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In biological taxonomy — also called scientific classification or biological classification — monophyly means a group exclusively includes a species and all its ancestors, while polyphyly means a group may contain a "grab bag" of different families. These are called monophyletic and polyphyletic groups respectively. An example of a polyphyletic group would be "worms" or "warm-blooded animals," while a monophyletic group would be "mammals" or "crustaceans."
The history of biological taxonomy has been one of trying to eliminate polyphyletic groups in favor of monophyletic ones. Since the 1970s, this has been made much easier by genetic methods — also called phylogenetic analysis or "molecular studies" — which study similar lengths of DNA to find how animals are related to one another. Many groups which look superficially similar may be found to be entirely unrelated in practice. For instance, Pygopodidae, a family of legless lizards, appear similar to snakes but are distinguished from them by eyelids that blink (which snakes lack), external ear holes, flat, unforked tongues, and vestigal limbs. To the amateur, telling the difference can be somewhat difficult, but to a professional biologist it can be clear.
One of the most standard examples of a polyphyletic group are warm-blooded animals, which include both birds and mammals. Both species have a common ancestor that lived during the Paleozoic, a very long time ago. Birds, although warm-blooded, evolved from cold-blooded ancestors, the dinosaurs, which are hardly monophyletic with mammals. Thus birds and mammals are from entirely different groups, but both fall in the general category of warm-blooded animals.
Specific biological taxonomy is the goal of taxonomists, but often at odds with common wisdom or lack of rigor. For instance, when asked "Are there sea insects?" some might answer, "Well, sure, lobsters are kind of like insects." Although to a casual observer, this answer might seem sufficient, it's enough to cause a career taxonomist to practically spit out her coffee.
Insects are members of class Insecta, while lobsters are a member of subphylum Crustacea, an entirely different group. Although both are arthropods, and are probably related, the groups are quite different, with one being primarily terrestrial and the other aquatic. Casually referring to them as a monophyletic group is the sort of habit that taxonomists are fighting against by being more specific about relationships between animals, which helps us understand them better.
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