 48 50 41 55 41 </li></ul><ul><li>65 55 50 50 75 80 59 58 60 </li></ul><ul><li>85 50 65 60 41 70 50 41 50 </li></ul><ul><li>60 75 80 55 49 80 52 68 90 </li></ul><ul><li>90 70 60 59 55 65 50 53 74 </li></ul>
60. 61. <ul><li>Class interval Tally f % M cf c% </li></ul><ul><li>41-45 </li></ul><ul><li>46-50 </li></ul><ul><li>51-55 </li></ul><ul><li>56-60 </li></ul><ul><li>61-65 </li></ul><ul><li>66-70 </li></ul><ul><li>71-75 </li></ul><ul><li>76-80 </li></ul><ul><li>81-85 </li></ul><ul><li>86-90 </li></ul>
61. 62. <ul><li>POST TEST 2 </li></ul><ul><li>From the table below representing the academic performance of students from a rural area and an urban area, find the </li></ul><ul><li>1.1. Percent of students from a rural area whose level of achievement is high </li></ul><ul><li>1.2. Percent of students from an urban area whose level of achievement is high </li></ul><ul><li>1.3. Proportion of students from a rural area whose level of achievement is high </li></ul><ul><li>1.4. Proportion of students from an urban area whose level of performance is low </li></ul>
62. 63. <ul><li>ACADEMIC PERFORMANCE OF STUDENTS FROM A RURAL AREA AND AN URBAN AREA </li></ul><ul><li>AREA </li></ul><ul><li>Achievement Level Rural Urban </li></ul><ul><li> f f </li></ul><ul><li>High 83 146 </li></ul><ul><li>Low 140 227 </li></ul><ul><li>Total 223 373 </li></ul>
63. 64. <ul><li>2. Convert the following distribution of scores into a grouped frequency distribution containing five class intervals, and </li></ul><ul><li>2.1. Determine the size of class intervals </li></ul><ul><li>2.2. Indicate the upper and lower limits of each class interval </li></ul><ul><li>2.3. Identify the midpoint of each class interval </li></ul><ul><li>2.4. Find the percentage for each class interval </li></ul><ul><li>2.5. Find the cumulative frequency for each class interval </li></ul><ul><li>2.6. Find the cumulative percentage for each class interval </li></ul>
64. 65. <ul><li>SCORE VALUE frequency </li></ul><ul><li>21 3 </li></ul><ul><li>20 4 </li></ul><ul><li>19 2 </li></ul><ul><li>18 2 </li></ul><ul><li>17 1 </li></ul><ul><li>16 5 </li></ul><ul><li>15 3 </li></ul><ul><li>14 2 </li></ul><ul><li>13 1 </li></ul><ul><li>12 1 </li></ul><ul><li>11 3 </li></ul><ul><li>10 2 </li></ul><ul><li>9 3 </li></ul>
65. 66. <ul><li>8 4 </li></ul><ul><li>7 3 </li></ul><ul><li>6 2 </li></ul><ul><li>5 4 </li></ul><ul><li>4 5 </li></ul><ul><li>3 2 </li></ul><ul><li>2 3 </li></ul><ul><li>N = 55 </li></ul>
66. 67. <ul><li>GRAPHIC PRESENTATION OF DATA </li></ul><ul><li>(MODULE THREE) </li></ul><ul><li>Introduction: </li></ul><ul><li>The previous module dealt with the organization of data employing the tabular form. With this of approach, columns of numbers can evoke fear, boredom, a partly, and misunderstanding on the part of the reader. While some people seem to “tune out” statistical information presented in tabular form, they may pay close attention to the same data presented in graphic or picture form. In fact, with the access of people to computer, the graphic or picture form of organizing and presenting data can easily be prepared. As a result, many researchers and authors prefer to use graphs as opposed to tables. For similar reasons, researchers often use visual aids as pie charts, bar graphs, frequency polygons, and line graphs in an effort to increase and ensure the readability of findings. </li></ul>
67. 68. <ul><li>Objective: </li></ul><ul><li>At the end of this module, you are expected to: </li></ul><ul><li>Organize and present data in graphic presentation such as: </li></ul><ul><li>1.1. Pie charts </li></ul><ul><li>1.2. Bar graphs </li></ul><ul><li>1.3. Frequency polygons </li></ul><ul><li>1.4. Line graphs </li></ul><ul><li>2. Operationally define the following concepts used in graphic presentation of data: </li></ul><ul><li>2.1. Circular chart or pie chart </li></ul><ul><li>2.2. Histogram or bar graph </li></ul><ul><li>2.3. Line graph </li></ul><ul><li>2.4. Frequency polygon </li></ul>
68. 69. <ul><li>PRE-TEST </li></ul><ul><li>Choose the letter of the best answer and write it on the blank before the number. </li></ul><ul><li>_____1. A graph whose pieces add up to 100 percent is called ___ </li></ul><ul><li>a. pie chart c. line graph </li></ul><ul><li>b. map d. bar graph </li></ul><ul><li>_____ 2. A graphic illustration in which rectangular bars indicate the frequencies for the range of score values or categories. </li></ul><ul><li>a. pie chart c. line graph </li></ul><ul><li>b. bar graph d. none of these </li></ul><ul><li>_____ 3. A graph in which frequencies are indicated by a series of points placed over the score values of each class interval and connected with a straight line dropped to the base line at either end. </li></ul>
69. 70. <ul><li>a. picture graph c. bar grahp </li></ul><ul><li>b. histogram d. frequency polygon </li></ul><ul><li>Study the figure below and answer the questions that follow: </li></ul><ul><li>Above Average </li></ul><ul><li>Average </li></ul><ul><li> Below Average </li></ul>
70. 71. <ul><li>___ 4. If there are 20 students representing their IQ, then how many belong to average IQ? </li></ul><ul><li>a. 50 c. 100 </li></ul><ul><li>b. 75 d. 150 </li></ul><ul><li>___ 5. How many belong to below average? </li></ul><ul><li>a. 50 c. 90 </li></ul><ul><li>b. 75 d. 125 </li></ul><ul><li>___ 6. What level of measurement must a researcher have in order to make use of pie chart? </li></ul><ul><li>a. ordinal c. ratio </li></ul><ul><li>b. nominal d. interval </li></ul>
71. 72. <ul><li>Study the illustration below and answer the questions that follow: </li></ul><ul><li>600 </li></ul><ul><li>500 </li></ul><ul><li>400 </li></ul><ul><li>300 </li></ul><ul><li>200 </li></ul><ul><li>100 </li></ul><ul><li>0 </li></ul><ul><li> Catholic Protestant Iglesia Methodist </li></ul><ul><li>Bar graph of student’s religions </li></ul>
72. 73. <ul><li>____ 7. What is the frequency of students belonging to Catholic? </li></ul><ul><li>a. 200 c. 400 </li></ul><ul><li>b. 300 d. 500 </li></ul><ul><li>____ 8. What is the total population represented by the graph? </li></ul><ul><li>a. 1000 c. 1200 </li></ul><ul><li>b. 1100 d. 1300 </li></ul><ul><li>Study the bar below and answer the questions that follow: </li></ul><ul><li>500 </li></ul><ul><li>Legend: 400 </li></ul><ul><li>male 300 </li></ul><ul><li>female 200 </li></ul><ul><li>100 </li></ul><ul><li> 0 </li></ul><ul><li> </li></ul><ul><li>Never seldom Sometimes Frequent Always </li></ul>
73. 74. <ul><li>Bar graph of students using library facilities </li></ul><ul><li>____ 9. How many students particularly males are frequently using the library facilities? </li></ul><ul><li>a. 100 c. 300 </li></ul><ul><li>b. 200 d. 400 </li></ul><ul><li>____ 10. What is the population of female students represented by the graph? </li></ul><ul><li>____ 11.What does the figure below show? </li></ul><ul><li>freq. </li></ul><ul><li>midpoint </li></ul>
74. 75. <ul><li>a. frequency polygon c. map </li></ul><ul><li>b. histogram d. none of these </li></ul><ul><li>____ 12. What level of measurement must a researcher have in order to make use of histogram of frequency polygon? </li></ul><ul><li>a. ordinal c. interval </li></ul><ul><li>b. nominal d. all of these </li></ul><ul><li>PRESENTING DATA IN A PIE CHART & BAR GRAPH </li></ul><ul><li>What is a pie chart? </li></ul><ul><li>When do you use a pie chart? </li></ul><ul><li>The pie chart , a circular graph whose pieces sum up to 100%, is one of the simplest methods of presenting data in graphical form. Pie chart is p[particularly useful for depicting the differences in frequencies or percents among categories of a nominal-level variable. The pie chart depicts the most noteworthy part. For instance, a researcher would want you to </li></ul>
