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Journal Article

Exploring Students' Conceptions of the Standard Deviation

01 May 2005-Statistics Education Research Journal (International Association for Statistical Education (IASE))-Vol. 4, Iss: 1, pp 55-82
TL;DR: This article investigated introductory statistics students' conceptual understanding of the standard deviation and found that students moved from simple, one-dimensional understandings of standard deviation that did not consider variation about the mean to more mean-centered conceptualizations that coordinated the effects of frequency (density) and deviation from the mean.
Abstract: SUMMARY This study investigated introductory statistics students’ conceptual understanding of the standard deviation. A computer environment was designed to promote students’ ability to coordinate characteristics of variation of values about the mean with the size of the standard deviation as a measure of that variation. Twelve students participated in an interview divided into two primary phases, an exploration phase where students rearranged histogram bars to produce the largest and smallest standard deviation, and a testing phase where students compared the sizes of the standard deviation of two distributions. Analysis of data revealed conceptions and strategies that students used to construct their arrangements and make comparisons. In general, students moved from simple, one-dimensional understandings of the standard deviation that did not consider variation about the mean to more mean-centered conceptualizations that coordinated the effects of frequency (density) and deviation from the mean. Discussions of the results and implications for instruction and further research are presented.

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01 Jan 2015
TL;DR: The results suggest that the LJQ is a reliable and valid instrument for evaluating LJ.
Abstract: Objectives: Lao Juan (LJ, 劳倦) is a syndrome described in Chinese medicine (CM) that manifests with : Lao Juan (LJ, 劳倦) is a syndrome described in Chinese medicine (CM) that manifests with fatigue, fever, spontaneous sweating, indigestion, work-induced pain, weakness of the limbs, and shortness of breath. fatigue, fever, spontaneous sweating, indigestion, work-induced pain, weakness of the limbs, and shortness of breath. The present study was conducted to examine the reliability and validity of a Lao Juan Questionnaire (LJQ). The present study was conducted to examine the reliability and validity of a Lao Juan Questionnaire (LJQ). Methods: A total of 151 outpatients and 73 normal subjects were asked to complete the LJQ. Seventy-three normal subjects A total of 151 outpatients and 73 normal subjects were asked to complete the LJQ. Seventy-three normal subjects were additionally asked to complete the Chalder Fatigue Scale (CFS). Twelve clinicians determined whether the were additionally asked to complete the Chalder Fatigue Scale (CFS). Twelve clinicians determined whether the 151 outpatients exhibited LJ or not. The internal consistency and construct validity for the LJQ were estimated using 151 outpatients exhibited LJ or not. The internal consistency and construct validity for the LJQ were estimated using data from the outpatient subjects. The CFS data were used to examine the concurrent validity of the LJQ. Total LJQ data from the outpatient subjects. The CFS data were used to examine the concurrent validity of the LJQ. Total LJQ scores and the clinicians' diagnoses of the outpatients were used to perform receiver operating characteristics (ROC) scores and the clinicians' diagnoses of the outpatients were used to perform receiver operating characteristics (ROC) curve analyses and to defi ne an optimum cut-off score for the LJQ. curve analyses and to defi ne an optimum cut-off score for the LJQ. Results: The 19-item LJQ had satisfactory internal : The 19-item LJQ had satisfactory internal consistency (α=0.828) and concurrent validity, with signifi cant correlations between the LJQ and the CFS subscales. consistency (α=0.828) and concurrent validity, with signifi cant correlations between the LJQ and the CFS subscales. In the test of construct validity using principal component analysis, a total of six factors were extracted, and the overall In the test of construct validity using principal component analysis, a total of six factors were extracted, and the overall variance explained by all factors was 59.5%. In ROC curve analyses, the sensitivity, specifi city, and area under the variance explained by all factors was 59.5%. In ROC curve analyses, the sensitivity, specifi city, and area under the curve were 76.0%, 59.2%, and 0.709, respectively. The optimum cut-off score was defi ned as six points. curve were 76.0%, 59.2%, and 0.709, respectively. The optimum cut-off score was defi ned as six points. Conclusions: Our results suggest that the LJQ is a reliable and valid instrument for evaluating LJ. Our results suggest that the LJQ is a reliable and valid instrument for evaluating LJ. KEYWORDS Chinese medicine, chronic fatigue syndrome, Chinese medicine-pattern Chinese medicine, chronic fatigue syndrome, Chinese medicine-pattern

3,787 citations


Cites background from "Exploring Students' Conceptions of ..."

