The product Bayes factor (PBF) synthesizes evidence for an informative hypothesis across heterogeneous replication studies. It can be used when fixed- or random effects meta-analysis fall short. For example, when effect sizes are incomparable and cannot be pooled, or when studies diverge significantly in the populations, study designs, and…

For ease of instruction in the classroom, the one-way analysis of variance F statistic is rewritten in terms of pairwise differences in individual sample means instead of differences of individual sample means from the overall sample mean. Likewise, the Kruskal-Wallis statistic may be rewritten in terms of pairwise differences in individual…

Statistics is enjoying some well-deserved limelight across mathematics curricula of late. Some statistical concepts, however, are not especially intuitive, and students struggle to comprehend and apply them. As an AP Statistics teacher, the author appreciates the central limit theorem as a foundational concept that plays a crucial role in…

In introductory statistics courses, the concept of power is usually presented in the context of testing hypotheses about the population mean. We instead propose an exercise that uses a binomial probability table to introduce the idea of power in the context of testing a population proportion. (Contains 2 tables, and 2 figures.)

Formal inference, which makes theoretical assumptions about distributions and applies hypothesis testing procedures with null and alternative hypotheses, is notoriously difficult for tertiary students to master. The debate about whether this content should appear in Years 11 and 12 of the "Australian Curriculum: Mathematics" has gone on…

Learning about statistics is a lot like learning about science: the learning is more meaningful if you can actively explore. This fifth installment of "Explorations in Statistics" revisits power, a concept fundamental to the test of a null hypothesis. Power is the probability that we reject the null hypothesis when it is false. Four…

Students learn to examine the distributional assumptions implicit in the usual t-tests and associated confidence intervals, but are rarely shown what to do when those assumptions are grossly violated. Three data sets are presented. Each data set involves a different distributional anomaly and each illustrates the use of a different nonparametric…

In this article, the authors address the practicalities of how data should be presented, summarized, and interpreted. There are no exact rules; indeed there are valid concerns that exact rules may be inappropriate and too prescriptive. New procedures evolve, and new methods may be needed to deal with new types of data, just as people know that new…

In experimental designs with nested structures, entire groups (such as schools) are often assigned to treatment conditions. Key aspects of the design in these cluster-randomized experiments involve knowledge of the intraclass correlation structure, the effect size, and the sample sizes necessary to achieve adequate power to detect the treatment…

Very little explanatory power is required in order for regressions to exhibit statistical significance. This article discusses some of the causes and implications. (Contains 2 tables.)

Statistics teaching should not be carried out in the same way for all kinds of university students. Instead, teaching statistics should take into account the different fields of study that students have chosen. For example, students of sciences or engineering have different interests and backgrounds compared to students of any social or juridical…

Unless the sample encompasses a substantial portion of the population, the standard error of an estimator depends on the size of the sample, but not the size of the population. This is a crucial statistical insight that students find very counterintuitive. After trying several ways of convincing students of the validity of this principle, I have…

From a priori and post hoc data collection perspectives, this paper describes the interrelations among (1) power, alpha, effect size, and sample size for hypothesis testing; and (2) precision, confidence, and sample size for interval estimation. Implications for special education researchers working with convenient samples of fixed size are…

Exploratory factor analysis (EFA) and confirmatory factor analysis (CFA) have contributed to test development and validation in counseling psychology, but additional applications have not been fully realized. The author presents an overview of the goals, terminology, and procedures of factor analysis; reviews best practices for extracting,…

Structural equation modeling (SEM) refers to a family of statistical techniques that explores the relationships among a set of variables. Structural equation modeling provides an extremely versatile method to model very specific hypotheses involving systems of variables, both measured and unmeasured. Researchers can use SEM to study patterns of…