75. 76. <ul><li>identify the course where most students are enrolled in a particular university. Looking at the illustration below. Which section of the pie chart is most noteworthy? Isn’t it the most students enrolled in a particular course? If the researcher would want the reader to note the least number of enrollees in a particular course, the section of the chart will be noted by the reader immediately. </li></ul><ul><li>Engr. Nursing </li></ul><ul><li>Educ. </li></ul><ul><li>Crim. Commerce </li></ul><ul><li>Figure 3.1 Pie Chart of Course of Students in a Particular University </li></ul>
76. 77. <ul><li>What is a bar graph? </li></ul><ul><li>When do you use graph? </li></ul><ul><li>The pie provides a quick and easy illustration of data. By comparison, the bar graph or histogram can accommodate any number of categories at any level of measurement. </li></ul><ul><li>The bar graph is constructed following the standard arrangement: a horizontal base line (or x-axis) along which the score values or categories are marked off, a vertical line (or y-axis) along the left side displays the frequencies for each score value or category. For grouped data, the midpoints of the class intervals are arranged along the base line. Study the following illustration: Notice in the illustration that the taller bar, the greater the frequency of the category. </li></ul>
77. 78. <ul><li>600 </li></ul><ul><li>500 </li></ul><ul><li>400 </li></ul><ul><li>300 </li></ul><ul><li>200 </li></ul><ul><li>100 </li></ul><ul><li>0 </li></ul><ul><li> BSE BSN AB Commerce Secretariat </li></ul><ul><li>Figure 3.2 Bar graph of students enrolled in different courses </li></ul><ul><li>In the illustration above, which course has the most number of enrollees? Least number of enrollees? Is it possible for you to arrange the data in an ordinal manner? What level of measurement do you have in the illustration? </li></ul>
78. 79. <ul><li>The course AB has the most number of enrollees while Commerce has the least, I.e., 600 and 200, respectively. It is also possible to arrange the data in an ordinal manner. In this case, the level of measurement is ordinal. However, the data presented is actually in the nominal-level variable. </li></ul><ul><li>Activity 3.1 </li></ul><ul><li>Given the following distribution, construct a pie chart. </li></ul><ul><li>Club Preference of 300 Students in a Secondary School </li></ul><ul><li>Club Frequency (f) </li></ul><ul><li>Art 100 </li></ul><ul><li>Dance 50 </li></ul><ul><li>Science 100 </li></ul><ul><li>Speech 50 </li></ul><ul><li>Total 300 </li></ul>
79. 80. <ul><li>2. Depict the following data in a pie chart </li></ul><ul><li>Religious Preference of Students </li></ul><ul><li>Religion Frequency (f) </li></ul><ul><li>Catholic 500 </li></ul><ul><li>Protestant 400 </li></ul><ul><li>Jewish 100 </li></ul><ul><li>Methodist 800 </li></ul><ul><li>Iglesia ni Christo 200 </li></ul><ul><li>Total 2000 </li></ul>
80. 81. <ul><li>3. On a graphing paper, draw a bar graph to illustrate the following distribution of IQ scores: </li></ul><ul><li>Class interval f </li></ul><ul><li>151-155 9 </li></ul><ul><li>146-150 7 </li></ul><ul><li>141-145 4 </li></ul><ul><li>136-140 3 </li></ul><ul><li>131-135 8 </li></ul><ul><li>126-130 9 </li></ul><ul><li> N = 40 </li></ul>
81. 82. <ul><li>4. Depict the following in a bar graph. </li></ul><ul><li>Socio-economic status of employees in a particular company </li></ul><ul><li> Status frequency (f) </li></ul><ul><li>Above average 800 </li></ul><ul><li>Average 1500 </li></ul><ul><li>Below Average 700 </li></ul><ul><li> N = 3000 </li></ul>
82. 83. <ul><li>PRESENTING DATA IN A FREQUENCY POLYGON </li></ul><ul><li>What is a frequency polygon? </li></ul><ul><li>Frequency Polygon is another graphic method employed in the presentation of data. Although the frequency polygon can accommodate a wide variety of categories, it tends to emphasize “ continuity” along a scale rather than differences. It is however, advantageous or useful for depicting ordinal and interval data. This is the reason why frequencies are indicated by a series of points placed over the score values or midpoints of each class interval. Adjacent points, however, are connected with a straight line, which is dropped to the base line at either end. In the figure below, the height of each point or dot indicates frequency of occurrence. </li></ul>
83. 84. <ul><li>Class interval f </li></ul><ul><li>136-140 10 f 40 </li></ul><ul><li>141-145 15 r 35 </li></ul><ul><li>146-150 30 e 30 </li></ul><ul><li>151-155 35 q 25 </li></ul><ul><li>156-160 30 u 20 </li></ul><ul><li>161-165 20 e 15 </li></ul><ul><li>166-170 15 n 10 </li></ul><ul><li>c 5 </li></ul><ul><li>y 0 </li></ul><ul><li> 138 143 148 153 158 163 </li></ul><ul><li>midpoints (IQ) </li></ul><ul><li>Figure 3.3 Frequency polygon of a distribution of IQ score </li></ul>
84. 85. <ul><li>Looking at the frequency polygon (figure 3.3), what class interval has the highest number of frequency? Is it not 136-140? </li></ul><ul><li>What is cumulative frequency polygon? </li></ul><ul><li>To construct a cumulative frequency polygon, cumulative frequencies (or cumulative percentages) are needed. Recall your previous module wherein you learned the meaning of cumulative frequency and how it is being presented in a frequency distribution. As shown in figure 3.4, cumulative frequencies are arranged along the vertical line of the graph and are indicated by the height of points above the horizontal base line. Cumulative frequencies are a product of successive additions. Any cumulative frequency is never less and is usually more than the preceding cumulative frequency. The points in a cumulative graph are plotted above the upper limits of class intervals rather at their midpoints. This is the reason why cumulative frequency represents the total number of cases both within and below a particular class interval. </li></ul>
85. 86. <ul><li>Using the data in figure 3, study how cumulative frequency polygon is being constructed. </li></ul><ul><li>cf </li></ul><ul><li>f 175 </li></ul><ul><li>r </li></ul><ul><li>e 150 </li></ul><ul><li>q </li></ul><ul><li>u 125 </li></ul><ul><li>e </li></ul><ul><li>n 100 </li></ul><ul><li>c </li></ul><ul><li>y 75 </li></ul><ul><li>50 </li></ul><ul><li>25 </li></ul><ul><li>0 </li></ul><ul><li>140.5 145.5 150.5 155.5 160.5 165.5 170.5 </li></ul><ul><li>Upper Class Boundaries </li></ul><ul><li>Figure 3.4 Cumulative Frequency Polygon </li></ul>
86. 87. <ul><li>Activity 3.2. </li></ul><ul><li>On a graphing paper, draw a cumulative frequency polygon to represent the following grades on a final examination in Research with Statistics: </li></ul><ul><li>Class interval f cf </li></ul><ul><li>91-100 5 36 </li></ul><ul><li>81-90 8 31 </li></ul><ul><li>71-80 10 23 </li></ul><ul><li>61-70 8 13 </li></ul><ul><li>51-60 5 5 </li></ul><ul><li> N = 36 </li></ul>
87. 88. <ul><li>2. Construct a cumulative frequency polygon based on the following frequency distribution </li></ul><ul><li>Class interval f cf </li></ul><ul><li>11-20 3 </li></ul><ul><li>21-30 6 </li></ul><ul><li>31-40 7 </li></ul><ul><li>41-50 9 </li></ul><ul><li>51-60 8 </li></ul><ul><li>61-70 5 </li></ul><ul><li>71-80 2 </li></ul><ul><li> N = 40 </li></ul>
88. 89. <ul><li>Hint: You cannot construct the cumulative frequency polygon not unless you have the cumulative frequencies. Therefore, you have to fill up first the column for cumulative frequencies. If you can notice the class interval in item # 1 are arranged from highest to lowest. In contrast, the class intervals in item #3 are arranged from lowest to highest. So you have to very careful in filling up the column for cumulative frequencies in this item. </li></ul><ul><li>3. Draw a frequency polygon for these data: </li></ul><ul><li>Class interval f </li></ul><ul><li> 11-15 3 </li></ul><ul><li> 16-20 5 </li></ul><ul><li> 21-25 8 </li></ul><ul><li> 26-30 10 </li></ul><ul><li> 31-35 7 </li></ul><ul><li> 36-40 5 </li></ul><ul><li> 41-45 2 </li></ul>
