Sensitivity to Sample Size in the Context of the Empirical Law of Large Numbers: Comparing the Effectiveness of Three Approaches to Support Early Secondary School Students

Abstract

AbstractA foundation for gathering and interpreting data is the empirical law of large numbers (eLLN). The eLLN has multiple aspects and can be regarded and used with multiple foci. However, it generally relates relative frequencies and probabilities or samples and the corresponding populations. Unfortunately, research has repeatedly revealed that students have problems with tasks including certain foci on the eLLN, particularly regarding their sensitivity to sample size when comparing smaller and larger samples.We first outline the eLLN with its central aspects and provide an overview of corresponding empirical findings. Subsequently, we use Stanovich’s (2018) framework of human processing in heuristics and biases tasks to (re-)interpret theoretical descriptions and prior empirical results to better understand and describe students’ problems with the eLLN. Subsequently, we present three main approaches to support students derived from prior research: A static-contrast approach, a dynamic approach, and a boundary-example approach.As currently no systematic and comparative evidence exists regarding the effectiveness of these approaches, we conducted a quasi-experimental intervention study (N = 256, 6th grade) which empirically compared three implementations of these approaches to a control group. Results underline significant positive short-term effects of each approach. However, the boundary-example intervention showed the highest pre-post effect, the only significant long-term effect, and also effectively reduced the common equal-ratio bias.Results are promising from a research perspective, as Stanovich’s framework proved very helpful and is a promising foundation for future research, and from an educational perspective, as the boundary-example approach is lightweight and easy to implement in classrooms.