The
objective of this study is to propose the Parametric Seven-Number Summary
(PSNS) as a significance test for normality and to verify its accuracy and
power in comparison with two well-known tests, such as Royston’s W test and
D’Agostino-Belanger-D’Agostino K-squared test. An experiment with 384
conditions was simulated. The conditions were generated by crossing 24 sample
sizes and 16 types of continuous distributions: one normal and 15 non-normal.
The percentage of success in maintaining the null hypothesis of normality
against normal samples and in rejecting the null hypothesis against non-normal
samples (accuracy) was calculated. In addition, the type II error against
normal samples and the statistical power against normal samples were computed.
Comparisons of percentage and means were performed using Cochran’s Q-test,
Friedman’s test, and repeated measures analysis of variance. With sample sizes of 150 or greater, high accuracy and mean power or
type II error (≥0.70 and ≥0.80, respectively) were achieved. All three normality tests were similarly accurate; however,
the PSNS-based test showed lower mean power
than K-squared and W tests, especially against non-normal samples of
symmetrical-platykurtic distributions, such as the uniform, semicircle, and
arcsine distributions. It is concluded that the PSNS-based omnibus test is
accurate and powerful for testing normality with samples of at least 150
observations.
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