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The Challenge of Developing Statistical Reasoning

This newspaper defines statistical intelligent and reviews research on this subject. Types of right and incorrect reason are summarized, and statistical reasoning about sampling distributions is examined in more detail. A model of statistical intelligent is presented, and suggestions are offered for assessing statistical argue. The paper concludes with implications for teaching students in ways that will facilitate the exploitation of their statistical argue .

1. What is Statistical Reasoning?

statistical reason may be defined as the room people reason with statistical ideas and make sense of statistical information ( Garfield and Gal 1999 ). This involves making interpretations based on sets of data, graphic representations, and statistical summaries. much of statistical reasoning combines ideas about data and find, which leads to making inferences and interpreting statistical results. Underlying this reason is a conceptual sympathy of authoritative ideas, such as distribution, concentrate, diffuse, association, doubt, randomness, and sample. statistical argue is a topic of concern to many types of people, including :

  • Psychologists, who study how people make judgments and decisions involving statistical information ( frequently using incorrect intuitions or misconceptions ),
  • Doctors and others in the medical profession, who need to understand and interpret risks, chances of different medical outcomes, and test results ,
  • Journalists and skill writers, who are interested in how to best excuse and review statistical information in the media ,
  • political analysts, who are concern in studying and interpreting polls and elections, and
  • Statistics teachers, who want to teach students not only a set of skills and concepts but besides how to reason about data and prospect .

The formulation “ statistical reasoning ” is widely used and appears in many different context. A Web search using the idiom “ Statistical Reasoning ” produced a list of about three thousand Web pages that contain the words “ statistical ” and “ reasoning. ” This list revealed the stick to categories of Web pages :

  • Advertisements for statistics textbooks ( that have “ statistical reasoning ” in their titles or promotional materials ) ,
  • Materials from Colleges ‘ or instructors ‘ Web pages for statistics courses offered in a assortment of different disciplines ( such as mathematics, statistics, psychology, education, engineer, physical therapy, and the health sciences ), and
  • Presentations, grant proposals, and papers that include discussions of statistical reason .

The Web research besides produced the base page of the electronic Journal of Applied Statistical Reasoning and a koran ( not a statistics casebook ) devoted to the topic of improving statistical reason ( Sedlmeier 1999 ). A quick scan of the materials on the Web that describe courses or textbooks suggests that people are using the term “ statistical reasoning ” to represent the desire outcomes of a statistics course, and that this saying is used interchangeably with “ statistical thinking. ” There were no clear definitions offered regarding what statistical intelligent means and there did not appear to be clear connections between what was in a path or textbook and the development of finical reasoning skills. For exemplar, some courses were traditional statistics courses with a focus on computations and no use of computing packages. Some courses were more focus on concepts and big ideas, while other courses combined concepts, calculation, and computing. overall, there did not appear to be a consensus in the broad statistics community as to what statistical intelligent means and how to develop this type of reasoning in statistics courses. The adjacent segment reviews some of the research literature on statistical reason and the ways this term has been used in research studies .

2. Statistical Reasoning in the Research Literature

Chervaney, Collier, Fienberg, Johnson, and Neter ( 1977 ) and Chervaney, Benson, and Iyer ( 1980 ) defined statistical reason as what a student is able to do with statistical content ( recalling, recognizing, and discriminating among statistical concepts ) and the skills that students demonstrate in using statistical concepts in particular trouble solving steps. They viewed statistical reason as a three-step process :

  • Comprehension ( seeing a particular problem as like to a class of problems ) ,
  • Planning and execution ( applying appropriate methods to solve the problem ), and
  • evaluation and interpretation ( interpreting the result as it relates to the original problem ) .

