# Chapter

• ### How to make standard deviation meaningful

Variability is a somewhat difficult concept. There is no internal relation in it which gives you insight into the concept of standard deviation at an intuitive level. Some relations that are very important to standard deviation are discussed. They relate standard deviation to mean and other notions at an abstract level. In a broader context of "statistics (mathematics ) as a tool to communicate" they loose their fascination. Maybe we will overcome this problem by techniques of exploratory data analysis and thus cope with the idea of variability more successful in the future.

• ### A note on the birthday problem

Formal calculations often do not yield insight into "how your model solves your problem". A strategy in connection to the birthday problem is discussed that does give intuitive orientation. Furthermore some situations do not seem "stochastical" at first sight but can be structured by stochastical models. Even in these models do not fit the situation you can get sensible results from that models. The birthday problem is shown to be a problem of that type.

• ### Research in probability and statistics: Reflections and directions

Probability and statistics (stochastics) are viewed as necessary for all students no matter their ambitions. However, there are barriers to the effective teaching of both stochastics and problem solving: 1) getting stochastics into the mainstream of the mathematical science school curriculum; 2) enhancing teachers' background and conceptions of probability and statistics; 3) confronting students' and teachers' beliefs about probability and statistics. Psychologists and mathematics educators should work collaboratively to diminish misconceptions. Doing so combines the roles of observer, describer, and intervener. Research in stochastics suggests that heuristics that are used intuitively by learners impede the conceptual understanding of concepts such as sampling. This paper reviews the research on judgemental heuristics and biases, conditional probability and independence (i.e., causal schemes), decision schema (i.e., outcome approach), and the mean. Learners have difficulties in these areas, however, evidence is contradictory as to whether training in stochastics improves performance and decreases misconceptions. The conclusion emerging from this research is that probability concepts can and should be introduced into the school at an early age. Instruction that is designed to confront misconceptions should encourage students to test whether their beliefs coincide with those of others, whether they are consistent with their own beliefs about other related things, and whether their beliefs are born out with empirical evidence. Computers can be used to provide both an exploratory and representational aspect of the discipline. The role of teachers in this type of environment and the issue of whether stu- dents should use artificial or real data sets should be considered.

• ### The psychology of inference and the teaching of probability and statistics: Two sides of the same coin?

In this paper we explore the interrelationships of research in judgment and decision making with research in mathematical education on the learning of probability concepts. The psychological literature demonstrates that people are subject to heuristic and biases when making inferences or probabilistic estimates. The literature of mathematics education indicates that many people are statistically illiterate. Thus, central motivating questions for the paper are: Can research in the learning and teaching of probability and statistics help the statistically naive judge? Can research in the psychology of inference help the naive statistician? How can research from both these disciplines aid the teacher of probability and statistics? The paper consists of three main parts. Part one investigates obstacles to the use of statistics when making judgments or inferences. Part two discusses some suggestions from psychologists and from mathematics educators for increasing people's reliance upon statistics when making inferences. In part three, suggestions for further research are discussed. It is suggested that cooperative research efforts between psychologists and mathematics educators be conducted in order to further investigate these questions.

• ### The child's intuition of probability

The main aim of our research has been to contribute to an elucidation of the following problem: from what age, and in what sense, is it possible to speak of an intuition of chance and probability in the child? Taking Piaget's view as our starting-point, we have assumed that the central feature is that of the relationship between the possible and the necessary. At the same time, we have looked at other aspects which seemed likely to throw light on the problem as a whole: (a) Probability can be expressed either as a prediction of a single isolated event (without specifying the hypothetical multiplicity of its origin), of as a prediction of several events - which may be repetitions of a single event, or a succession of different events. We therefore need to know to what extent children of different ages understand the concept of relative frequency, and the extent to which this enters into their understanding of probabilistic situations. (b) How far is the child able to abstract a common probabilistic structure from different specific contexts and situations? It seemed important to us to include series of events described by unequal probabilities. Reprinted from Enfance 2 (1967), 193 - 206.

• ### The ups and downs of the hope function in a fruitless search

In order to investigate the nature of probabilistic reasoning, we devised two experimental problems, each of which involved two hope questions (long- and short-term). We will present the two standard problems along with the Bayesian solution. Then we will discuss a number of features of the solution by applying it to a variety of situations. After describing how our subjects reasoned about the standard problems, we will present a didactic device we developed to make the search problem more conducive to resolution. Finally, we will explore subjects' ability to transfer the lesson learned from the didactic device to the analogous wait problem.

• ### Heuristics and biases

This article described three heuristics that are employed in making judgments under uncertainty: (i) representativeness, which is usually employed when people are asked to judge the probability that an object or event A belongs to class or process B; (ii) availability of instances or scenarios, which is often employed when people are asked to assess the frequency of a class or the plausibility of a particular development; and (iii) adjustment from an anchor, which is usually employed in numerical prediction when a relevant value is available. These heuristics are highly economical and usually effective, but they lead to systematic and predictable errors. A better understanding of these heuristics and of the biases to which they lead could improve judgments and decisions in situations of uncertainty.

• ### On the psychology of prediction

In this paper, we explore the rules that determine intuitive predictions and judgments of confidence and contrast these rules to the normative principles of statistical prediction.

• ### Studies of representativeness

Daniel Kahneman and Amos Tversky have proposed that when judging the probability of some uncertain event people often resort to heuristics, or rulers of thumb, which are less than perfectly correlated (if, indeed, at all) with the variables that actually determine the event's probability. One such heuristic is representativeness, defined as a subjective judgment of the extent to which the event in question "is similar in essential properties to its parent population" or "reflects the salient features of the process by which is is generated" (Kahneman &amp; Tversky, 1972b, p. 431, 3). Although in some cases more probable events also appear more representative, and vice versa, reliance on the representativeness of an event as an indicator of its probability may introduce two kinds of systematic error into the judgment. First, it may give undue influence to variables that effect the representativeness of an event but not its probability. Second, it may reduce the importance of variables that are crucial to determining the event's probability but are unrelated to the event's representativeness.

• ### Judgments of and by representativeness

The first part of this chapter is concerned with the nature of the representativeness relation and and also with the conditions on which the concept of representativeness is usefully invoked to explain intuitive predictions and judgments of probability. In the second part of the chapter we illustrate the contrast between the logic of representativeness and the logic of probability in judgments of the likelihood.