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MAKING A GROUPED-DATA FREQUENCY TABLE: DEVELOPMENT AND
EXAMINATION OF THE ITERATION ALGORITHM
A dissertation presented to the faculty of the College of Education of Ohio University
In partial fulfillment of the requirements for the degree Doctor of Philosophy
Hippolyte O. Lohaka November 2007
This dissertation titled Making a Grouped-data Frequency Table: Development and Examination of the Iteration Algorithm
by HIPPOLYTE O. LOHAKA
has been approved for the Department of Educational Studies and the College of Education by
Gordon P. Brooks Associate Professor of Educational Studies
Renée A. Middleton Dean, College of Education
This dissertation is dedicated to the memories of my late father César Lohaka and my late daughter Sibyl Walo Lohaka.
ACKNOWLEDGMENTS
Throughout the entire process of my doctoral program, I have relied on the support of so many tremendous people. First of all, and most significantly, I would like to offer my heartfelt gratitude to the Almighty God who created me and gave me all the intellectual and physical abilities, with which I have been able to achieve whatever life and educational goals I set for myself. Second, I wish to express my utmost gratitude and appreciation to my academic advisor and dissertation committee chair Dr. Gordon Brooks for his unremitting backing, feedback, insight, and guidance through my unending years of uncertainty and ambiguity. I could not have completed this without his encouragement, challenge, and mentoring. I am also grateful to my other dissertation committee members: Dr. Teresa Franklin, Dr. George Johanson, and Dr. Robert Klein for their helps particularly with sampling plan and data collection procedures. The contributions they made to this research study are incommensurable. In addition, my many thanks go to the Mathematics and Psychology department professors who kindly accepted to administer my questionnaires to their elementary statistics students. Without their cooperation, it would not have been possible to complete, as they are, the last two chapters of this project. A special thank you goes to Dr. Timothy Kayworth, my current Department Chair, for providing continual support and reducing my teaching load during my first year. Last but not least, special thanks are due to Helene Obonga Lohaka, my wife and life partner. All her emotional supports, encouragements, and caring love were constant sources of strength and inspiration during the difficult times of this study.
LIST OF FIGURES
- Abstract…………………………………………………………………………………....
- Dedication………………………………………………………………………………....
- Acknowledgements………………………………………………………………………..
- List of Tables.……………………………………………………………………………
- List of Figures……………………………………………………………………………
- Chapter 1: Introduction…………….....….……………..……………………………….. - Forms of Tabulation Arrangement…………...…..………….…………..………. - Presenting Data in an Ordered Array.….….………....…..………………. - Example of an Ordered Array……………..…..…………………. - Presenting Ordered Array Data in a Stem-and-Leaf Plot………………… - Example of a Stem-and-Leaf Plot..…..……...………………….... - Presenting Data in a Simple Frequency Table..…………..……………… - Example of a Simple Frequency Table..……..…..……………… - Presenting Data in a Grouped-data Frequency Table.…………..……..… - Example of a Grouped-data Frequency Table……..……………... - Presenting Data in a Histogram…………………………..……………… - Example of a Histogram…………………………..…………..…. - Rationale for the Study…………………………..………..………………………
- Statement of the Problem…..……………………………………..……………….
- Research Questions.…….………….……….…………………....…….…………. - Monte Carlo Simulations…………………….……….………..…………
- Survey of Attitudes towards the Method Used……….…………..………
- Research Hypotheses………………….……………………………….………… - Monte Carlo Simulations……………………….……..….………………
- Survey of Attitudes towards the Method Used………..….………………
- Significance of the Study………………..….……………………….…………… - Difference between the Iteration and Traditional Methods…..….….…… - Benefits expected to be gained over the Traditional Methods.……..…… - Summary of Major Differences and Benefits……………………….……
- Limitations and Delimitations of the Study…………………………………...….
- Definitions of Terms…………………………………………………………..…. - Attitude……………………....…………………………………….…….. - Class Boundary……………....…………………………………….…….. - Class Frequency...…………….…………………………………….……. - Class Interval…...………………………………………….…………….. - Class Limits....………………..………………………………………….. - Class Midpoint…………….….…………………………….……………. - Class Width…………………………………………………...………….. - Coefficient of Variation .……..………………………………………….. - Frequency Distribution………..…………...…………………………….. - Grouped Data………………….....………...…………………………….. - Grouped-Data Frequency Distribution..…...…………………………….. - Iteration Algorithm….…………..……………………………………….. - Range…….……………………..…………………………….………….. - Raw Data..……..……………….……………….……………………….. - Simple Frequency Table…..…….……………………………………….. - Traditional Method…………..…..………………………………………..