The authors proposed a systems approach path for teaching and assessing statistical reasoning based on this model. however, there is no print literature to describe or support the use of this model. In their unique volume on teaching statistics, Hawkins, Jolliffe, and Glickman ( 1992 ) discuss statistical argue and thinking together, summarizing some of the research in this area, but not distinguishing between the two types of processes. These authors acknowledge that little is known about the statistical reason process or about how students actually learn statistical ideas. Nisbett ( 1993 ) edited a collection of research studies conducted by himself and many of his colleagues on statistical reasoning and ways to train people to follow “ rules for reasoning. ” early in this collection he offers a set of generalizations based on the research in this sphere which are summarized below :

  1. People have intuitive rules that apply to some statistical problems in everyday liveliness .
  2. There are individual differences in the degree to which people understand these rules and apply them to concrete problems .
  3. instruction in statistics changes the way people view the worldly concern .
  4. The rule systems people use are at the “ dinner dress operations ” level of abstraction .
  5. People may be able to apply statistical rules in one jell ( for model, to random generating devices such as die ) but rarely or never to like problems that involve social content .
  6. Training people to apply rules by coding a world in terms of the rule can improve their use of these rules across domains .

Nisbett concludes that people can improve their statistical reasoning if they learn statistical rules, that some people will understand these rules better than others, that everyone ‘s use of statistical rules can by improved by direct education, that this teaching can be abstract, and that people should be teach how to “ decode the world ” to make it easier for them to apply these rules. He admits that instructors can do a much better job of teaching statistical rules than they presently do. research by this author on assessing statistical reason ( see Garfield 1998a, 1998b ), revealed that students can often do well in a statistics course, earning good grades on homework, exams, and projects, however still perform ill on a measure of statistical reasoning such as the Statistical Reasoning Assessment ( Garfield 1998b ). These results suggest that statistics instructors do not specifically teach students how to use and apply types of reasoning. alternatively, most instructors tend to teach concepts and procedures, provide students opportunities to work with data and software, and hope that reasoning will develop as a leave. however, it appears that reasoning does not actually develop in this way. current research ( see delMas, Garfield and Chance 1999 ) is focused on exploring and describing the development ( and assessment ) of statistical reason skill, particularly in the area of statistical inference. Sedlmeier ( 1999 ) claims that statistical intelligent is rarely teach and when it is teach ( that is, training people to use specific rules such as those described by Nisbett and colleagues ), it is rarely successful. He discusses “ everyday statistical reasoning, ” summarizes the research on coach to improve statistical intelligent, and presents some aim programs of his own designed to teach people to correctly consumption specific types of reason ( involving conditional probabilities, samples, and bayesian inference, for examples ). Lovett ( 2001 ) provides a detailed review of the inquiry on statistical reason, which she views as falling into one of three approaches : theoretical studies ( in the 1970 ‘s ), empirical studies ( in the 1980 ‘s ), and classroom-based studies ( in the 1990 ‘s ). She describes her own employment at Carnegie Mellon University, which is aimed at both understanding and improving students ‘ statistical argue, and which integrates all three approaches. She suggests a model for a learn environment to help students develop adjust statistical reasoning which will be evaluated in future research studies.

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To summarize the research studies on statistical reasoning described above, it appears that inquiry in this area is still evolving. There is no clear consensus about how to help students develop statistical reason or how to determine the tied and correctness of their reasoning. possibly with more classroom-based studies that examine finical types of reason, the prerequisite cognition and skills for each type of argue, and the affect of different instructional activities on intelligent, researchers may be better able to understand the process of how compensate statistical reasoning develops .