89. 90. <ul><li>POST TEST 3 </li></ul><ul><li>You can use extra sheet/s of paper preferably graphing paper for your answer in this posttest. </li></ul><ul><li>Display the following unemployment rates both in a bar graph and as a line chart. Draw the two graphs separately. </li></ul><ul><li>AGE MALE UNEMPLOYMENT RATE </li></ul><ul><li>21-25 20.5 </li></ul><ul><li>26-30 15.25 </li></ul><ul><li>31-35 10.50 </li></ul><ul><li>36-40 10.00 </li></ul><ul><li>41-45 8.75 </li></ul><ul><li>46-50 6.50 </li></ul><ul><li>51-55 5.25 </li></ul><ul><li>56-60 3.50 </li></ul>
90. 91. <ul><li>2. depict the following data in a bar graph </li></ul><ul><li>Employment Status </li></ul><ul><li>Probationary Permanent </li></ul><ul><li>Age f f </li></ul><ul><li>21-25 20 15 </li></ul><ul><li>26-30 10 20 </li></ul><ul><li>31-35 20 10 </li></ul><ul><li>36-40 5 5 </li></ul><ul><li>41-45 25 30 </li></ul><ul><li>46-50 40 50 </li></ul><ul><li>51-55 30 45 </li></ul>
91. 92. <ul><li>3. Construct a cumulative frequency polygon for the following data: </li></ul><ul><li>CI f cf </li></ul><ul><li>52-55 10 </li></ul><ul><li>46-50 18 </li></ul><ul><li>41-45 20 </li></ul><ul><li>36-40 30 </li></ul><ul><li>31-35 15 </li></ul><ul><li>26-30 10 </li></ul><ul><li>21-25 5 </li></ul>
92. 93. <ul><li>4. Construct a pie chart for the following data; </li></ul><ul><li>Percentage of PBET Board passers </li></ul><ul><li>Year % </li></ul><ul><li>1989 10 </li></ul><ul><li>1990 20 </li></ul><ul><li>1991 25 </li></ul><ul><li>1992 30 </li></ul><ul><li>1993 10 </li></ul><ul><li>1994 5 </li></ul>
93. 94. <ul><li>COPUTER APPLICATION </li></ul><ul><li>MICROSTAT is a program used to prepare statistical computations. With the use of computers and this program the statistical information can be prepared at once with accuracy and speed. </li></ul><ul><li> MICROSTAT has a command to prepare or construct frequency polygon and bar graph or histogram. In the discussion portion of the module, specifically on presenting data into graphic form, the histogram or the bar graph is constructed vertically. The construction of histogram in this sense is done manually, I.e., using pencil and paper. On the other hand, the construction and presentation of histogram employing computer and the MICROSTAT program, the manner in done horizontally. </li></ul><ul><li>NOTE: Where can you secure a copy of this software? You can ask your professor or the school to provide you a copy so that you will be able tp prepare all the necessary statistical computations. </li></ul>
94. 95. <ul><li>APPLICATION </li></ul><ul><li>Encode the following set of score in the computer using the MICROSTAT program and prepare a histogram. </li></ul><ul><li>25 55 54 78 45 88 78 55 51 </li></ul><ul><li>65 45 25 89 47 74 82 77 70 </li></ul><ul><li>32 85 44 93 28 65 46 99 61 </li></ul><ul><li>36 35 22 12 10 22 58 22 53 </li></ul><ul><li>66 48 56 75 20 37 66 58 11 </li></ul>
95. 96. <ul><li>IS THIS YOUR FIRST TIME TO USE COMPUTER? </li></ul><ul><li>If this is your time to make use of computer, I suggest if you can have a little practice of typing. You don’t have to worry regarding the use of this program. All you have to do is to the understand what the computer tells and commands you to do. The first things you have to learn in this program is the DATA MANAGEMENT SYSTEM which will enable you to encode, edit, list, etc. the data. The instructions in this program are simple and easy to grasp and follow. In fact, high school and college students can learn the commands in this program even without the assistance or guidance of their teacher. You have more intelligence than these high school and college students. So, you can do it! </li></ul><ul><li>During your free time, you are advised to think or create your own sets of numbers or scores. Then encode these in the computer. Prepare the computer print outs. </li></ul>
96. 97. <ul><li>However, if there is really a need for you to consult your professor regarding the use of his program, please don’t hesitate to ask questions from your professor. At any rate, you have to submit the computer print outs to your professor for checking and recording the results of your computer task/activities. </li></ul>
97. 98. Module four <ul><li>MEASUREMENT OF CENTRAL TENDENCY </li></ul><ul><li>Introduction </li></ul><ul><li>Researchers in many fields have used the term “ average” in asking questions such as What is the average grade of students in fourth year? How many bottles of beer are consumed by the average teenager? What is the grade-point average of a doctoral students? On the average, how many crimes are committed every quarter in the Philippines? </li></ul><ul><li>A useful way to describe a group as a whole is to find a single number that represents what is “average” of that set of data. In research, such a value is known as a measure of central tendency. Measures of central tendency are generally located toward the center of distribution where most of the data tend to be concentrated. </li></ul>
98. 99. <ul><li>The three best known measures of central tendency are discussed in this module: the mean, the median, and the mode. </li></ul><ul><li>GENERAL OBJECTIVES </li></ul><ul><li>At the end of this module, you are expected to: </li></ul><ul><li>Calculate the arithmetic mean, median, and mode; </li></ul><ul><li>Explain the characteristics, use, advantages, and disadvantages of using the mean, mode, and median in researches; </li></ul><ul><li>Identify the position of arithmetic mean, median, and mode for both a systematical and a skewed distribution. </li></ul><ul><li>TIME FRAME </li></ul>
99. 100. <ul><li>PRE-TEST 4 </li></ul><ul><li>Read each item carefully. Choose the best answer. Write only the letter of the correct answer on the blank before the number. </li></ul><ul><li>_____ 1. The sum of a set of scores divided by the total number of scores in the set </li></ul><ul><li>A. mode C. mean </li></ul><ul><li>B. median d. standard deviation </li></ul><ul><li>_____ 2. The middlemost point in a frequency distribution is know as </li></ul><ul><li>A. mode C. mean </li></ul><ul><li>B. median D. range </li></ul>
100. 101. <ul><li>_____ 3. The most frequent score in a distribution is known as __. </li></ul><ul><li>A. mode C. mean </li></ul><ul><li>B. median D. interval </li></ul><ul><li>_____ 4. What is the mode in this set of scores? 1,2,3,1,1,6,4,3,1,2,2,4,5, </li></ul><ul><li>A. 1 B. 2 C. 3 D.4 </li></ul><ul><li>_____ 5. What is the average of this set of scores? 50,45,60,55,50, </li></ul><ul><li> 35,65,50,40,60 </li></ul><ul><li>A. 35 B. 40 C. 45 D. 50 </li></ul><ul><li>_____ 6. What is the median in this set of scores? 3,5,7,9,11,15,20, </li></ul><ul><li> 30,50 </li></ul><ul><li>A. 9 B. 11 C. 20 D. 21 </li></ul><ul><li> </li></ul>
101. 102. <ul><li>_____ 7. Given the following data, compute the mean: </li></ul><ul><li>Respondent X (IQ) </li></ul><ul><li>Allan 90 </li></ul><ul><li>Buddy 110 </li></ul><ul><li>Cathy 100 </li></ul><ul><li>Daniel 112 </li></ul><ul><li>Efren 75 </li></ul><ul><li>Felipe 130 </li></ul><ul><li>Garry 85 </li></ul><ul><li>Harold 93 </li></ul><ul><li>A. 99.375 C. 132.49 </li></ul><ul><li>B. 113.57 D. 158.99 </li></ul>
102. 103. <ul><li>____ 8. Using the formula X = fX/N, compute the mean from the following distribution: </li></ul><ul><li>X f fX </li></ul><ul><li>5 10 50 </li></ul><ul><li>4 8 32 </li></ul><ul><li>3 7 21 </li></ul><ul><li>2 9 18 </li></ul><ul><li>1 5 5 </li></ul><ul><li>A. 3.23 B. 7.8 C. 8.4. D. 25.20 </li></ul>
103. 104. <ul><li>____ 9. Find the mode of the data below: </li></ul><ul><li> Class interval f </li></ul><ul><li>25-29 3 </li></ul><ul><li>30-24 2 </li></ul><ul><li>35-39 5 </li></ul><ul><li>40-44 8 </li></ul><ul><li>45-49 8 </li></ul><ul><li>50-54 9 </li></ul><ul><li>55-59 8 </li></ul><ul><li>60-64 6 </li></ul><ul><li>65-69 4 </li></ul><ul><li>70-74 3 </li></ul><ul><li> N = 56 </li></ul><ul><li>A. 55 B. 56 C. 57 D. 58 </li></ul>