  • ...…are experiencing difficulties with introductory statistics, and emerge with a lack of understanding of core concepts, such as statistical variation (delMas & Liu, 2005), chance (Garfield, 2003), graphical representations of distributions (Bakker & Gravemeijer, 2004), sampling variation (Reading &…...

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Journal ArticleDOI
TL;DR: The authors provide an overview of current research on teaching and learning statistics, summarizing studies that have been conducted by researchers from different disciplines and focused on students at all levels, and suggest what can be learned from the results of each of these questions.
Abstract: Summary This paper provides an overview of current research on teaching and learning statistics, summarizing studies that have been conducted by researchers from different disciplines and focused on students at all levels. The review is organized by general research questions addressed, and suggests what can be learned from the results of each of these questions. The implications of the research are described in terms of eight principles for learning statistics from Garfield (1995) which are revisited in the light of results from current studies.

418 citations

Journal Article
TL;DR: The CAOS test as discussed by the authors is designed to measure students' conceptual understanding of important statistical ideas across three years of revision and testing, content validation, and realiability analysis, and results reported from a large scale class testing and item responses are compared from pretest to posttest in order to learn more about areas in which students demonstrated improved performance from beginning to end of the course, as well as areas that showed no improvement or decreased performance.
Abstract: This paper describes the development of the CAOS test, designed to measure students’ conceptual understanding of important statistical ideas, across three years of revision and testing, content validation, and realiability analysis. Results are reported from a large scale class testing and item responses are compared from pretest to posttest in order to learn more about areas in which students demonstrated improved performance from beginning to end of the course, as well as areas that showed no improvement or decreased performance. Items that showed an increase in students’ misconceptions about particular statistical concepts were also examined. The paper concludes with a discussion of implications for students’ understanding of different statistical topics, followed by suggestions for further research.

251 citations

Journal ArticleDOI
TL;DR: In this article, a systematic review of publications that provide empirical evidence of these misconceptions was presented. But, the methodology used in the empirical studies was very diverse and the sources and possible solutions for these misconceptions were not identified.

190 citations


Cites background or methods or result from "Exploring Students' Conceptions of ..."

  • ...Third, with regard to the data gathering methodology, interviews (e.g., delMas & Liu, 2005), and mixture of multiple-choice and open-answer items can be found (e.g., Vallecillos, 1995, 1996, 2000, 2002)....

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  • ...Our search criteria identified five publications documenting empirical evidence about misconceptions regarding the idea behind the law of large numbers, more specifically: Well et al. (1990), Finch (1998), Sedlmeier (1998), Chance et al. (2004; see next section), and delMas and Liu (2005)....

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  • ...…(2004) More than 100 Variety None assumed 3 similar settings: activity with PC + assessment Software: simulation assessment: posttest and final exam delMas and Liu (2005) 12 – Pre-university 1 group: interview Interview while interaction with computer program Falk and Greenbaum (1995) 53…...

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  • ...Before being able to understand the concept and features of sampling distributions, students should be able to develop a good understanding of the idea of variability (delMas & Liu, 2005)....

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  • ...Another factor making it difficult to compare results of these studies is the differences in sample size, which range from very small (e.g., delMas & Liu, 2005; Vallecillos & Batanero, 1997) to large numbers of participants (e.g., Vallecillos, 1995, 1996, 2000)....

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Journal ArticleDOI
TL;DR: In this article, the first studies on the teaching and learning of statistics appeared in the research literature, the scholarship in this area has grown dramatically, and given the diversity of disciplines, meth...
Abstract: Since the first studies on the teaching and learning of statistics appeared in the research literature, the scholarship in this area has grown dramatically. Given the diversity of disciplines, meth...

134 citations


Cites background or methods from "Exploring Students' Conceptions of ..."

  • ...In a related study, delMas and Liu (2005) studied students’ development of reasoning about standard deviation....

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  • ...The types of tasks used by delMas and Liu (2005) about what makes the standard deviation larger or smaller can be used to promote this type of discussion....

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  • ...Students have shown inconsistencies in their reasoning about the most "elementary" concepts of measures of central tendency and spread (e.g., Chance, delMas, and Garfield, 2004; delMas and Liu, 2005; Groth and Bergner, 2006; Noss, Pozzi, and Hoyles, 1999)....

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  • ...Researchers have found that use of specific technology tools can help illuminate difficulties in understanding particular concepts such as sampling distributions (Chance, delMas and Garfield, 2004) and standard deviation (delMas and Liu, 2005)....

