The following formulas are used to convert the ranks into expected normal probability values, that is, the respective normal z values.

Normal probability plot. The normal probability value zj for the jth value (rank) in a variable with N observations is computed as:

zj = F-1 [(3*j-1)/(3*N+1)]

where F-1 is the inverse normal cumulative distribution function (converting the normal probability p into the normal value z).

Half-normal probability plot. Here, the half-normal probability value zj for the jth value (rank) in a variable with N observations is computed as:

zj = F-1 [(3*N+3*j-1)/(6*N+1)]

where F-1 is again the inverse normal cumulative distribution function.

Detrended normal probability plot. In this plot each value (xj) is standardized by subtracting the mean and dividing by the respective standard deviation (s). The detrended normal probability value zj for the jth value (rank) in a variable with N observations is computed as:

zj = F-1 [(3*j-1)/(3*N+1)] - (xj -mean)/s

where F-1 is again the inverse normal cumulative distribution function.

See also, Multiple Normal Probability Plots.