104. 105. <ul><li>____ 10. Find the median of the following distribution: </li></ul><ul><li> Class interval f cf </li></ul><ul><li>25-29 3 3 </li></ul><ul><li>30-34 2 5 </li></ul><ul><li>35-39 5 10 </li></ul><ul><li>40-44 8 18 </li></ul><ul><li>45-49 8 26 </li></ul><ul><li>50-54 8 34 </li></ul><ul><li>55-59 9 43 </li></ul><ul><li>60-64 6 49 </li></ul><ul><li>65-69 6 55 </li></ul><ul><li>70-74 3 58 </li></ul><ul><li>75-79 3 61 </li></ul><ul><li>80-84 3 64 </li></ul><ul><li>A. 53.25 B. 52.25 C. 51.5 D. 50.25 </li></ul>
105. 106. <ul><li>THE MEAN FOR THE UNGROUPED DATA </li></ul><ul><li>What is an average </li></ul><ul><li>You often need a certain number to represent a set of data. This one number can be thought of as being “typical” of all the data. If the annual salary of a school supervisor is P150,000.00, then P200,000.00 would be above average . Likewise, if the annual salary of a classroom teacher is P75,000.00, a salary of 74,500.00 is about average . What is an average? </li></ul><ul><li>Average is a single value or number that represents a set of data. It pinpoints a center of the values. </li></ul><ul><li>The Sample Mean </li></ul><ul><li>The measure of central tendency (average) most widely used is the arithmetic mean, usually shortened to the mean. For raw data, I.e., the ungrouped data, the mean is the sum of all the values divided by the total number of values. To find the </li></ul>
106. 107. <ul><li>Mean for a sample, you can use the following formula: </li></ul><ul><li>Sum of all the Values in the Sample </li></ul><ul><li>Sample Mean = </li></ul><ul><li>Number of Values in the Sample </li></ul><ul><li>Instead of writing the formula in words, it is convenient to use the shorthand notation of algebra. Thus, in symbol the formula is: </li></ul><ul><li>X  X </li></ul><ul><li> N </li></ul><ul><li> </li></ul>
107. 108. <ul><li>Where: X stands for the sample mean – it is read as “X bar” </li></ul><ul><li> X stands for a particular value </li></ul><ul><li> is the Greek capital sigma and indicates the operation of </li></ul><ul><li>adding </li></ul><ul><li>So: </li></ul><ul><li> X stands for the sum of all the Xs </li></ul><ul><li>N is the total number of values in the sample </li></ul><ul><li>Try this example: </li></ul><ul><li>What is the sample mean of the following ungrouped data? </li></ul><ul><li>25,15,35,24,26,12,38,30,20,11,39. </li></ul>
108. 109. <ul><li>What is your sample mean? Did you get 25? The correct sample mean for these data is 25. What is a statistical term appropriately applied to the sample mean of 25? The mean of a sample, or any other measure based on a data, is called a statistic. Thus, statistic is a measurable characteristic of a sample. </li></ul><ul><li>Lets have another example </li></ul><ul><li>The results of the test of five students were taken randomly. The scores are as follows: 55, 76.5,55.25,75, and 88. What is the arithmetic mean of these five scores of the students? </li></ul>
109. 110. <ul><li>Solution: </li></ul><ul><li>X = 55 + 76.5 + 55.25 + 75 +88 </li></ul><ul><li>5 </li></ul><ul><li>X = 349.75 </li></ul><ul><li> 5 </li></ul><ul><li>X = 69.95 </li></ul><ul><li>Therefore, the arithmetic mean is 69.95 </li></ul><ul><li>The Population Mean </li></ul><ul><li>The population mean is computed in the same manner as to how the sample mean is computed. The measurable characteristic of the mean is known as statistic. In contrast measurable characteristic of a population is called parameter . The formula used for computing the population parameter is as follows: µ =  X </li></ul><ul><li> N </li></ul>
110. 111. <ul><li>Where: </li></ul><ul><li> µ stands for the population mean </li></ul><ul><li> N stands for the total number of observation in the </li></ul><ul><li>population </li></ul><ul><li> X stands for the sum of all the Xs </li></ul><ul><li>The Properties of the Arithmetic Mean </li></ul><ul><li>1. Every set of interval-level and ratio-level data has a mean. </li></ul><ul><li>2. All the values are included in computing the mean. </li></ul><ul><li>3. A set of data has only one mean. </li></ul><ul><li>4. The mean is a very useful measure for comparing two or more </li></ul><ul><li>populations. </li></ul><ul><li>5. The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean will always be zero. In symbols, it is expressed as follows: </li></ul>
111. 112. <ul><li>  (X – X) = 0 </li></ul><ul><li>Let’s take an example: </li></ul><ul><li>The mean of 3, 7, and 5 is 5. The: </li></ul><ul><li>  (X- X) = (3 – 5) + (7-5) + (5 – 5) </li></ul><ul><li>= -2 + 2 + 0 </li></ul><ul><li>= 0 </li></ul><ul><li>In algebra, this is called additive inverse. This means that the sum of two values having opposite signs will always be equal to zero. Like (-4) + (+4) = 0, (-25) + (25) = 0 etc. </li></ul><ul><li>Now, do you understand this property of the mean? Do you still need some example? You can do the following examples on your own: Find the mean of the following sets of scores and show that the sum of the deviations from the mean is equal to zero: </li></ul>
112. 113. <ul><li>33,27,18,32,50,0,35, and 10 </li></ul><ul><li>12,8,10,5,15,10,9,and 11 </li></ul><ul><li>6,3,7,8,2,7,3,and 5 </li></ul><ul><li>After answering these drill, check your own work. The key to correction is as follows: </li></ul><ul><li>1. 25 2. 10 3. 5 </li></ul><ul><li>Of course the sum of the deviations from the mean in all items is zero. Now, you are ready to answer the following more challenging exercises if you got all the answers correctly. </li></ul>
113. 114. <ul><li>Activity 4.1 </li></ul><ul><li>Five scores from a Statistics test were taken randomly. The test results of the five students are as follows: 85,93,78,65,and 93. Find the average or the arithmetic mean and show that the sum of deviations from the mean is zero. </li></ul><ul><li>What would be the grade-point average of a doctoral student after a semester if he had the following grades: </li></ul><ul><ul><li>Philosophy of Education 1.75 </li></ul></ul><ul><ul><li>Research 1.25 </li></ul></ul><ul><ul><li>Foundations of Education 2.50 </li></ul></ul><ul><ul><li>Educational Statistics 3.00 </li></ul></ul><ul><ul><li>3. The average number of crimes committed every quarter in a particular country is 15. For the first quarter there were 13 crimes committed; second quarter 17; and third quarter 8. Find the number of crimes committed for the last quarter. </li></ul></ul>
114. 115. <ul><li>4. For the past five years, the enrolment in University XYZ were as follows: </li></ul><ul><li>School Year: Enrolment : </li></ul><ul><li>1991-1992 2,050 </li></ul><ul><li>1992-1993 2,500 </li></ul><ul><li>1993-1994 2,450 </li></ul><ul><li>1994-1995 2,780 </li></ul><ul><li>1995-1996 3,050 </li></ul><ul><li>Find the mean annual number of enrolment in XYZ university. </li></ul><ul><li>5. The U.S. Education Department reported that for the past several years 5,033; 5,652; 6, 407; 7,201; 8,719; 11,154; and 15,121 people received bachelor’s degree in computer and information sciences. What is the mean annual number receiving this degree? </li></ul>
115. 116. <ul><li>is it a sample mean or a popular mean? (Mason, Robert et al. 1990, p.79). </li></ul><ul><li>THE MEAN OF THE GROUPED DATA </li></ul><ul><li>The computation of mean for the grouped data is being done when there is a large number of scores to be treated in statistics. For instance, you want to obtain the arithmetic mean or average performance of the 10,000 examinees in Professional Board Examination for Teachers.Of course, it is very tedious to add all the scores and then divide by the total number of examinees. Also, the speed and accuracy will of great problem on the part of the researcher. Instead, the scores or raw data are grouped in order to make the computation convenient. </li></ul><ul><li>Recall the procedure on how to prepare the frequency distribution. In this case, you need to obtain the highest and lowest scores in order to get the range; the desired number of steps; the class intervals; and the frequency for each class interval. </li></ul>
