Typical Yet Unlikely and Normally Abnormal: The Intuition Behind High-Dimensional Statistics

Vowels Matthew J. ()
Additional contact information
Vowels Matthew J.: University of Lausanne, Lausanne, Switzerland

Statistics, Politics and Policy, 2024, vol. 15, issue 1, 87-113

Abstract: Normality, in the colloquial sense, has historically been considered an aspirational trait, synonymous with ideality. The arithmetic average and, by extension, statistics including linear regression coefficients, have often been used to characterize normality, and are often used as a way to summarize samples and identify outliers. We provide intuition behind the behavior of such statistics in high dimensions, and demonstrate that even for datasets with a relatively low number of dimensions, data start to exhibit a number of peculiarities which become severe as the number of dimensions increases. Whilst our main goal is to familiarize researchers with these peculiarities, we also show that normality can be better characterized with ‘typicality’, an information theoretic concept relating to entropy. An application of typicality to both synthetic and real-world data concerning political values reveals that in multi-dimensional space, to be ‘normal’ is actually to be atypical. We briefly explore the ramifications for outlier detection, demonstrating how typicality, in contrast with the popular Mahalanobis distance, represents a viable method for outlier detection.

Keywords: statistics; information theory; outlier; normality; typicality (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1515/spp-2023-0028 (text/html)
For access to full text, subscription to the journal or payment for the individual article is required.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bpj:statpp:v:15:y:2024:i:1:p:87-113:n:3

Ordering information: This journal article can be ordered from
https://www.degruyter.com/journal/key/spp/html

DOI: 10.1515/spp-2023-0028

Access Statistics for this article

Statistics, Politics and Policy is currently edited by Joel A. Middleton

More articles in Statistics, Politics and Policy from De Gruyter
Bibliographic data for series maintained by Peter Golla ().