Saurophaganax. David K. Smith, examining Allosaurus fossils by quarry, found that the Cleveland-Lloyd Dinosaur Quarry (Utah) specimens are generally smaller than those from Como Bluff (Wyoming) or Brigham Young University's Dry Mesa Quarry (Colorado), but the shapes of the bones themselves did not vary between the sites. A later study by Smith incorporating Garden Park (Colorado) and Dinosaur National Monument (Utah) specimens found no justification for multiple species based on skeletal variation; skull variation was most common and was gradational, suggesting individual variation was responsible. Further work on size-related variation again found no consistent differences, although the Dry Mesa material tended to clump together on the basis of the astragalus, an ankle bone. Kenneth Carpenter, using skull elements from the Cleveland-Lloyd site, found wide variation between individuals, calling into question previous species-level distinctions based on such features as the shape of the horns, and the proposed differentiation of A. jimmadseni based on the shape of the jugal. A study published by Motani et al., in 2020 suggests that Allosaurus was also sexually dimorphic in the width of the femur's head against its length. Skull The skull and teeth of Allosaurus were modestly proportioned for a theropod of its size. Paleontologist Gregory S. Paul gives a length of for a skull belonging to an individual he estimates at long. Each premaxilla (the bones that formed the tip of the snout) held five teeth with D-shaped cross-sections, and each maxilla (the main tooth-bearing bones in the upper jaw) had between 14 and 17 teeth; the number of teeth does not exactly correspond to the size of the bone. Each dentary (the tooth-bearing bone of the lower jaw) had between 14 and 17 teeth, with an average count of 16. The teeth became shorter, narrower, and more curved toward the back of the skull. All of the teeth had saw-like edges. They were shed easily, and were replaced continually, making them common fossils. Its skull was light, robust and equipped with dozens of sharp, serrated teeth. The skull had a pair of horns above and in front of the eyes. These horns were composed of extensions of the lacrimal bones, and varied in shape and size. There were also lower paired ridges running along the top edges of the nasal bones that led into the horns. The horns were probably covered in a keratin sheath and may have had a variety of functions,