116. 117. <ul><li>Consider the following example: </li></ul><ul><li>Fifty students took an entrance test in University ABC. The scores obtained by the students from a 100-item test are as follows: </li></ul><ul><li>88,75,77,90,56,66,28,84,39,40 </li></ul><ul><li>88,30,50,66,30,39,56,76,44,88 </li></ul><ul><li>71,73,92,99,55,87,38,40,20,10 </li></ul><ul><li>90,22,55,76,20,55,44,10,29,0 </li></ul><ul><li>57,39,74,39,67,55,39,82,40,23. </li></ul><ul><li>What is the highest score? Lowest score?They are 99 and 0, respectively. How about the range? Obviously, it is 99 because 99-0 = 99. How many steps do you want to apply? Suppose you decide to have 10, what will be the size of the class interval? You simply divide 99 by 10 and the size of the class interval becomes 9.9 or 10 as the rounded value. Therefore, what will be </li></ul>
117. 118. <ul><li>the first class interval? The first class interval will be 0-9; second class interval will be 10-19; and so on. Thus, you are now ready to tally the scores and prepare the frequency distribution. </li></ul><ul><li>Class interval Tally frequency (f) M fM </li></ul><ul><li>0-9 </li></ul><ul><li>10-19 </li></ul><ul><li>20-29 </li></ul><ul><li>30-39 </li></ul><ul><li>40-49 </li></ul><ul><li>50-59 </li></ul><ul><li>60-69 </li></ul><ul><li>70-79 </li></ul><ul><li>80-89 </li></ul><ul><li>90-99 </li></ul><ul><li>N = </li></ul>
118. 119. <ul><li>The formula for computing the mean of the grouped data is as follows: </li></ul><ul><li> fm </li></ul><ul><li> N </li></ul><ul><li>Where: X stands for the mean </li></ul><ul><li>fM stands for the product of the frequency and midpoint </li></ul><ul><li> fM stands for the sum of the product of all fs and Ms </li></ul><ul><li>N stands for the total number of scores or cases </li></ul><ul><li>You will notice that in the table of the frequency distribution, there is a column for M (or midpoint) and another column for fM (or product of frequency and midpoint). These two columns are left for your own computation. This is to apply what you have learned in the previous modules. Do you still know how to get the midpoint? After you have all the midpoints </li></ul>
119. 120. <ul><li>for all the class intervals, you multiply them with their corresponding frequency then get the summation of the column for fM. </li></ul><ul><li>Your summation for the column of fM should be 2715 . If you divide this by 50 which is the total number of scores, you will obtain a mean of 54.3. Therefore, the mean performance of the fifty students who took the entrance test of University ABC is 54.3. Now, are you ready for the activity? </li></ul><ul><li>Activity 4.2 </li></ul><ul><li>Given the set of scores below, find the following: </li></ul><ul><li>Highest and lowest scores </li></ul><ul><li>Range </li></ul><ul><li>Size of the class interval (use 10 steps) </li></ul><ul><li>Class intervals </li></ul>
120. 121. <ul><li>5. Determine the frequency in each class interval </li></ul><ul><li>6. Identify the midpoints </li></ul><ul><li>7. Determine the summation of the product of all fs and Ms </li></ul><ul><li>8. Solve for the mean </li></ul><ul><li>Scores: 10, 9, 20, 7, 18, 40, 25, 18, 20, 23 </li></ul><ul><li> 4, 1, 49, 25, 33, 23, 19, 23, 47, 33 </li></ul><ul><li>48, 44, 25, 37, 13, 34, 30, 20, 10, 27. </li></ul><ul><li>The next lesson for you to learn is median which is another measure of central tendency . Since median is another measure of centrality, you will notice that value of it is close to the value of the mean. However, you have to learn first how to obtain the median of the ungrouped data then the median of the grouped data. </li></ul>
121. 123. <ul><li>THE MEDIAN OF UNGROUPED AND GROUPED DATA </li></ul><ul><li>What is median? </li></ul><ul><li>Median is know as the middlemost point in a frequency distribution. In an ungrouped data, median can be defined as the centermost score in a distribution. For instance, you want to obtain the median from a 10-item test taken by 5 students. The scores obtained are as follows: 7, 10, 5,3, and 1. What is the median for this ungrouped data. To get the median, arrange the scores from lowest to highest or vice-versa. Thus, the order of the scores will be 1,3,5,7, and 10 or 10,7,5,3, and 1. Looking at the order of the scores, the centermost score is 5 either from the other of lowest to highest or vice-versa. </li></ul>
122. 124. <ul><li>Try these examples: Find the median: </li></ul><ul><li>73,54,33,12,0,23, and 41 </li></ul><ul><li>6,8,29,44,19,23,and 5 </li></ul><ul><li>14,34,97,33,and 67 </li></ul><ul><li>10,45,66,78,and 44 </li></ul><ul><li>27,87,and 230 </li></ul><ul><li>What are your answer? Did you get 33,19,34,45 , and 87 for item 1,2,3,4, and 5, respectively? If you have noticed the number of scores in every item is odd. For items 1 and 2, there are 7 scores; in items 3 and 4,5 and so on. How about if the scores are even-numbered. For instance, you have 4 scores, 10 scores, etc. How will you obtain the median? To find the median of even-numbered scores, simply add the two centermost scores then divide by 2. For example: What is the median of these scores: 5,24,12, and 40. There are four scores </li></ul>
123. 125. <ul><li>And if you arrange them, say from lowest to highest, you will have 5,12,24 , and 40. Obviously the two centermost scores are 12 and 24 . The sum of these two centermost scores is 36. If you divide 36 by 2, then you will have a quotient of 18. Therefore, the median is 18. </li></ul><ul><li>Try solving the median of these sets of scores: </li></ul><ul><li>1. 24,90,156,76 </li></ul><ul><li>2. 25, 78, 12, 56, 77, 90 </li></ul><ul><li>3. 45, 98, 12, 89, 34, 33 </li></ul><ul><li>4. 256, 345, 897, 234 </li></ul><ul><li>5. 45, 88, 72, 13 </li></ul><ul><li>Let checked if you got the correct answers. You should have the following median: 50,66.5,39.5,400.5, and 58.5 for item 1,2,3,4, and 5, respectively. </li></ul>
124. 126. <ul><li>The Median of Grouped Data </li></ul><ul><li>As mentioned earlier in this module, the mean for a large group of numbers is computed by organizing the data in a way that these data are grouped accordingly. In like manner, the median for a large number of scores is also computed by organizing and grouping the data. Also, as mentioned earlier, the purpose of grouping large number of scores is to make the computation easy and to ensure accuracy and speed. It is accurate as far as the employment of formula and mathematical operations are concerned. However, with the use of grouped data for the computation of the median and even the mean, you should bear in mind that the obtained mean and median from grouped frequency distributions are only approximations of what you would get if they were calculated from the raw scores. </li></ul><ul><li>Suppose a set of 100 scores were grouped and translated into frequency distribution. How will you obtain the median? So, take a look at this example and examine how the median is obtained? </li></ul>
125. 127. <ul><li>Class interval f cf </li></ul><ul><li>20-24 3 3 </li></ul><ul><li>25-29 10 13 </li></ul><ul><li>30-34 12 25 </li></ul><ul><li>35-39 15 40 </li></ul><ul><li>40-44 18 58 </li></ul><ul><li>45-49 14 72 </li></ul><ul><li>50-54 10 82 </li></ul><ul><li>55-59 9 91 </li></ul><ul><li>60-64 6 97 </li></ul><ul><li>65-69 3 100 </li></ul><ul><li> </li></ul><ul><li> 100 </li></ul>
126. 128. <ul><li>The formula for finding the median of grouped data is given as follows: </li></ul><ul><li>N/2 – F </li></ul><ul><li>Mdn = L + (i) </li></ul><ul><li> fm </li></ul><ul><li>Where: </li></ul><ul><li>Mdn = median </li></ul><ul><li>L = lower boundary of interval containing the median class </li></ul><ul><li>F = the sum of all the frequencies below L </li></ul><ul><li>fm = frequency of interval containing the median class </li></ul><ul><li>N = the total number of cases </li></ul><ul><li>i = class interval </li></ul>
