Abstract
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.
Highlights
IntroductionIn the first edition of the Elements of Statistics, Bowley [1] introduced
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
The present study takes up the proposal to test whether a sequence of sample data has been drawn from a normal distribution using seven quantiles that are spaced, at a distance of two-thirds, from the central point in a standard normal distribution [4]
Summary
In the first edition of the Elements of Statistics, Bowley [1] introduced. In the first edition of the Elementary Manual of Statistics, Bowley [3] expanded the summary from five to seven numbers, by including the first and last deciles, allowing two extremes of deviation to be defined. Very low scores are below the 10th percentile and very high scores are above the 90th percentile. The central tendency or shoulder zone extends into the interquartile range, low scores are between the 10th and 25th percentiles, and high scores are between the 75th and 90th percentiles. The seven-number summary is useful for calculating statistics on central tendency (median), variation (absolute and relative ranges), skewness (quartile coefficient) and kurtosis (percentile coefficient), and to interpret the scores [4]
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