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Biometrics is the quantitative analysis of ontogenetic ("of or relating to life cycle or development") parameters such as height, weight, shape, morphology, age, etc. Biometrics can be applied to groups of organisms (such as a population of Douglas fir trees, or all butterflies, or all mammals) to gain insights into fundamental principles of growth or behavior in those groups.

Schoener (1965) assembled data (from previously published sources) on bill size ratios for populations of several hundred different species of birds. For each bird sampled from a population, the length of the culmen (the upper ridge of the upper bill) was measured. Many bird bills are curved, but the length of the culmen is often measured as the "chord" from the tip of the beak to where the bill meets the fleshy tissue or "cere".

The bill ratio is the ratio of the largest bill to the smallest bill measured in each population. For example, one population of Cooper's hawks (Accipiter cooperi) showed a mean culmen length of 15.2 mm with a ratio of 1.41; the largest bill measured was 1.41 times that of the smallest measured. A ratio of 1 indicates all the birds in that sample had the same bill size.

The bill-size ratio is essentially a measure of bill variability within the population. Not surprisingly, populations with radically varying bill sizes are few, whereas populations with all one bill size are common, as can be seen in the histogram. What conditions could lead to variability in bill sizes in a population? Large ratios are found among highly specialized feeders (woodpeckers) whose prey is not abundant, either because the prey's environment is not abundant (rotten logs) or the prey itself is not abundant (certain grub). Large ratios are found among birds whose body size is large in proportion to the total food abundance; certain hawks, for example. In these and other cases, there is a "forcing" in bill size variability among the species by external pressures (such as food availability). External forcing can lead, in the long term, to evolution of a species.

The data show a good fit to either a power law or exponential relationship between bill size ratio and frequency.

Data sieved from a large number of independent publications often have unique problems. For example, the sample size varies by an unknown amount for each bird species. Is the bill ratio dependent upon sample size? If so, are the data reliable?

Schoener, T. W. (1965), The evolution of bill size differences among sympatric congeneric species of birds; Evolution, vol. 19, pp. 189-213.