127. 129. <ul><li>To solve for the median of the data above, the following values must be determined first then substitution follows: </li></ul><ul><li>The lower boundary of the interval containing the median class or the L. </li></ul><ul><li>The sum of all the frequencies below the exact lower limit or the F. </li></ul><ul><li>The frequency of the interval containing the median class or the fm </li></ul><ul><li>The total number of cases or the N. </li></ul><ul><li>The class interval or the i. </li></ul><ul><li>The 50% or half of N or th N/2. </li></ul><ul><li>Based on the data presented, what is </li></ul><ul><li>L? L is equal to ________________ </li></ul><ul><li>F? F is equal to ________________ </li></ul>
128. 130. <ul><li>Fm? fm is equal to ____________ </li></ul><ul><li>N? N is equal to _____________ </li></ul><ul><li>i? i is equal to ______________ </li></ul><ul><li>N/2? N/2 is equal to _____________ </li></ul><ul><li>Checked if you got the answers correctly. The answers are as follows: explain how to set this L = 39.5; F = 40; fm = 18; N = 100; i = 5; and N/2 = 50. If you got them correctly, you are ready to substitute the values to the formula. </li></ul>
129. 131. <ul><li>To substitute, you will have the following: </li></ul><ul><li>100 - 40 </li></ul><ul><li>Mdn = 39.5 + 2 </li></ul><ul><li> .5 </li></ul><ul><li> 18 </li></ul><ul><li>= 39.5 + 50 </li></ul><ul><li> 18 </li></ul><ul><li>= 39.5 + 2.9 </li></ul><ul><li>= 42.4 </li></ul>
130. 132. <ul><li>Now, do you understand the steps on how to solve for the median of grouped data. Try this. </li></ul><ul><li>Class interval f cf </li></ul><ul><li>20-24 7 7 </li></ul><ul><li>25-29 10 17 </li></ul><ul><li>30-34 15 32 </li></ul><ul><li>35-39 9 41 </li></ul><ul><li>40-44 7 48 </li></ul><ul><li>45-49 5 53 </li></ul><ul><li>50-54 3 56 </li></ul><ul><li>55-59 4 60 </li></ul>
131. 133. <ul><li>Based on the data presented above, what is: </li></ul><ul><li> L? L is equal to ________________ </li></ul><ul><li>F? F is equal to ________________ </li></ul><ul><li>fm? fm is equal to ____________ </li></ul><ul><li>N? N is equal to _____________ </li></ul><ul><li>i? i is equal to ______________ </li></ul><ul><li>N/2? N/2 is equal to _____________ </li></ul><ul><li>This time you have to discover the answer to this exercise. Anyway,for purposes of checking your answer, the median or the final answer is 34.0. </li></ul>
132. 134. <ul><li>Activity 4.3. </li></ul><ul><li>Find the median of the following sets of scores: </li></ul><ul><li>1. 34,98,56,99,23,and 12 </li></ul><ul><li>2. 55,90,23,45, and 77 </li></ul><ul><li>3. 40,34,78,22,12,and 66 </li></ul><ul><li>4. 45,8,35,18,39,55,and 89 </li></ul><ul><li>5. 9,0,3,12,5,4,67,71, and 35 </li></ul><ul><li>Compute the median of the following frequency distribution: </li></ul><ul><li>Class interval f cf </li></ul><ul><li>45-49 10 10 </li></ul><ul><li>50-54 29 39 </li></ul><ul><li>55-59 32 71 </li></ul><ul><li>60-64 40 111 </li></ul><ul><li>65-69 35 146 </li></ul><ul><li> </li></ul>
133. 135. <ul><li>70-74 30 176 </li></ul><ul><li>75-79 25 201 </li></ul><ul><li>80-84 15 216 </li></ul><ul><li>85-89 9 225 </li></ul><ul><li>THE MODE OF THE UNGROUPED AND GROUPED DATA </li></ul><ul><li>What is mode? </li></ul><ul><li>What is unimodal? Bimodal? Trimodal? </li></ul><ul><li>How will you find for the mode of ungrouped data? Grouped data? </li></ul><ul><li>For ungrouped data, the mode is defined as that datum value or specific score which the highest frequency. In other words, the most frequency occurring score is the mode . If there is one </li></ul>
134. 136. <ul><li>frequently occurring score in a set of values, the mode is classified as unimodal. If the are two, the classification is known as bimodal and if there are three, the classification is known as trimodal, and so on. </li></ul><ul><li>Example: Find the mode of the following set of score: 1,2,2,2,6,6,8,8,8,8,8,9, </li></ul><ul><li>By inspection, the mode (Mo0 is 8 because it appears the most number of times. </li></ul><ul><li>How about this example? 5,5,5,5,5,6,6,6,6,6,7,7,9 </li></ul><ul><li>What value or values appear the most number of times? Are there more one values? What are they? What is the classification of this mode? The values that appear most number of times are 5 and 6. There are two values, therefore the classification of mode in this case is bimodal. </li></ul>
135. 137. <ul><li>The mode for the grouped data is defined as the midpoint of the interval containing largest number of cases. </li></ul><ul><li>Example: Find the mode of the data below: </li></ul><ul><li>Class interval f </li></ul><ul><li>34-39 15 </li></ul><ul><ul><ul><ul><ul><li>40-44 11 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>45-49 20 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>50-54 34 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>55-59 50 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>60-64 39 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>65-69 17 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>70-74 10 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>74-79 9 </li></ul></ul></ul></ul></ul>
136. 138. <ul><li>The mode (Mo) is 57 because the class interval 55-57 has the largest frequency and 57 is the midpoint of this class interval </li></ul><ul><li>Activity 4.4. </li></ul><ul><li>A. Find the mode of the following sets of scores: </li></ul><ul><li>1. 23,34,34,55,55,55,55,60,90,100 </li></ul><ul><li>2. 12,34,12,55,12,67,23,11,10,200 </li></ul><ul><li>3. 25,66,66,75,66,66,13,13,10,44 </li></ul><ul><li>4. 4,5,7,2,5,5,5,5,8,9 </li></ul><ul><li>5. 7,34,34,3,12,34,8,9,10,40 </li></ul><ul><li>B.Find the mode of the following: </li></ul><ul><li>1. Class interval f 2. Class interval f </li></ul><ul><li>10-14 14 30-34 9 </li></ul><ul><li>15-19 10 35-39 15 </li></ul><ul><li>20-24 17 40-44 60 </li></ul><ul><li>25-29 40 45-49 23 </li></ul><ul><li>30-34 55 50-54 7 </li></ul>
137. 139. <ul><li>3. Class interval f 4. Class interval f </li></ul><ul><li>45-49 1 55-59 3 </li></ul><ul><li>50-54 8 60-64 10 </li></ul><ul><li>55-59 25 65-69 45 </li></ul><ul><li>60-64 17 70-74 9 </li></ul><ul><li>65-69 9 75-79 5 </li></ul><ul><li>70-74 5 80-84 2 </li></ul><ul><li>5. Class interval f </li></ul><ul><li>34-39 17 </li></ul><ul><li>40-44 23 </li></ul><ul><li>45-49 78 </li></ul><ul><li>50-54 54 </li></ul><ul><li>55-59 35 </li></ul><ul><li>60-64 22 </li></ul>
138. 140. <ul><li>POST TEST 4 </li></ul><ul><li>Test I. Write Mn if the statement tells something about the mean, Mo if it is about the mode and Md if it is about the median. Write your answer on the blank before the number. </li></ul><ul><li>___ 1. It is used for interval data. </li></ul><ul><li>___ 2. It is used for ordinal or interval data. </li></ul><ul><li>___ 3. It is used for nominal, ordinal or interval data. </li></ul><ul><li>___ 4. It is most appropriate for unimodal systematical distribution </li></ul><ul><li>___ 5. It is most appropriate for bimodal distribution </li></ul><ul><li>___ 6. It is most appropriate for highly skewed distribution </li></ul><ul><li>___ 7. It is used when the purpose is to consider the value of each </li></ul><ul><li>score </li></ul><ul><li>___ 8. It is used when the center score is needed to avoid the influence of extreme values </li></ul><ul><li>___ 9. It is used when the researcher wants to identify the most frequency occurring score </li></ul>
139. 141. <ul><li>___ 10. This value is referred as to the result of driving the sum of scores by the total number of cases. </li></ul><ul><li>Test II. Find the mean for the following grouped and ungrouped data. </li></ul><ul><li>89 45 57 34 23 90 56 44 34 78 </li></ul><ul><li>134 345 879 345 223 590 276 </li></ul><ul><li>87 34 77 40 122 45 66 88 39 </li></ul><ul><li>Class interval f M fM </li></ul><ul><li>20-22 33 </li></ul><ul><li>23-25 25 </li></ul><ul><li>26-28 40 </li></ul><ul><li>29-31 37 </li></ul><ul><li>32-34 20 </li></ul><ul><li>35-37 14 </li></ul><ul><li>N= </li></ul>