- Organization of the Study……………………….………………………………..
- Chapter 2: Review of the Literature………………….…………………………..…….... - Introduction……………….…………….………………………….…………….
- Review of the Relevant Literature….…….…..…………………….……………. - Vague Guidelines……………………….……………………………….. - Presenting Data in a Frequency Table…..……………………………..… - Presenting Data in a Simple Frequency Table..…………………. - Presenting Data in a Frequency Table for Grouped Data..….…… - Advantages…………………………….…..………….…. - Disadvantages………………………………….………… - Summary of Advantages and Disadvantages…………..… - Presenting Data in a Graph…………..………………………………..…. - Histogram………………………………………………………… - Traditional Methods……………………………………………………… - Inclusive Range…….……………………………………………. - Number of Class Intervals……………………………………….. - Class Interval Size……………………………………………….. - Starting Point…………………………………………………….. - Tallies……………………………………………………………. - Summary Statistics ..………………..…………………………..……….. - From the Teaching and Learning Statistics’ Perspectives…………....….. - Definitions of Teaching, Learning, and Education……………… - Various Teaching Methods Documented…………….……….... - Teaching and Learning Statistics………………………………. - Factors Affecting Attitudes towards Statistics.………………… - Unique Solution Feature………….…………………..………… - Structural and Organizational Issues…..……………………….. - Reasoning about Graphical Representation of Data….………… - Attitudes towards Statistics………..……………..………….….. - Summary and Conclusion…………..……………..………….…
- Chapter 3: Iteration Algorithm Development and Examination….………...……….…
- Development of a Traditional Method………..………..…………………….… - Determining the Range…..…………………….…….…………. - Choosing a Number of Class Intervals………..……..…………. - Computing a Class Width…….……………….…….…………. - Picking up a Starting Point……….………………….…………. - Table Preparation and Construction……………………….……
- Development of the Iteration Algorithm….………..……..…………………… - Determination of the Inclusive Range of Grouped Data….……. - Selecting a Number of Class Intervals………..…….……..……. - Fixing on the Class Interval Size....………….…….…………… - Setting up the Starting Point………………………..…………… - Table Preparation and Construction…………………………..…
- Iteration Algorithm versus Traditional Method…….……………………..…….
- Monte Carlo Simulations…………………………………………..……………. - Random Generation of Normal Data…………………………..……..… - Main Outcomes………………………………………………..……..… - Conclusion…………….……………..………………………………….
- Chapter 4: Methodology..………….….….…..………………..………………………. - Research Design……………………….……………………………………….. - Operational Definitions of the Variables……………….…………..…………... - Survey of Attitudes of the Method Used…………….…………………. - Measures to Represent the Student Perceived Ability in Statistics…..… - Identification of Population……….….…………………….…………………... - Sampling Plan……………………..…………………………………..………...
- Instrumentation……………………………………………………………..…… - Selection/Development of Instrument………………………….………. - Pilot Studies…………….…………………………………………….………… - Reliability Issues……………..………………………………………… - Cronbach’s Alpha as an Index of Reliability…..………………… - Pilot Studies Results……………………………………………… - Data Collection Procedures…………………………………………….. - Data Analysis Procedures……………………….…………….………..………. - Statistical Methods…………………………………….……………….. - Power and Effect Size…………………………………………………..
- Chapter 5: Data Analysis…………………………..…………..………………………. - Results………………………………….……………………………………….. - Data Analysis……….………………….……….…………..…………... - Survey of Attitudes towards Method Used (SAMU)...……….… - MANCOVA Results………………………….…………………. - Assumptions when Using MANCOVA.……………………….... - Analyzing and Interpreting MANCOVA Results……………..… - Assumptions when Using ANCOVA.………………………........ - Univariate ANCOVA Follow-up Analyses…………………….... - Conclusion………….………………………..……………………………..……
- Chapter 6: Discussion, Conclusions, Recommendations…………………..…………... - Summary of the Results and Findings………………………………………….. - Discussion of the Results and Findings ….…………….…………..…………... - Explanations of the Results………………..…..…………….……….… - Implications of the Study…..……………………………..….…………………. - Implications for Students and Teachers..………………….………….... - Implications for Researchers and Practitioners…………………...……. - Summary of the Implications……………………………….…………... - Limitations……………………………………………………………………… - Conclusion………….……….…….….…………………….…………………... - Recommendations for Future Research…………………….…………………...
- References…....……………...………………………………………………………….
- Appendix A: Informed Consent form (Protocol #05E080).............................................
