Research

  • Almost all studies of adult notions of correlation between dichotomous variables show that people do not incorporate two conditional probabilities as they should according to normative definitions. However, these studies disagree considerably about what correlational notions people do have. This paper identifies three factors that contribute to the variability in research results. The first two factors were mentioned in the literature, and the evidence concerning them is summarized: (1) the way data are presented and (2) the instructions subjects receive. A third factor is suggested and studied; the type of variables between which correlation is judged may affect subjects' notion of correlation. Specifically, asymmetric, present/absent variables (e.g., symptom: present, absent) may strengthen the incorrect notion of correlation as the tendency of two events to coexist (e.g., presence of symptom and presence of disease) disregarding the complementary events. In three experiments, subjects were asked to choose among five interpretations of the sentence "A strong (or no) relationship exists between (two variables)." The above prediction was confirmed.

  • The present study used both judgments of strength of relationship and measures of the ability to predict one variable from another to assess subjects' sensitivity to the covariation of two continuous variables. In addition, one group of subjects judged strength of relationship after merely observing the presentation of 60 pairs of two-digit numbers, and a second group made strength judgments after being actively engaged in predicting one member of a pair when given the other. The prediction and judgment data provide different pictures of subjects' sensitivity to covariation. The subjects were quite poor at estimating strength of relationship but, by some measures, good at predicting one variable from another. Judgments were not strongly influenced by whether subjects had previously engaged in overt prediction. The implications of these results for the literature on covariation estimation are discussed.

  • In the first three experiments, we attempted to learn more about subjects' understanding of the importance of sample size by systematically changing aspects of the problems we gave to subjects. In a fourth study, understanding of the effects of sample size was tested as subjects went through a computer-assisted training procedure that dealt with random sampling and the sampling distribution of the mean. Subjects used sample size information more appropriately for problems that were stated in terms of the accuracy of the sample average or the center of the sampling distribution than for problems stated in terms of the tails of the sampling distribution. Apparently, people understand that the means of larger samples are more likely to resemble the population mean but not the implications of this fact for the variability of the mean. The fourth experiment showed that although instruction about the sampling distribution of the mean led to better understanding of the effects of sample size, subjects were still unable to make correct inferences about the variability of the mean. The appreciation that people have for some aspects of the law of large numbers does not seem to result from an in-depth understanding of the relation between sample size and variability.

  • Subjects were given an experimental task in which they had to play the role of a quality-control researcher for a company. They had to consider a hypothetical experiment that involves testing a sample of batteries from a truck load, which may or may not be substandard. In the main experiment, subjects were given information about the prior probability of substandard truck loads (base rate), the degree of variability of battery life, and the mean difference between standard and substandard batteries, all of which are formally relevant to the decision, and they were also told the number of batteries in the truck (population size) that is formally irrelevant. The task was to decide (intuitively) how many batteries to test to achieve a specified error rate using a specified decision rule. In a second study, subjects were given a similar scenario, but asked to rate which pieces of information would be relevant to the decision. Subjects showed themselves to be sensitive to the effects of sample variability and base rate when making intuitive design decisions, though an odd effect of the mean difference factor was observed. There is also clear confirmation of a bias-to-weight sample size by population size as reported in earlier research using different kinds of judgment tasks.

  • In this paper an experimental study of students' strategies in solving a judgment of association in scatter plots is presented. The classification of these strategies from a mathematical point of view allows us to determine concepts and theorems in action and to identify students' conceptions concerning statistical association in scatter plots. Finally, correspondence analysis is used to show the effect of task variables of the items on students' strategies.

  • During the period 1991 to 1993 a new junior high school curriculum was introduced in many South African schools. This curriculum is fairly strongly constructivist in design. A study of probability was included for the first time in any ordinary South African curriculum, this being at the Standard 7 (Grade 9) level. The approach is initially experimental but continues into the more formal presentation in terms of sample spaces. This situation presented the researchers with an opportunity of looking at the unschooled understanding of probability concepts amongst South African children before the curriculum was actually implemented. Data were also collected once some of the children had been taught about probability according to the new curriculum. It was anticipated that analysis of results would enable the researchers to identify prevalent misconceptions; to ascertain the effects of the reaching of probability according to the new curriculum; to compare the intuitive understanding of various groups (male and female, urban and rural); to offer suggestions for teaching on the basis of the findings and to compare the intuitive understanding of South African children with that of children from other countries such as Britain, Canada and Brazil. In this paper we look at the pre- and post-testing done in a selection of schools in the Johannesburg region and, for the Johannesburg and Umtata samples, present an innovative analysis of data from a selection of items from the instrument used.

  • This study reviews research on cultural differences in "probabilistic thinking" and presents some intra- and inter-cultural findings. Strong differences are shown to exist between people raised under Asian and British cultures on measures of this ability. These differences were found to out-weigh any influence of subculture, religion, occupation, arts/science orientation and sex. Generally, Asians were found to adopt a less finely differentiated view of uncertainty both numerically and verbally than did the British sample. Possible antecedents of these differences are outlined, and cultural differences in probabilistic thinking are shown to be compatible with descriptions of cultural differences in business decision making. It is argued that there are qualitative cultural differences in ways of dealing with uncertainty.

  • This review grows out of a strong conviction on three points: 1. Statistics is fundamentally and primarily concerned with analyzing real data. 2. Data analysis, including inference, is both intellectually challenging and intrinsically interesting. 3. Until recently, most authors of introductory statistics textbooks have managed to do a superb job of concealing from their readers the truth of the first two points. Fortunately, the last decade has seen the arrival of a number of innovative introductory textbooks, so I now find it much more reasonable than in the past to apply high standards in judging an elementary book. In preparing this review, I have tried to present these standards systematically; I use them as an organizing frame for comparing 11 new books (or new editions) with 5 favorites from the past 10 years.

  • Students in our first year probability and statistics course typically experience problems in learning formal probability. They also often fail to grasp the logic behind confirmatory methods. The premise of this paper is as follows: to enable students to understand and be comfortable with inferential (or even exploratory) statistics, they must be allowed to (1) experience the omnipresence of variation and (2) experience probability as a means to describe and quantify that variation. A pilot study to investigate the understanding of variability and probability of a small group of students enrolled in the 1994 course is described. These students have a strong tendency to think deterministically (especially in real world settings); they have little understanding of variability and its relationship to sample size; and they are generally unable to reconcile their intuitions with the formal probability they are taught. There were some initial indications that allowing students to experience variation personally made aware of their over-emphasis on causal explanations of variability. Lastly, it appears that students' awareness about probabilistic thinking can be raised by actively challenging and discussing their tacit intuitive models about chance.

  • This paper is a pilot study for a major investigation into children's understanding of statistical graphics, using a written test paper incorporating a hierarchy of assumed question difficulty. Four provisional levels of understanding of statistical graphics are established from the test paper results. This might permit formulation of recommendations for development of approaches encouraging better understanding of statistical graphics in pupils. The four provisional levels would then be used as a basis for a main investigation with wider scope subject to the adjustments to the sample, test paper, and administration recommended in the conclusions to this pilot study paper. It seems particularly important to put an unambiguous carefully structured test paper to a more stringently selected and representative sample, with more computer support for data analysis.

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