TITLE:
Standardized Distance from the Mean to the Median as a Measure of Skewness
AUTHORS:
José Moral de la Rubia
KEYWORDS:
Symmetry, Shape Measures, Normal Distribution Hypothesis, Confidence In-terval Calculation Methods
JOURNAL NAME:
Open Journal of Statistics,
Vol.13 No.3,
June
20,
2023
ABSTRACT: The normal distribution, which has a symmetric and middle-tailed profile,
is one of the most important distributions in probability theory, parametric
inference, and description of quantitative variables. However, there are many
non-normal distributions and knowledge of a non-zero bias allows their
identification and decision making regarding the use of techniques and
corrections. Pearson’s skewness coefficient defined as the standardized signed
distance from the arithmetic mean to the median is very simple to calculate and
clear to interpret from the normal distribution model, making it an excellent
measure to evaluate this assumption, complemented with the visual inspection by
means of a histogram and a box-and-whisker plot. From its variant without tripling the numerator or
Yule’s skewness coefficient, the objective of this methodological article is to
facilitate the use of this latter measure, presenting how to obtain asymptotic
and bootstrap confidence intervals for its interpretation. Not only are the formulas shown, but they are applied with an example
using R program. A general rule of interpretation of ∓0.1 has been suggested, but this can only become relevant when
contextualized in relation to sample size and a measure of skewness with a
population or parametric value of zero. For this purpose, intervals with
confidence levels of 90%, 95% and 99% were estimated with 10,000 draws
at random with replacement from 57 normally distributed samples-population with
different sample sizes. The article closes with suggestions for the use of this
measure of skewness.