3. Types of Correct and Incorrect Statistical Reasoning

There is an abundance of research on incorrect statistical reasoning, indicating that statistical ideas are frequently misunderstood and misused by students and professionals alike. Psychologists ( such as Kahneman, Slovic, and Tversky 1982 ) and educators ( such as Garfield and Ahlgren 1988 ) have collected convert information that shows how people often fail to use the methods learned in statistics courses when interpreting or making decisions involving statistical information. This body of research indicates that inappropriate reasoning approximately statistical ideas is far-flung and dogged, similar at all old age levels ( even among some experience researchers ), and quite difficult to change. Some of the types of errors and misconceptions are described below : Misconceptions involving averages: Averages are viewed as the most coarse number ( the value that occurs more much than the others ). People often believe that to find an average one must always add up all the numbers and separate by the phone number of data values ( careless of outliers ). A mean is viewed as the lapp thing as a medial, and there is a impression that one should always compare groups by focusing entirely on the dispute in their averages. The outcome orientation: An intuitive model of probability that leads students to make yes or no decisions about single events rather than looking at the series of events ( Konold 1989 ). For example, a weather forecaster predicts the chance of rain to be 70 % for 10 days. On 7 of those 10 days it actually rained. How good were his forecasts ? many students will say that the forecaster did not do a commodity job, because it should have rained on all days on which he gave a 70 % find of rain. These students appear to focus on outcomes of single events preferably than being able to look at series of events. To students with an result orientation, a 70 % find of rain means that it should rain. similarly, a calculate of 30 % rain would mean it will not rain. Good samples have to represent a high percentage of the population: Most people have impregnable intuitions about random sample and most of these intuitions are ill-timed ( Kahneman, et aluminum. 1982 ). While the Law of Large Numbers guarantees that large samples will be example of the population from which they are sampled, students ‘ intuitions tell them that it is the ratio of the sample size to the population that is more crucial to consider. many believe that it does not matter how big a sample is or how well it was chosen, but that it must represent a big share of a population to be a good sample. consequently, they may be doubting about a sample that is very large, but represents a belittled percentage of the population. They do not realize that happy samples do a good job of representing a population, even if the proportion sample size to population size is small. The “law of small numbers”: People believe that samples should resemble the populations from which they are sampled. many people besides believe that any two random samples, careless of how modest they are, will be more alike to each other and to the population than sampling theory would predict. This misconception has led even experienced researchers to use little samples for making inferences and generalizations about populations ( Kahneman, et aluminum. 1982 ). The representativeness misconception: People estimate the likelihood of a sample based on how closely it resembles the population. consequently, a particular sequence of n tosses of a fairly mint that has an approximately even mix of heads and tails is judged more likely than a sequence with more heads and fewer tails. For example, the solution HTHHTT is judged as a more likely succession of 6 tosses of a carnival coin than HTHHHH ( Kahneman, et alabama. 1982 ). Another example of this misconception is the Gambler ‘s Fallacy, which is found in people who believe that after a long series of heads when tossing coins, a tail is more likely to occur on the future flip than another oral sex. This is a fallacy, because if the coin is average, then the probability of getting a head or a tail on the future discard is equally likely. The Equiprobability bias: Different outcomes of an experiment tend to be viewed as equally probable. For exemplar, if there are different numbers of skill majors and business majors in a class, some students may view as evenly probable the outcomes of selecting a science major or selecting a occupation major when one students is randomly drawn from the class list. Another example is when students are asked to compare the chances of getting different outcomes of three die rolls, students tend to judge as equally likely the prospect of rolling three fives and the prospect of obtaining precisely one five. however, the probability of rolling one five is higher than the probability of obtaining three fives, because there are several ways to roll one five, and merely one way to roll three fives ( Lecoutre 1992 ). In contrast to the shape of psychologists, who have focused on errors in reasoning and misconceptions, educational researchers have focused on the types of adjust reasoning they would like students to develop. Some have designed instructional materials or activities to develop reasoning skills such as the ones described below : Reasoning about data: Recognizing or categorizing data as quantitative or qualitative, discrete or continuous ; and knowing why the type of data leads to a detail type of table, graph, or statistical standard. Reasoning about representations of data: Understanding the way in which a plat is meant to represent a sample distribution, understanding how graph may be modified to better represent a data set ; being able to see beyond random artifacts in a distribution to recognize general characteristics such as supreme headquarters allied powers europe, center and spread. Reasoning about statistical measures: Understanding why measures of concentrate, spread, and position tell different things about a data set ; knowing which are best to use under different conditions, and why they do or do not represent a datum set ; knowing why using summaries for predictions will be more accurate for big samples than for little samples ; knowing why a thoroughly compendious of data includes a meter of center vitamin a well as a measure of spread and why summaries of center field and spread can be useful for comparing data sets. Reasoning about uncertainty: Correctly using ideas of randomness, opportunity, and likelihood to make judgments about uncertain events, knowing why not all outcomes are evenly likely, knowing when and why the likelihood of unlike events may be determined using different methods ( such as a probability tree diagram, a model using coins, or a calculator program ). Reasoning about samples: Knowing how samples are related to a population and what may be inferred from a sample, knowing why a well chosen sample will more accurately represent a population and why there are ways of choosing a sample distribution that make it unrepresentative of the population ; knowing to be disbelieving of inferences made using little or bias samples. Reasoning about association: Knowing how to judge and interpret a relationship between two variables, knowing how to examine and interpret a bipartite postpone or scatterplot when considering a bivariate kinship, knowing why a solid correlation coefficient between two variables does not mean that one causes the early. The adjacent section gives an exercise of one of these areas by describing a series of educational research studies investigating students ‘ reasoning about samples and sampling .

