Adaptive plotting position and test of normality

Abstract

The quantile–quantile plot is widely used to check normality. The plot depends on the plotting positions. Many commonly used plotting positions do not depend on the sample values. We propose an adaptive plotting position that depends on the relative distances of the two neighbouring sample values. The correlation coefficient obtained from the adaptive plotting position is used to test normality. The test using the adaptive plotting position is better than the Shapiro–Wilk W test for small samples and has larger power than Hazen's and Blom's plotting positions for symmetric alternatives with shorter tail than normal and skewed alternatives when n is 20 or larger. The Brown–Hettmansperger T* test is designed for detecting bad tail behaviour, so it does not have power for symmetric alternatives with shorter tail than normal, but it is generally better than the other tests when β₂ is greater than 3.25.

Keywords:

• Blom's plotting position
• Hazen's plotting position
• correlation coefficient test
• empirical distribution function
• Q–Q plot
• Shapiro–Wilk W test
• Brown–Hettmansperger T* test