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References
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Book
24 May 2004
TL;DR: The research presented in this thesis is a sequel to design research carried out by Cobb, McClain, and Gravemeijer (2003) and aims at contributing to an empirically grounded instruction theory for early statistics education in which educational computer tools are used.
Abstract: The present knowledge society requires statistical literacy-the ability to interpret, critically evaluate, and communicate about statistical information and messages (Gal, 2002). However, research shows that students generally do not gain satisfactory statistical understanding. The research presented in this thesis is a sequel to design research carried out by Cobb, McClain, and Gravemeijer (2003) and aims at contributing to an empirically grounded instruction theory for early statistics education in which educational computer tools are used. Computer software allows users to dynamically interact with large data sets and to explore different representations in ways that are impossible by hand. The computer tools used were the Statistical Minitools (Cobb et al., 1997), which have been designed for middle school students. One important end goal of instruction was that students would gain understanding of sampling and learn to use 'shape' to reason about distributions. In line with the theory of Realistic Mathematics Education, a central tenet was that learning mathematics should be a meaningful activity. The research questions were: 1. How can students with little statistical background develop a notion of distribution? 2. How does the process of symbolizing evolve when students learn to reason about distribution? In the latter question, 'symbolizing' refers to the reflexive process of making symbols and mentally constructing the objects which they represent. The design research consisted of five cycles of three phases: a design phase, a teaching experiment, and a retrospective analysis. Prior to these cycles, historical and didactical phenomenological analyses (Freudenthal, 1991) were carried out as well as exploratory interviews. The historical study gave rise to hypotheses that were partially tested in the teaching experiments carried out in grades 7 and 8 (12 to 14 years old). In the design phase, a so-called hypothetical learning trajectory (Simon, 1995) was formulated, which was tested and revised in the subsequent cycles of design research. The recurring activity of discussing growing samples proved useful to support reasoning about distribution and sampling. For the analyses of students' learning, a method was used similar to the constant comparative method (Strauss & Corbin, 1998). It turned out that students conceived distributions as 'bumps' consisting of a small group of low values, a large group of 'average' values, and a small group of high values. Peirce's semiotics proved useful for analyzing students' process of symbolizing, in particular, his notions of hypostatic abstraction and diagrammatic reasoning. Hypostatic abstraction refers to the transition from a predicate to a new abstract object ("the dots are spread out" to "the spread is large"). Diagrammatic reasoning consists of three steps: making a diagram, experimenting with it, and reflecting on the results. The research shows the importance of letting students make their own diagrams and discussing these. The computer tools seemed most useful during the experimentation phase. Remarkably, the best diagrammatic reasoning occurred only during class discussions without computers around. One recommendation is: only invest in using computer tools if all educational factors such as teaching, end goals, instructional activities, tools, and assessment are tuned to each other.

230 citations

Journal ArticleDOI
TL;DR: In this paper, the authors emphasize the need for acquiring a greater understanding of statistical thinking and its role in total quality, and present some ideas on how statistical thinking can be used to help achieve total quality.
Abstract: Much of American business has begun to respond to the new economic era in which we live. This response takes many forms. Much of it is aimed at improving the quality of everything we do. Many of us call this effort total qualitv. By aiming at total quality, our businesses will be competitive in global markets, the United States will continue to be a world leader, and we will continue to enjoy a high standard of living. This attention to total quality has brought with it a renewed emphasis on statistics, and in particular, on statistical thinking. The purpose of this article is to emphasize our need for acquiring greater understanding of statistical thinking and its role in total quality. It is my view that this understanding is not adequately developed today and, as a result, we cannot successfully respond to our changing world and the need for total quality. I will present some ideas on how statistical thinking can be used to help achieve total quality.

130 citations


"Exploring Students' Conceptions of ..." refers background in this paper

  • ...Very little research has been conducted on students’ understanding of variability (Reading & Shaughnessy, 2004; Shaughnessy, 1997), despite the central role the concept plays in statistics (Hoerl & Snee, 2001; Moore 1990; Snee, 1990) and an apparent conceptual gap in students’ understanding of...

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Journal ArticleDOI
TL;DR: The authors distinguish two conceptions of sample and sampling that emerged in the context of a teaching experiment conducted in a high school statistics class and argue that the former conception is a powerful one to target for instruction.
Abstract: We distinguish two conceptions of sample and sampling that emerged in the context of a teaching experiment conducted in a high school statistics class. In one conception ‘sample as a quasi-proportional,small-scale version of the population’ is the encompassing image. This conception entails images of repeating the sampling process and an image of variability among its outcomes that supports reasoning about distributions. In contrast, a sample may be viewed simply as ‘a subset of a population’ – an encompassing image devoid of repeated sampling, and of ideas of variability that extend to distribution. We argue that the former conception is a powerful one to target for instruction.

