# Approximate order statistics for Half-normal probability plot

To construct the half-normal probability plot, plot the absolute values in a certain statistical diagnostic (residual, leverage, Cook distance and others) versus $z_i$ where:

$\displaystyle z_{i} = \Phi^{-1} \left(\frac{k + n - \frac{1}{8}}{2n +\frac{1}{2}}\right)$

$\therefore$ Applied Linear Statistical Models, (Neter, 1996) Fifth Edition, Pag 596

or

$\displaystyle z_{i} = \Phi^{-1} \left(\frac{k + n - \frac{1}{2}}{2n +\frac{9}{8}}\right)$

$\therefore$ Generalized Linear Models, (Nelder, 1989) Second Edition, Pag 407

where $\Phi$ is the cumulative distribution function of the standard normal distribution.

$k$ is the $kth$ ordered absolute residual

$n$ is the sample size

I could not locate the mathematical justification for the difference of the $z_i$, and more importantly I don't have any intuition about where it comes from.

Any help? Could you please guide me?