5. Assessing Statistical Reasoning

Most judgment instruments used in research studies of statistical reasoning and understand consist of items presented to students or adults individually as part of clinical interviews or in little groups. In contrast, traditional paper-and-pencil assessment instruments much focus on computational skills or trouble solving rather than on reason and understand. Questions and task formats that culminate in elementary “ right or wrong ” answers do not adequately reflect the nature of students ‘ thinking and problem resolve, and consequently provide only limited information about students ‘ statistical intelligent processes and their ability to construct or interpret statistical arguments ( Gal and Garfield 1997 ). Garfield and Chance ( 2000 ) offer some suggestions for classroom assessment techniques to evaluate students ‘ statistical reason. These include : Case studies or authentic tasks: detailed problems based on a real context that uncover students ‘ strategies and interpretations as they solve the problem. Concept maps: ocular representations of connections between concepts that students may either complete or construct on their own. Critiques of statistical ideas or issues in the news: short written reports to reveal how well students reason about information provided in a newsworthiness article, including comments on missing information a well as conclusions and interpretations offered in the article. Minute papers: brief, anonymously written remarks provided by students that may include explanations of what they have learned, comparisons of concepts or techniques, etc. Enhanced multiple-choice items: items that require students to match concepts or questions with appropriate explanations, may be used to capture students ‘ reason and measure conceptual agreement The Statistical Reasoning Assessment: a multiple-choice test designed to assess students ‘ chastise and incorrect reason for a sample distribution of statistical concepts ( Garfield 1998a, 1998b ). informal methods may besides be used during course activities, such as asking students to provide written or verbal interpretations of data, explanations of concepts, or matching different types of representations ( for example, matching boxplots to histograms, or matching graph to statistics, as shown in Scheaffer et aluminum. 1996 ). These methods can help inform an teacher about the level of students ‘ statistical reasoning about particular concepts or procedures, which may be quite different from students ‘ ability to compute and carry out procedures. The contrast in these two types of assessments can be quite startle angstrom well as enlighten, and may suggest that extra instructional activities are needed .

6. Summary: Implications for Teachers

Although the term “ statistical argue ” is frequently used in different ways, it appears to be universally accepted as a goal for students in statistics classes. It has been shown that statistical argue used in casual life angstrom well as in classes is much incorrect, ascribable to the different intuitions and rules that people use when evaluating statistical information.

While most instructors would like their students to develop statistical reason, research shows that it is not enough to instruct students about the right rules and concepts in order for them to develop an integrate understand to guide their argue. It may be tempting to conclude that if students have been well teach and have performed well on exams, that they are able to reason correctly about statistical information. however, unless their reasoning is carefully examined, specially in apply context, these students may only be at the early stages of reasoning and not have an integrated sympathy needed to make correct judgments and interpretations. Activities specifically designed to help develop students ‘ statistical intelligent should be carefully integrated into statistics courses. For exemplar, activities such as having students match verbal descriptions to plots of data ( see Rossman and Chance 2001 ) can help develop reasoning about data and distribution, and activities that challenge students to consider what makes a standard deviation larger or smaller ( see delMas 2001 ) can help students develop reasoning about variability. similarly, activities that steer students through ocular simulations of sampling distributions, varying sample size and population parameters, can help develop students ‘ reasoning about sampling distributions. In order to better understand how well students are developing and using statistical reason, statistics teachers may include assessments that meter students ‘ reasoning so that they are not misled in their evaluation of student eruditeness ( that is, high scores on computational procedures ). holocene inquiry on developmental models of statistical reason may help instructors better understand the process of developing decline statistical reason and guide them in developing instructional methods and assessments. More research is needed, particularly classroom research conducted in a variety of settings, to help determine how instructional methods and materials may best be used to help students develop correct statistical reason .

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