Blom rank-normal scores for pooled draws

Let $x_1,\dots ,x_S\in \mathbb{R}$ be a nonempty finite pooled collection, where $S\ge 1$. Give each observation its average rank $r_i$ among all $S$ values; equal values receive the average of the ranks they occupy. Let $\mathrm{\Phi }$ be the standard-normal CDF from Standard normal and normal laws. Define the Blom normal score

$z_i={\mathrm{\Phi }}^{-1}\text{\hspace{0.17em}⁣}\left(\frac{r_i-3\mathrm{/}8}{S+1\mathrm{/}4}\right)\mathrm{.}$

The standard-normal CDF is continuous, strictly increasing, and has limits $0$ and $1$ at the two infinities by The standard normal CDF is continuous and strictly increasing. The published quantile result The standard normal CDF has unique symmetric quantiles supplies the existence and uniqueness of a finite inverse for every probability in $(0,1)$. Since $1\le r_i\le S$, the displayed Blom fraction lies strictly between $0$ and $1$, so each score is well defined. This convention averages ties before applying the score transform.