Two qualitative taxonomic characters are potentially compatible if the stat
es of each can be ordered into a character state tree in such a way that th
e two resulting character state trees are compatible. The number of potenti
ally compatible pairs (NPCP) of qualitative characters from a data set may
be considered to be a measure of its phylogenetic randomness. The value of
NPCP depends on the number of evolutionary units (EUs), the number of chara
cters, the number of states in the characters, the distributions of EUs amo
ng these states, and the amount and distribution of missing information and
so does not directly indicate degree of phylogenetic randomness. Thus, for
an observed data set, we used Monte Carlo methods to estimate the probabil
ity that a data set chosen equiprobably from among those identical (with re
spect to all the other above determining features) to the observed data set
would have as high (or low) an NPCP as the observed data set. This probabi
lity, the realized significance of the observed NPCP, is attractive as an i
ndication of phylogenetic randomness because it does not require the assump
tions made by other such methods: No character state trees are assumed and
consequently, only potential compatibility can be determined; no particular
method of phylogenetic estimation is assumed; and no phylogenetic trees ar
e constructed. We determined the values and significances of NPCP for analy
ses of 57 data sets taken from 53 published sources. All data sets from 37
of those sources exhibited realized significances of <0.01, indicating high
levels of phylogenetic nonrandomness. From each of the remaining 16 source
s, at least one data set was more phylogenetically random. Inclusion of out
groups changed significance in some cases, but not always in the same direc
tion. Data sets with significantly low NPCP may be consistent with an ancie
nt hybrid origin (or other ancient polyphyletic gene exchange, crossing ove
r, viral transfer, etc.) of the study group.