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The rank of an observation records where that observation lies among the order statistics of a statistical sample. When all values are distinct, $r_i=1+\#\{j:x_j<x_i\}$ and $x_i=x_{(r_i)}$. For $(0.7,0.1,0.9,0.4,0.2)$, the ranks of observations in original order are $(4,1,5,3,2)$: the first observation is fourth smallest, while the fourth observation is third smallest.

Ties require a convention. A tied block occupying sorted positions $a,\dots ,b$ can assign each member the average rank of an observation $(a+b)/2$, the minimum position $a$, or the maximum position $b$; an explicit rule can also break ties into distinct positions. R's official documentation describes these choices and uses average ranks by default. ^[ref].

For example, $(0.1,0.4,0.4,0.9)$ has order statistics $(0.1,0.4,0.4,0.9)$. The two $0.4$ observations occupy positions $2$ and $3$; average ranks of observations are $(1,2.5,2.5,4)$, whereas minimum-position ranks are $(1,2,2,4)$. A fractional average rank is a summary of tied positions, rather than an index of one order statistic. Without omissions, distinct ranks and average ranks sum to $n(n+1)/2$, since averaging each tied block preserves its sum; minimum-position ranks need not.

Wilcoxon signed-rank tests rank the absolute nonzero paired differences, while Wilcoxon rank sum tests rank the pooled observations from two samples; these are different rankings.