129 citations


"Exploring Students' Conceptions of ..." refers background in this paper

  • ...An understanding of statistical variation and of measures of variation is also needed to understand conceptually complex concepts such as sampling distributions, inference, and p-values (Chance, delMas, & Garfield, 2004; delMas, Garfield, & Chance, 1999; Saldahna & Thompson, 2002; Thompson, Saldahna, & Liu, 2004)....

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Book
09 Apr 2012
TL;DR: This chapter discusses the Six Sigma Improvement Approach, which focuses on the development of a model for Statistical Thinking in the context of Six Sigma Process Improvement.
Abstract: Part I Statistical Thinking Concepts 1 The Need for Business Improvement Overview Today's Business Realities and the Need to Improve We Now Have Two Jobs: A Model for Business Improvement New Management Approaches Require Statistical Thinking Principles of Statistical Thinking Applications of Statistical Thinking Summary of Key Points Exercises References 2 The Overall Statistical Thinking Approach Overview Case Study: The Effect of Advertising on Sales Case Study: Improvement of a Soccer Team's Performance A Model for Statistical Thinking Variation in Business Processes The Synergy Between Data and Subject Matter Knowledge The Dynamic Nature of Business Processes Summary Project Update Exercises References 3 Understanding Business Processes Overview Examples of Business Processes The SIPOC Model for Processes Identifying Business Processes Analysis of Business Processes Systems of Processes The Measurement Process Summary Project Update Exercises References Part II Improvement Strategies And Basic Tools 4 Process Improvement And Problem-Solving Strategies Overview Case Study: Reducing Resin Output Variation Case Study: Reducing Telephone Waiting Time at a Bank The Process Improvement Strategy Case Study: Resolving Customer Complaints of Baby Wipe Flushability Case Study: The Realized Revenue Fiasco The Problem-Solving Strategy Six Sigma Process Improvement Strategy Summary Project Update Exercises References Introduction to Microsoft Excel 5 Process Improvement And Problem-Solving Tools Introduction Relationship of the Tools to the Strategies Data Collection Tools Data Analysis Tools Knowledge-Based Tools Summary Project Update Exercises References Part III Formal Statistical Methods Introduction to Minitab Introduction to JMP 6 Building And Using Models Overview Examples of Business Models Types and Uses of Models The Regression Modeling Process Building Models With One Predictor Variable Building Models With Several Predictor Variables Multicollinearity, Another Model Check Some Limitations of Using Existing Data Summary Project Update Exercises References 7 Using Process Experimentation To Build Models Overview Why Do We Need a Statistical Approach? Examples of Process Experiments The Statistical Approach to Experimentation Two-Factor Experiments: A Case Study Three-Factor Experiments: A Case Study Larger Experiments Blocking, Randomization, and Center Points Summary Project Update Exercises References 8 Applications Of Statistical Inference Tools Overview Examples of Statistical Inference Tools The Process of Applying Statistical Inference Statistical Confidence and Prediction Intervals Statistical Hypothesis Tests Test for Continuous Data Test for Discrete Data Test for Regression Analysis Sample Size Formulas Summary Project Update Exercises References 9 The Underlying Theory Of Statistical Inference Overview Applications of the Theory The Theoretical Framework of Statistical Inference Types of Data Probability Distributions Sampling Distributions Linear Combinations Transformations Summary Project Update Exercises References 10 Summary And Path Forward Overview A Personal Case Study by Tom Pohlen Case Study: Newspaper Accuracy Review of the Statistical Thinking Approach Text Summary Potential Next Steps to Deeper Understanding of Statistical Thinking Project Summary and De-Briefing Exercises References Appendix A Effective Teamwork Appendix B Presentations and Report Writing Appendix C More on Surveys Appendix D More on the Six Sigma Improvement Approach Appendix E More on Design of Experiments Appendix F More on Inference Tools Appendix G More on Probability Distributions Appendix H Process Design (Reengineering) Appendix I t Critical Values Appendix J Standard Normal Probabilities (Cumulative z Curve Areas) Index

129 citations


"Exploring Students' Conceptions of ..." refers background in this paper

  • ...Shaughnessy, 2004; Shaughnessy, 1997), despite the central role the concept plays in statistics ( Hoerl & Snee, 2001; Moore 1990; Snee, 1990) and an apparent conceptual gap in students’ understanding of...

    [...]

  • ...Very little research has been conducted on students’ understanding of variability (Reading & Shaughnessy, 2004; Shaughnessy, 1997), despite the central role the concept plays in statistics (Hoerl & Snee, 2001; Moore 1990; Snee, 1990) and an apparent conceptual gap in students’ understanding of...

    [...]