140. 142. <ul><li>5. Class Interval f M fM </li></ul><ul><li>15-17 45 </li></ul><ul><li>18-20 78 </li></ul><ul><li>21-23 90 </li></ul><ul><li>24-26 80 </li></ul><ul><li>27-29 65 </li></ul><ul><li>30-32 30 </li></ul><ul><li> N= </li></ul><ul><li>6. Class interval f M fM </li></ul><ul><li>35-39 8 </li></ul><ul><li>40-44 10 </li></ul><ul><li>50-54 15 </li></ul><ul><li>55-59 9 </li></ul><ul><li>60-64 7 </li></ul><ul><li>65-69 4 </li></ul><ul><li>70-74 1 </li></ul><ul><li>N= </li></ul>
141. 143. <ul><li>Test III. Find the mode of the following grouped and ungrouped data. Write your answer on the blank before the number. </li></ul><ul><li>___ 1. 34,56,56,56,78,89,90,and 45 </li></ul><ul><li>___ 2. 45,12,10,10,34,10,78,23,and 10 </li></ul><ul><li>___ 3. 16,39,345,100,39,39,23,and 39 </li></ul><ul><li>___ 4. 23,90,23,88,23,67,23,23,12,and 23 </li></ul><ul><li>___ 5. 10,20,30,20,20,50,60,70,80,and 40 </li></ul><ul><li>___ 6. Class interval f </li></ul><ul><li>44-46 12 </li></ul><ul><li>47-49 9 </li></ul><ul><li>50-52 34 </li></ul><ul><li>53-55 18 </li></ul><ul><li>56-58 10 </li></ul>
142. 144. <ul><li>___ 7. Class interval f </li></ul><ul><li>1-2 8 </li></ul><ul><li>3-4 10 </li></ul><ul><li>5-6 15 </li></ul><ul><li>7-8 9 </li></ul><ul><li>9-10 3 </li></ul><ul><li>Test III . Find the median of the following grouped and ungrouped and ungrouped data. Write your answer on the blank before the number. </li></ul><ul><li>34,78,90,12,45,76,34, and 56 </li></ul><ul><li>10,45,20,55,30,67,90,and 40 </li></ul><ul><li>16,34,88,77,23,76,and 45 </li></ul><ul><li>66,90,34,22,19,30,and 79 </li></ul><ul><li>44,90,56,34,67,21,43,and 45 </li></ul>
143. 145. <ul><li>6. Class interval f cf </li></ul><ul><li>10-14 9 9 </li></ul><ul><li>15-19 12 12 </li></ul><ul><li>20-24 15 36 </li></ul><ul><li>25-29 30 66 </li></ul><ul><li>30-34 14 80 </li></ul><ul><li>35-39 10 90 </li></ul><ul><li>40-44 10 100 </li></ul><ul><li>7. Class interval f cf </li></ul><ul><li>33-36 9 9 </li></ul><ul><li>37-40 13 22 </li></ul><ul><li>41-44 18 40 </li></ul><ul><li>45-48 30 70 </li></ul><ul><li>49-52 17 87 </li></ul><ul><li>53-56 9 96 </li></ul><ul><li>57-60 4 100 </li></ul>
144. 146. <ul><li>COMPUTER APPLICATIONS </li></ul><ul><li>Using the MICROSTAT software, encode the following data: </li></ul><ul><li>564 879 998 243 456 789 345 101 </li></ul><ul><li>238 875 345 897 345 234 675 234 </li></ul><ul><li>689 409 497 100 346 368 203 234 </li></ul><ul><li>409 567 290 567 298 456 109 309 </li></ul><ul><li>208 390 476 190 267 457 333 289 </li></ul><ul><li>390 386 509 390 520 345 987 289 </li></ul><ul><li>With the data that you encoded, compute the mean and prepare a computer print out. </li></ul>
145. 147. <ul><li>________ DESCRIPTIVE STATISTICS_________ </li></ul><ul><li>HEADER DATA FOR : A: PHILIP LABEL : mean </li></ul><ul><li>NUMBER OF CASES : 54 NUMBER OF VARIABLES : 1 </li></ul><ul><li>NO. NAME N MEAN STD.DEV. MINIMUM MAXIMUM </li></ul><ul><li>1 SCORES 54 428.7073 224.3080 100.000 998.0000 </li></ul><ul><li>PRESS ANY KEY TO CONTINUE. </li></ul>
146. 148. <ul><li>MODULE FIVE </li></ul><ul><li>MEASURES OF VARIATION </li></ul><ul><li>Introduction </li></ul><ul><li>In module number 4, you learned that the mode, median, and mean could be utilized to summarize in a single number what is “typical” or “average” of a particular distribution. However, when these measures of central tendency are employed alone, they can be misleading because they do not depict the complete picture of a set of data. </li></ul><ul><li>In addition to the measures of central tendency, you need an index of how the scores are scattered the distribution. Thus, this will introduce you to the concepts on variety. </li></ul>
147. 149. <ul><li>OBJECTIVES </li></ul><ul><li>At the end of this module, you are expected to: </li></ul><ul><li>Explain the uses of measures of variation </li></ul><ul><li>Compute the range, mean deviation, variance and standard deviation from a grouped data and ungrouped data; and </li></ul><ul><li>Compare the range, mean deviation, variance and standard deviation. </li></ul>
148. 150. <ul><li>PRE-TEST 5 </li></ul><ul><li>Read each item carefully. Choose the best answer from the given four options. Write only the letter of your answer on the blank before the number. </li></ul><ul><li>____ 1. The manner in which the score are scattered around the center of the distribution. Also known as dispersion or spread. </li></ul><ul><li>A. variability C. standard deviation </li></ul><ul><li>B. range D. mean deviation </li></ul><ul><li>____ 2. The mean of the squared deviation from the mean of a distribution. A measure of variability in a distribution. </li></ul><ul><li>A. deviation C. variance </li></ul><ul><li>B. range D. standard deviation </li></ul><ul><li>____3. The difference between the highest and lowest scores in distribution </li></ul><ul><li>A. deviation C. range </li></ul><ul><li>B. mean deviation D. variance </li></ul>
149. 151. <ul><li>____ 4. The sum of the absolute deviations from the mean divided by the number of scores in a distribution. </li></ul><ul><li>A. mean deviation C. standard </li></ul><ul><li>B. variance D. none of these </li></ul><ul><li>____ 5. The square root of the mean of the squared deviations from the mean of a distribution. </li></ul><ul><li>A. standard deviation C. mean deviation </li></ul><ul><li>B. variance D. none of these </li></ul><ul><li>____ 6. Suppose the highest grade of a student is 98 and his lowest grade is 94. Find the range between the two grade values. </li></ul><ul><li>A. 4 C. 6 </li></ul><ul><li>B. 5 D.7 </li></ul>
150. 152. <ul><li>____ 7. The average of 5,8,9, and 6 is 7. Find the mean deviation. </li></ul><ul><li>A. 1.5. C. 2.0 </li></ul><ul><li>B. 1.75 D. none of these </li></ul><ul><li>____ 8. The standard deviation of a certain distribution is 5.6. What is its variance? </li></ul><ul><li>A. 29.36 C. 31.36 </li></ul><ul><li>B. 30.36 D. 32.36 </li></ul><ul><li>____ 9. The variance of a distribution is 50.41. What is its standard deviation? </li></ul><ul><li>A. 7.30 C. 7.10 </li></ul><ul><li>B. 7.20 D. 7.00 </li></ul><ul><li>____ 10. The following are measures of variability except one. </li></ul><ul><li>A. variance C. range </li></ul><ul><li>B. mean deviation D. mean </li></ul>
151. 153. <ul><li>THE RANGE AND THE MEAN DEVIATION </li></ul><ul><li>What is range </li></ul><ul><li>What is mean deviation? </li></ul><ul><li>To get quick but rough measure of variability, you might find what is known as the range. The range is the difference between the highest and lowest score. Some teachers usually find the difference between these two scores, say from a mid term test results, to set the passing score. </li></ul><ul><li>Study these examples : </li></ul><ul><li>Five students in Statistics class took a special midterm test. The scores obtained by the students were as follows: 76,45,90,55, and 85. Find the range. </li></ul><ul><li>The temperature in a certain place was recorded for one week. The record shows the following temperature expressed in degree Celsius: 45,55,39,40,44,49, and 26. Find the range. </li></ul>
152. 154. <ul><li>Obviously, the highest and lowest scores in example 1 are 90 and 45, respectively.Applying the equation R=HS-LS, the range will be 45. In example 2, the highest and lowest scores and 55 and 26, respectively. Thus, the range for this is 29 because 55-26 = 29. Suppose you have the following grades during the first semester: 1.00,2.00, 3.00,2.75, and 5.00. Find the range. The range for this example is 4.00 because the two extreme grades are 5.00 and 1.00 and finding the difference between these two, the range becomes 4.00 </li></ul><ul><li>What is mean deviation? </li></ul><ul><li>Mean deviation is defined as the distance of any given raw score from the mean. To find for such deviation, subtract the mean from any raw score. For instance, the mean of 5 scores is 7 Find the deviation from the mean of score whose magnitude is 10. Obviously, the answer is 2. How about a score whose magnitude is 3? What is the absolute value of its deviation from the mean? The absolute value of any raw score is simply referred to its </li></ul>
