# Theory

• ### Concept or computation: Students' understanding of the mean

In statistics, and in everyday life as well, the arithmetic mean is a frequently used average. The present study reports data from interviews in which students attempted to solve problems involving the appropriate weighting and combining of means into an overall mean. While mathematically unsophisticated college students can easily compute the mean of a group of numbers, our results indicate that a surprisingly large proportion of them do not understand the concept of the weighted mean. When asked to calculate the overall mean, most subjects answered with the simple, or unweighed, mean of the two means given in the problem, even though these two means were from different-sized groups of scores. For many subjects, computing the simple mean was not merely the easiest or most obvious way to initially attack the problem; it was the only method they had available. Most did not seem to consider why the simple mean might or might not be the correct response, nor did they have any feeling for what their results represented. For many students, dealing with the mean is a computational rather than a conceptual act. Knowledge of the mean seems to begin and end with an impoverished computational formula. The pedagogical message is clear: Learning a computational formula is a poor substitute for gaining an understanding of the basic underlying concept.

• ### Principles of Learning Probability and Statistics

Probability judgments may have to be revised if new information is available. From the mathematical perspective probability revisions are intimately connected to the notion of conditional probability and Bayes' formula, a subsidiary concept and a trivial theorem respectively. Nevertheless empirical investigations in subjects' understanding of probability do indicate that people do not cope adequately with situations involving probability revisions, if they have been taught the mathematical concepts or not does not matter. In what follows I will try to sketch some phenomena of misunderstanding, give some comments on the interplay between mathematics and intuitions which I think represents the origin of lack of comprehension. A brief overview on the favor concept should enable the impression that by way of teaching this concept probabilistic reasoning could be improved.

• ### The Role of Statistics in Revitalizing Precollege Mathematics and Science Education

We live in an information society. We are confronted, in fact, inundated, with quantitative information at all levels of endeavor, Charts, graphs, figures, rates, percentages, probability, averages, forecasts, trend lines, etc., are in inescapable part of our everyday lives that affect our decisions on health, citizenship, parenthood, jobs, financial concerns and many other important matters. In order to be called for dealing with data and making intelligent decisions based on quantitative arguments. We live in a scientific age. We are confronted with arguments that demand logical, scientific reasoning even if we aren't trained scientists. We must be able to clearly see our way through a maze of reported "facts" in order to separate credible conclusions from specious ones. We must be able to intelligently weigh the evidence on the cause of cancer, the effects of pollutants on the environment, or the results of a limited nuclear war. Teachers, and then students, must be trained to make intelligent decisions based on numerical information if our society is to grow and prosper. We live amidst burgeoning technology. We are confronted with a job market that demands scientific and technological skills, and our students must be trained to deal with the tools of this technology in productive, efficient, and correct ways. Much of this new technology is concerned with information processing and dissemination and proper use of this technology requires statistical skills. These skills are in demand in engineering, business management, data management, and economic forecasting, just to name a few.

• ### Misconceptions of probability: From systematic errors to systematic experiments and decisions

Some of our misconceptions of probability may occur just because we haven't studied much probability. However, there is considerable recent evidence to suggest that some misconceptions of probability are of a psychological sort. Mere exposure to the theoretical laws of probability may not be sufficient to overcome misconceptions of probability. Cohen and Hansel [20], Edwards [29], and Kahneman and Tversky [63-66] are among those psychologists who have investigated the understanding of probability from a psychological point of view. The work of Daniel Kahneman and Amos Tversky is especially fascinating, for the attempt to categorize certain types of misconceptions of probability which they believe are systematic and even predictable. Kahneman and Tversky claim that people estimate complicated probabilities by relying on certain simplifying techniques. Two of the techniques they have identified are called representativeness and availability. We shall explore these two techniques in more detail and discuss some implications for teaching probability and statistics in the schools.

• ### The psychology of learning probability

This paper discusses the importance of including probability in the teaching of statistics.

• ### Statistics and society

It is customary for non-statisticians to mock statistical descriptions of social and economic phenomena on the grounds that the statistics can be manipulated to indicate of support one's preconceived views. In other words, they highlight one particular chapter in the textbooks that all of us have read, namely "Uses and Abuses of Statistics". Perhaps we statisticians have invited this ridicule because quantification and hard data are do often used to support "facts", that facts have now become synonymous with statistics. Yet we know, at least in the social sciences, that numbers and measures are vulnerable. Cost of living indices, pure indices, and many other individual and composite numbers, are so value-loaded that we should simply admit the fact: then we would perhaps be less ridiculed. Oxford philosophers have even challenged the existence of anything called a "pure fact", suggesting that no observations are value-free. But however mocking those who do not deal with statistics, there is a deep and genuine unease about statistical descriptions of society and the economy among those who are struggling to remove poverty and inequality. This unease appears at many layers and levels.

• ### Statistical graphics: Developments from statistical practice into statistical education

This paper discusses several current topics in the areas of statistical graphics research and applications and suggests additional ways that graphical methods can be used to improve statistical education.

• ### Statistics in liberal arts education

Statistics has not been a part of the typical liberal arts curriculum. After an examination of some central features of both liberal arts education and statistics, it is argued that statistics can play a strong role in liberal arts education, in part because of the advances that have taken place in computer technology.

• ### Children's Concepts of Chance and probability

Interest in children's concepts of chance and probability has been prompted by several questions. Assuming that the development of a concept of chance and probability is influenced by experience, what are the conditions that bring it about? What are its precursors? Is it acquired all at once, or is it acquired gradually over a relatively long period of time? At what age is its development complete? Does every mature adult have a similarly functioning concept of chance, or are there individual differences? If so, how are they to be explained? To what extent is a concept of chance a result of formal instruction in school? What kinds of training are likely to improve upon immature or deficient concepts of chance or probability? When making probability judgments, is there a optimum strategy that can be said to be correct in each type of situation, or is there a variety of strategies more or less adequate or appropriate? To what extent is performance in a probability setting controlled by the reinforcing consequences of previous outcomes? What is the relationship between chance and probability concepts, on the one hand, and the development of linguistic ability to articulate them, on the other? In what ways are various probability tasks alike, and how do they differ? What makes some tasks seem harder than others? What is the relationship between the development of concepts of chance or probability and cognitive development in general? These do not seem to be trivial questions. Indeed, many of them have been addressed in published research reports and monographs. The purpose of this chapter is to review procedures that have been devised to investigate some of these questions and to evaluate the conclusions that have tentatively been drawn.

• ### Elementary teachers' conceptions of graphical representations of statistical data

Proposed reforms in the K-12 mathematics included incorporating data analysis and probability into the mathematics curriculum. It was proposed that elementary school students engage in experiences to: a) collect, organize, and describe data; b) construct, read and interpret displays of data; and c) formulate and solve problems that involve collecting and analyzing data. However, the research literature contains few studies about the teaching and learning of statistical concepts, especially for elementary school students and teachers.