TITLE:
Shape Measures for the Distribution of a Qualitative Variable
AUTHORS:
José Moral de la Rubia
KEYWORDS:
Symmetry, Peakedness, Descriptive Measures, Nominal Measurement Scale, Qualitative Variables
JOURNAL NAME:
Open Journal of Statistics,
Vol.13 No.4,
August
23,
2023
ABSTRACT: There are several shape
measures for quantitative variables, some of which can also be applied to
ordinal variables. In quantitative variables, symmetry, peakedness, and
kurtosis are essential properties to evaluate the deviation from assumptions,
particularly normality. They aid in selecting the most appropriate method for
estimating parameters and testing hypotheses. Initially, these properties serve
a descriptive role in qualitative variables. Once defined, they can be
considered to check for non-compliance with assumptions and to propose
modifications for testing procedures. The objective of this article is to
present three measures of the shape of the distribution of a qualitative
variable. The concepts of qualitative asymmetry and peakedness are defined. The
measurement of the first concept involves calculating the average frequency
difference between qualitative categories matched by frequency homogeneity or
proximity. For the second concept, the peak-to-shoulder difference and the
qualitative percentile kurtosis are taken into consideration. This last
measurement is a less effective option than the peak-to-shoulder difference to measure
peakedness. A simulated example of the application of these three measures is
given and the paper closes with some conclusions and suggestions.