- Appendix B: Ohio University Consent Form Template………….…………….………
- Appendix C: Exercise I (with Integers)..……………………………………………….
- Appendix D: Construction of the Grouped-data Frequency Table (Traditional)………
- Appendix E: Construction of the Grouped-data Frequency Table (Iteration)….………
- Appendix F: Survey of Attitudes towards the Method Used (SAMU)………………… - Appendix G: Exercise II (with Decimals)....…………………………………………...
- Appendix H: Construction of the Grouped-data Frequency Table (Iteration)….………
- Appendix I: Construction of the Grouped-data Frequency Table (SPSS)….….……….
- Appendix J: A Copy of the E-Mail Sent to the SPSS Corporate Headquarters………...
- Appendix K: List of Abbreviations………………………………………….………….
- Appendix L: Sample Set of Scatter Plots for Pairs of Dependent Variables..………….
- Appendix M: Additional Output of the Exploratory Factor Analysis…............……….
- Table 1. Grade of 80 Students in the Final Statistics Exam...…………………....……… LIST OF TABLES
- Table 2. Ordered Grades of 80 Students in the Final Statistics Exam..……………….… - Table 3. Simple Frequency Distribution of the Grades of 80 Students..…………..……
- Table 4. Grouped-data Frequency of the Grades of 80 Students….………………….….
- Table 5. Differences and Benefits of Iteration Algorithm over Traditional Methods..….
- Table 6. Appropriate Number of Classes in Frequency Distributions as per Keller.....…
- Table 7. Value of k for Different n Numbers according to Each Formula……………… - (Traditional Method with First Option)….…………………………..…. Table 8. Grouped-data Frequency Table for the Final Statistics Grades of 80 Students - (Traditional Method with Third Option)……….…………………….…. Table 9. Grouped-data Frequency Table for the Final Statistics Grades of 80 Students - Sincich et al…….……………………………………………………….. Table 10. General Rule for Determining the Number of Classes in a Histogram by - Iteration Algorithm)…………………………………………………….. Table 11. Grouped-data Frequency Table for the Final Statistics Grades of 80 Students
- Table 12. Overall Summary Table of Square Roots of Deviations..……………...........
- Table 13. Summary Table by Sample Size of Square Roots of Deviations…….…...…
- Table 14. Reliability Statistics between Iteration and Traditional Algorithms………...
- Table 15. Inter-item Correlation Matrix for the Iteration Algorithm.…………………..
- Table 16. Inter-item Correlation Matrix for the Traditional Algorithm.………….........
- Table 17. Reliability Statistics for Iteration Algorithm and Traditional Method……… - (Principal Component Analysis)……………………………………….. Table 18. Partial outputs of the Initial Exploratory Factor Analysis Summary
- Table 19. Means and Standard Deviations of the SAMU……….….…..………...…….
- Table 20. Inter-correlations of the SAMU (Iteration – Mathematics)…...……...……...
- Table 21. Inter-correlations of the SAMU (Iteration – Psychology)……........………...
- Table 22. Inter-correlations of the SAMU (Tradition – Mathematics)….…..……….....
- Table 23. Inter-correlations of the SAMU (Traditional – Psychology)…....…..…….....
- Table 24. Selected Output for Multivariate Analysis of Covariance (PMA).......……...
- Table 25. Correlations between Dependent Variables and Covariate…………….........
- Table 26. Adjusted Means of the SAMU for Method…………………..………….......
- Table 27. Adjusted Means of the SAMU for Department……………..….....................
- Table 28. Adjusted Means of the SAMU for Method by Department…………….....…
- Table 29. Levene’s Test of Equality of Error Variances……..………………..……….
- Table 30. Selected Output for Univariate F-Tests……………………………….…..…
- Table 31.Elementary Statistics Final Exam Grades of 65 students………….........……
- Table 32. Grouped-data frequency Table for the Grades (Traditional)………………...
- Table 33. Grouped-data Frequency Table for the Weights (Iteration)…..…….…..…...
- Table 34. Thickness (in mm) of 100 Mass-Produced Metal Blocks…...……………… - of Metal Blocks Random Sample SAMPLEQ3 (Iteration Algorithm)… Table 35. Grouped-Data Frequency and Cumulative Frequency Table for the Thickness - of Metal Blocks Random Sample SAMPLEQ3 (SPSS Algorithm)……. Table 36. Grouped-Data Frequency and Cumulative Frequency Table for the Thickness
- Table 37. Comparison of the Basic Statistics from the Raw Data……………………...
- Figure 1. Grades stem-and-leaf plot……………………….….………...………………..
- Figure 2. Histogram produced by the SPSS default format.....………......……………....
