Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.22918

Oh doughnuts, it's the variance! A visual metaphor for the variance, standard deviation, and related quantities

Eugene D. Hahn
View original source ↗Read source document ↗Export metadata ↓

Abstract

Despite the rise of artificial intelligence, human intuition endures as both a foundation for education and also (for now) a primarily human experience. We re-examine a formula for the sum of squares that is sometimes set aside because of possible numerical issues. The formula can be used to create square doughnuts, a delicious regional favorite. The variance becomes the dough, and we can examine the variance by either comparing doughnuts or by making different sizes of doughnuts with the same amount of dough. In doing so, we provide a new visualization of a cornerstone formula, understandable by students at almost any level. We further provide insight into related topics such as the coefficient of variation as well as a visualization of when numerical issues are likely to be problematic. We also provide a new vocabulary for understanding the variance formula, with `dough' and `hole' taking the place of less edible concepts from the typical recipe.

Explore related subjects

stat.OT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Eugene D. Hahn. 2026-10-16. Oh doughnuts, it's the variance! A visual metaphor for the variance, standard deviation, and related quantities. https://arxiv.org/abs/2610.22918

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Framework for Teaching the Responsible and Effective Use of Generative AI in Statistics and Data Science

The rapid rise of generative artificial intelligence (AI) following the public release of tools such as ChatGPT has begun to reshape how students engage with coding, statistical analysis, and written communication in higher education. Growing use of conversational AI systems creates opportunities for learning alongside challenges around reliability, transparency, privacy, and appropriate academic and professional practice, highlighting the need for structured education in their responsible use. This paper presents a framework for teaching the responsible and effective use of generative AI in statistics and data science. The framework is organised around four areas: Understanding, Development, Reporting, and Safety, covering conceptual foundations, practical interaction within analytical workflows, documentation and transparency, and issues including reliability, bias, privacy, and ethics. Its design draws on emerging work in AI literacy and established principles from statistics and data science education, and was further refined through workshop and conference activities and participant feedback. The framework has been implemented across multiple settings, including a massive open online course, short training sessions, and undergraduate and postgraduate skills-based teaching. We use these experiences to identify broader design considerations for educators, including the value of authentic analytical tasks, critical evaluation of generated outputs, integrated consideration of responsible practice, and flexibility as generative AI technologies continue to evolve.

stat.OT↗
arXiv2026

Differentially Private Learning of Markov Chains: Near-Optimal Sample Complexity for Item and User Level Privacy

Sequential data such as documents, conversations, and trajectories are naturally partitioned across users, motivating privacy guarantees that protect each user's entire trajectory. We initiate the study of privately estimating the transition matrix of a discrete ergodic Markov chain from N independent trajectories of length T, with arbitrary initial distributions, under maximum row-wise ell 1 error. Besides N, T, and the number of states, our bounds depend on the minimum stationary probability and the pseudo-spectral gap, which govern state coverage and mixing. Under item-level privacy, we give upper and lower bounds that separate the costs of sampling, state occupancy and mixing, and privacy. Under user-level privacy, we reduce the problem for sufficiently long trajectories to private product distribution estimation. Along the way, we establish optimal user-level private bounds for estimating products of discrete distributions, which may be of independent interest. In the user level setting, our bounds are nearly tight in the large T regime, whereas in the item-level setting, our bounds as tight up-to a non-private term.

stat.OT↗
arXiv2026

AI-Guided Assignments in Practice: Students' Experiences in a Large Introductory Statistics Course

Large language models (LLMs) raise new questions for instructors, among them how to integrate LLMs productively into coursework and what to evaluate in students' work when an LLM can produce much of it directly. LLM-guided homework assignments offer one answer. Instructors configure the LLM to guide students in ways that achieve specific pedagogical goals, such as recalling material, discovering new concepts, or critically evaluating deliberately flawed reasoning, and grade the full student--LLM conversation, and with it the student's reasoning, rather than only the final answer. In this paper, we report on the large-scale deployment of such assignments, implemented through LLteacher, an open-source tool that supports recall, discovery, and critical-thinking assignments, in a large introductory statistics course serving mostly pre-major undergraduates. Results on 134 students show meaningful gains in self-reported confidence, with over 80\% agreeing that the assignments improved their comprehension of course material and their ability to interpret statistical results; gains were largest among students who started with lower confidence, suggesting this approach may help narrow gaps in preparation. Interaction patterns varied widely, from long single messages to rapid iterative exchanges, and hallucinations, when present, were rarely disruptive to learning.

stat.OT↗