153. 155. <ul><li>Magnitude, I. E., the sign whether positive or negative, is disregarded. Thus, the absolute value of –10 and + 10 is 10. </li></ul><ul><li>How will you compute for the mean deviation of a set of score? For instance, you have these scores: 9,8,6,4,2, and 7. </li></ul><ul><li>What will be the formula to solve for the mean deviation of these scores? The formula is as follows: </li></ul><ul><li>  /X – X/ </li></ul><ul><li>MD </li></ul><ul><li> N </li></ul><ul><li>Where: </li></ul><ul><li>MD = the mean deviation </li></ul><ul><li>  /X – X/ = the sum of the absolute deviation (disregarding plus the minus signs) </li></ul><ul><li>N = total number of scores or cases </li></ul>
154. 156. <ul><li>Activity 5.1. </li></ul><ul><li>Find the range of the following sets of scores. Write the answer on the blank before the the number. </li></ul><ul><li>____ 1. 36,75,16,95,80 </li></ul><ul><li>____ 2. 39,40,78,10,36 </li></ul><ul><li>____ 3. 84,85,79,64,78,85.5 </li></ul><ul><li>____ 4. 33,44,55,66,77,88 </li></ul><ul><li>____ 5. 2,5,9,18,1,3,7,90 </li></ul><ul><li>II. Compute the mean deviation or MD for the following distribution. </li></ul><ul><ul><li>X X-X /X-X/ </li></ul></ul><ul><li>7 ____ _____ </li></ul><ul><li>5 ____ _____ </li></ul><ul><li>9 ____ _____ </li></ul><ul><li>12 ____ _____ </li></ul><ul><li>9 ____ _____ </li></ul><ul><li>6 ____ _____ </li></ul><ul><li> X = </li></ul><ul><li>  /X-X/ </li></ul>
155. 157. <ul><li>Find the mean: </li></ul><ul><li>  X </li></ul><ul><li>X = = </li></ul><ul><li> N </li></ul><ul><li>Find the mean deviation: </li></ul><ul><li>MD =  /X –X = ______ </li></ul><ul><li> N </li></ul>
156. 158. <ul><li>X X-X /X-X </li></ul><ul><li>10 _10-10_____ ______ </li></ul><ul><li>9 ______ ______ </li></ul><ul><li>11 ______ ______ </li></ul><ul><li>5 ______ ______ </li></ul><ul><li>15 ______ ______ </li></ul><ul><li>7 ______ ______ </li></ul><ul><li>13 ______ ______ </li></ul><ul><li>  X  /X-X/ </li></ul>
157. 159. <ul><li>Find the mean: </li></ul><ul><li>  X  X </li></ul><ul><li>= = _______ = </li></ul><ul><li> N </li></ul><ul><li>Find the deviation: </li></ul><ul><li>MD  /X-X/ </li></ul><ul><li>= = _______ = </li></ul><ul><li> N </li></ul>
158. 160. <ul><li>THE VARIANCE AND STANDARD DEVIATIO N </li></ul><ul><li>What is variance? </li></ul><ul><li>In the previous learning activity, you learned that the mean deviation avoid the problem of negative values that cancel out positive values by simply ignoring positive (+) and negative (-) signs and getting the summation of absolute deviation from the mean. This procedure, however, has disadvantage in creating a measure of variability, I.e, such absolute values are not always useful in more advanced statistical analysis because they cannot easily be manipulated algebraically. </li></ul><ul><li>Thus, to overcome this kind of problem and obtain a measure of variability which is more accepted to advanced statistical procedure, you might square each of the actual deviations from the mean and add the squared values together. In symbols, it becomes: </li></ul><ul><li>  (X-X) ² </li></ul>
159. 161. <ul><li>Table 5.2.1 SQUARING Deviations to Eliminate Negative Values: An Illustration </li></ul><ul><li>X X-X (X-X) ² </li></ul><ul><li>5 -3 9 </li></ul><ul><li>8 0 0 </li></ul><ul><li>10 2 4 </li></ul><ul><li>12 4 16 </li></ul><ul><li>5 -3 9 </li></ul><ul><li> X = 40 0  (X-X) ² = 38 </li></ul>
160. 162. <ul><li>Under the column of X-X, how will you show the solution for the first value which is –3? Since the mean is 8, subtract it from the first value of X which is 5. Therefore, 5-8= -3. So, squaring –3 or multiplying by itself twice, it yields 9. </li></ul><ul><li>Now, what is the summation of all the squared deviations from the mean? If you will divide this by N, you will arrive at a measure of variability which is known as variance. Variance is actually the mean of the squared deviations. To illustrate the solution for finding the variance, you have to employ the following formula: </li></ul><ul><li>(X-X) ² </li></ul><ul><li>s ² = </li></ul><ul><li> N </li></ul>
161. 163. <ul><li>Where: </li></ul><ul><li> s ² is the variance </li></ul><ul><li>(X-X)² is the summation of all squared deviations </li></ul><ul><li>N is the total number of cases </li></ul><ul><li>To substitute the values you will have the following equation: </li></ul><ul><li>38 </li></ul><ul><li> s ² = = 7.6 </li></ul><ul><li>5 </li></ul>
162. 164. <ul><li>Therefore, the variance is 7.6. </li></ul><ul><li>Can you tell now the advantage of variance over the mean deviation? Variance given appropriately greater emphasis to extreme values. In other words, it is more sensitive to the degree of deviation in the distribution. </li></ul><ul><li>With the use of variance, however, another problem arises. As a direct result of having squared the deviations, the unit of measurement is altered. Thus, it makes the variance rather difficult to interpret. For instance, the variance is 7.6, but 7.6 of what? </li></ul><ul><li>What will you do with the variance in order to put the measure of variability in the right perspective? Isn’t it that you will find the square root of the variance? </li></ul><ul><li>What will you obtain when extracting the root is done? You will obtain another measure of variability which is called standard deviation. Standard deviations is the result of summing the squared deviations from the mean, dividing it by N, and finally, taking the square root. </li></ul>
163. 165. <ul><li>To translate this definition into symbols, you will have the formula: </li></ul><ul><li>s =  (X-X) ² </li></ul><ul><li> N </li></ul><ul><li>Where: </li></ul><ul><li>s is the standard deviation </li></ul><ul><li> (X-X) ² is the sum of squared deviations from the mean </li></ul><ul><li>N is the total number of scores/cases </li></ul><ul><li>Now, how will you summarize the procedure for computing the standard deviation? The procedure does not differ much from the method you learned earlier to obtain the mean deviation with reference to the example on page 6, table 7, the following steps are carried out. </li></ul>
164. 166. <ul><li>Step I Find the mean for the distribution. </li></ul><ul><li>X </li></ul><ul><li>5  X </li></ul><ul><li>8 X = </li></ul><ul><li>10 N </li></ul><ul><li>12 </li></ul><ul><li>5 40 </li></ul><ul><li> N= 40 = 5 </li></ul><ul><li> = 8 </li></ul><ul><li>Step 2 Subtract the mean from each raw score to obtain deviation </li></ul><ul><li>X X – X </li></ul><ul><li>5 -3 </li></ul><ul><li>8 0 </li></ul><ul><li>10 2 </li></ul><ul><li>12 4 </li></ul><ul><li>5 -3 </li></ul>
165. 167. <ul><li>Step 3 Square each deviation, then add the squared deviations. </li></ul><ul><li>X X-X (X-X) ² </li></ul><ul><li>5 -3 9 </li></ul><ul><li>8 0 0 </li></ul><ul><li>10 2 4 </li></ul><ul><li>12 4 16 </li></ul><ul><li>5 -3 9 </li></ul><ul><li>  (X-X) ² = 38 </li></ul><ul><li>Step 4 Divide the summation of squared deviations by N and get the square root of the results. </li></ul><ul><li> s =  (X-X) ² </li></ul><ul><li> N </li></ul>
166. 168. <ul><li>= 38 </li></ul><ul><li> 5 </li></ul><ul><li>= 7.6 </li></ul><ul><li>= 2.76 </li></ul><ul><li>You can now conclude that the standard deviation for the set of scores 5, & 10,12 and 5 is 2.76. </li></ul><ul><li>The Raw score Formula for Variance and Standard Deviation </li></ul><ul><li>The computation of variance and standard deviation employing the raw score formula is very simple and easy. </li></ul><ul><li>The formula for variance and standard deviation are as follows: </li></ul>
167. 169. <ul><li>X ² </li></ul><ul><li>s ² = - X² </li></ul><ul><li> N </li></ul><ul><li>s = X² </li></ul><ul><li>- X² </li></ul><ul><li>N </li></ul><ul><li>Where </li