- Figure 3. Histogram modified with a traditional method’s choices…...............................
- Figure 4. Histogram with the iteration algorithm’s choices..…………..………………
- Figure 5. SampleQ3…………….……………………………….…..…………….……
- Figure 6. Scatter Plot for Value versus Difficulty (Iteration – Mathematics)………….
- Figure 7. Scatter Plot for Cognitive versus Difficulty (Iteration – Mathematics)...........
- Figure 8. Scatter Plot for Affect versus Difficulty (Iteration – Psychology)..………….
- Figure 9. Scatter Plot for Cognitive versus Difficulty (Iteration – Psychology)..…...…
- Figure 10. Scatter Plot for Affect versus Cognitive (Traditional – Mathematics)...……
- Figure 11. Scatter Plot for Value versus Cognitive (Traditional – Mathematics)….......
- Figure 12. Scatter Plot for Value versus Difficulty (Traditional – Psychology)……….
- Figure 13. Scatter Plot for Value versus Affect (Traditional – Psychology)..………….
CHAPTER 1: INTRODUCTION
Once the data have been gathered from a population of interest, it is often difficult to get a sense of what the data indicate when they are presented in an unorganized fashion. Assembling the raw data into a meaningful form, such as a frequency distribution, makes the data easier to understand and interpret by a broad audience in a relatively short period of time. It is in the context of frequency distributions that the importance of conveying in a succinct way numerical information contained in the data to a general public is encountered. Communication is at the heart of statistics and it implies finding out significant patterns in a given dataset (Rangecroft, 2002). This view is shared by Chatfield (1995) who recognized communication as an important component of investigation that should be incorporated into statistics education. Brase and Brase (1999) also concurred by stressing “no matter what type of data is available, sample or population, it is important to be able to organize and present the data to other people” (p. 23). Brase and Brase went on to infer “for this purpose, tables and graphs are indispensable” (p. 23). As Hulsizer (2004) appositely expressed it, tables and graphs allow a researcher to “make sense of his/her own data, communicate his or her results to others who are interested in them, and actually see relationships in the data” (p. 12). Unfortunately, the nature of large data sets is difficult to either communicate to any intended audience or make a decision without considering some logical procedure for organizing and summarizing data for a meaningful representation. This can be done either in a tabulated arrangement, graphically, or statistically (Aron & Aron, 2003; Aryee, 2003; Roscoe, 1975).
done in 3 different forms, according primarily to the size of the data collected (Aryee, 2003; Hopkins & Glass, 1978). First, if the size of the data collected ranges from small to medium, the data collected may be arranged in an ordered array. It is an arrangement of data items in either an ascending (from lowest to highest value) or descending (from highest to lowest value) order of magnitude (refer, for instance, to Isaac & Michael, 1997, p. 166). Second, if the size ranges from medium to large, the data can be presented into a frequency distribution with an interval size (or interval width) of one. Frequency distributions with an interval width of one are referred to as simple frequency distributions. Frequency distributions with an interval width of one are used to report exactly each and every of the values observed of the variable that has been measured, traditionally, starting with the smallest value at the beginning and ending with the highest value. Third, if the sample size is large or very large, the “data are often summarized into a frequency distribution with an interval width greater than one, which reports the number of observations in a set of data within a given range of values of the variable that has been measured” (Raymondo, 1999, p. 48). Frequency distributions with an interval size bigger than one will be referred to as grouped-data frequency distributions. For comparison purposes, the same large dataset is carried throughout the first 3 chapters to exemplify the 3 different forms of tabulation arrangement (that is, ordered array, simple frequency table, and grouped-data frequency table). A frequency distribution is only a major method of data reduction. Another major method of data reduction is to present the data in some type of graphic display. Since the data used for
illustration purposes in this study are quantitative data, two types of graphic display retained are the stem-and-leaf plot for the ordered array data and the histogram for the grouped data. Presenting Data in an Ordered Array A simplest way for organizing raw data in a systematic order is usually the ordered array. The ordered array is perhaps more useful when only a few measures are involved (usually if there are 20 or fewer observations). The ordered array helps to see at once the maximum and minimum values. It also gives a rough idea of the distribution of the items over the range. In contrast, when the number of data items involved is significantly large, the ordered array appears a rather awkward data organization tool. “Listing the values individually for bigger datasets consumes too much space and makes it difficult to visualize the important patterns in the data” (Sanders, Murphy, & Eng, 1976, p. 45). Table 1 lists grades of 80 students obtained in the final exam of an elementary statistics class. The grades are scored out of 100 points and rounded to the nearest whole number.