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 species bill ratio (large to small culmen) Accipiter gundlachi 1.80 A. cooperi 1.41 A. gentilis 1.30 A. bicolar 1.39 Buteo jamaicensis 1.40 B. jamaicensis 1.37 B. lineatus 1.19 B. jamaicensis 1.18 B. lagopus 1.01 B. harlan 1.09 B. jamaicensis 1.02 B. jamaicensis 1.15 B. regalis 1.10 B. lineatus 1.06 B. nitidus 1.02 B. albicaudatus 1.16 B. albicaudatus 1.14 B. jamaicensis 1.02 B. nitidus 1.21 B. albicaudatus 1.13 Spizactus tyrannus 1.04 Leucopternis albicollis 1.41 Falco peregrinas 1.87 F. peregrinas 1.57 F. rusticolus 1.12 F. femoralis 1.40 F. deiroleucus 1.49 Micrastur semitorquatus 1.48 Asio stygius 1.25 A. flammeus 1.06 Aegolius funcreus 1.18 Otus trichopsis 1.19 O. asio 1.13 O. guatemalae 1.29 O. trichopsis 1.44 O. choliba 1.03 O. guatemalae 1.04 Ciccaba nigrolineata 1.19 Ninox scutulata 1.26 Dendrortyx barbatus 1.05 Odontophorus leucolaemus 1.02 Totanus melanoleucus 1.50 Erolia bairdi 1.28 E. fuscicollis 1.05 E. melanotos 1.25 E. alpina 1.25 E. canutus 1.16 E. mauri 1.21 Rallus elegans 1.45 R. longirostris 1.01 Fulica americana 1.12 Aramides cajanea 1.47 Laterallus albigularis 1.12 Amauronis olivacea 1.12 Geotrygon mrystacea 1.44 G. caniceps 1.09 G. chrysia 1.30 Zenaida aurita 1.11 Z. asiatica 1.31 Columba leucocephala 1.06 C. caribaea 1.00 C. inornata 1.06 C. squamosa 1.06 Geotrygon chrysia 1.48 G. versicolor 1.25 G. costaricensis 1.22 G. lawrenci 1.01 G. violacea 1.09 Columbigallina talpacoti 1.11 Columba subvinacea 1.03 C. cayennensis 1.20 C. speciosa 1.12 Claravis pretrosa 1.08 Leptotila plumbeiceps 1.05 L. verrcauxi 1.08 Phapitreron amethystina 1.50 Ptilinopus occipitalis 1.04 Ducula aenea 1.09 Amazona imperialis 1.20 A. collaria 1.04 A. finschi 1.20 A. ochrecephala 1.33 A. ochrecephala 1.13 A. farinosa 1.13 Aratinga finschi 1.45 A. holochlora 1.51 Ara macao 1.66 A. ambigua 1.03 A. chloroptera 1.20 Tanygnathus sumatranus 1.05 Caccyzus minor 1.06 Crotophaga ani 1.20 C. major 1.42 Centropus viridis 1.22 C. melanops 1.33 Nyctibius grandis 1.52 Chordeiles minor 1.10 Chaetura rinerciventris 1.06 C. rutilus 1.10 Collocalia inexpectata 1.28 C. esculenta 1.10 Anthracothorax viridis 1.04 Lophornis helenae 1.03 Selasphorus terridus 1.17 S. flammula 1.05 Eupherusa eximia 1.28 Calypte anna 1.06 C. thalassinus 1.28 Calothrax lucifer 1.18 Amazilia violiceps 1.16 A. rutila 1.01 A. virifrons 1.06 A. cyanocaphala 1.21 A. yucatanensis 1.03 A. tzacatl 1.05 A. amabilis 1.01 A. tzacatl 1.23 A. rutila 1.14 A. tzacatl 1.02 Lampornis demenciae 1.10 Campylopterus rulus 1.07 C. hemilcucurus 1.08 Heliomaster constanti 1.05 Pharthornis superciliosus 1.75 P. guy 1.11 Trogan collaris 1.02 T. citreolus 1.18 T. massena 1.34 T. violaceus 1.01 T. elegans 1.11 T. citreolus 1.16 T. massena 1.14 T. violaceus 1.01 T. clathratus 1.26 T. massena 1.17 T. collaris 1.02 T. aurantiiventris 1.03 T. rujus 1.00 T. viridis 1.31 T. malnurus 1.06 T. massena 1.04 Chloroceryle american 1.45 C. amazona 1.46 C. amierican 1.42 C. inda 1.13 C. amazona 1.30 Halcyon winchelli 1.03 H. smyrnensis 1.25 Ceyx melanurus 1.04 Todus subulatus 1.13 Notharchus pectoralis 1.45 N. macrorhynchos 1.35 Ramphastos swainsoni 1.30 Dendrocopos villosus 1.72 Picoides arcticus 1.23 Dendrocopos medius 1.44 D. major 1.33 D. albolarvatus 1.73 D. villosus 1.12 Sphyrapicus thyroides 1.05 Dendrocopos scalaris 1.27 D. villosus 1.15 D. arizonae 1.08 D. villosus 1.15 D. stricklandi 1.02 D. villosus 1.24 Colaptes cafer 1.14 Centurus chyrsogenys 1.14 C. aurifrons 1.25 C. aurifrons 1.48 C. aurifrons 1.64 C. pucherani 1.06 Piculus rubiginosus 1.23 Celeus castaneus 1.17 Phloeoceastes haematogaster 1.12 Dendrocincla anabatina 1.00 Xiphorhynchus lachrymosus 1.07 X. guttatus 1.08 Lepidocolaptes affins 1.03 Dendrocolaptes picumnus 1.10 Lepidocolaptes leucogaster 1.16 Xiphorhynchus flavigaster 1.16 Xenops minutus 1.06 Synallaxis brachyura 1.25 Sclerurus mexicanus 1.09 Myrmotherula axillaris 1.01 M. fulviventris 1.10 Cercomacra nigricans 1.09 Myrmeciza longipes 1.02 Thamnophilus punctatus 1.02 Formicarius nigricapillus 1.13 Grallaria fulviventris 1.02 G. guatimalensis 1.47 Pachyramphus cinnamomeus 1.08 Tityra semifasciata 1.04 Pipra mentalis 1.04 Tyrannus dominicensis 1.35 Elaenia flavogaster 1.26 Tyrannus dominicensis 1.01 T. cubensis 1.26 Myiarchus stolidus 1.16 M. validus 1.18 Empidonax flaviventris 1.05 E. trailli 1.07 Tyrannus dominicensis 1.37 T. vociferans 1.08 Myiobius sulphureipygius 1.06 Elaenia frantzi 1.03 Todirostrum sylvia 1.11 T. cinercum 1.09 Myiozetetes similis 1.06 Tolmomyias assimilis 1.05 Contopus cinereus 1.07 C. pertinax 1.31 C. ochraceus 1.03 Myiarchus nuttingi 1.01 M. tyrannulus 1.37 M. fcrox 1.17 Myiodynastes hemichrysus 1.03 M. maculatus 1.16 Tyrannus crassirostris 1.23 Pitta sordida 1.05 P. steeri 1.15 Coracina striata 1.40 Corvus leucognaphalus 1.20 C. nasicus 1.27 C. brachyrhynchos 1.21 C. corax 1.53 C. cryptoleucus 1.39 C. corax 1.31 Cyanolyoa cucullata 1.27 C. cucullata 1.08 Aphelocoma ultramarina 1.02 A. unicolor 1.05 Corvus macrorhynchus 1.18 Parus atricapillus 1.05 P. hudsonicus 1.03 P. bicolor 1.43 P. atricristatus 1.22 P. inornatus 1.22 P. palustris 1.19 P. ater 1.00 P. major 1.23 Sitta carolinensis 1.32 S. carolinesnsis 1.35 S. carolinesnsis 1.36 S. castanca 1.01 S. himalayensis 1.14 S. europea 1.15 S. tephronota 1.23 Stachyrus nigrocapitata 1.27 Pycnontus goiavier 1.12 Hypsipetes everetti 1.22 Troglodytes aedon 1.15 T. musculus 1.04 Thryothorus modestus 1.00 T. semibadius 1.08 T. rufalbus 1.04 T. fasciato-ventris 1.03 T. castaneus 1.20 T. atrogularis 1.02 T. zeledoni 1.00 Campylorhynchus rufinucha 1.04 Margarops fuscatus 1.56 Mimus gundlachi 1.28 Toxostoma curvirostre 1.32 T. lecontei 1.02 T. dorsale 1.08 T. curvirostre 1.08 T. ocellatum 1.12 T. redivivum 1.31 Turdus fumigatus 1.16 T. aurantius 1.27 Catharus minimus 1.02 C. fuscescens 1.08 C. guttatus 1.01 C. mexicanus 1.01 C. frantzi 1.01 C. fuscater 1.02 Sialis sialis 1.01 Myadestes obscurus 1.02 Turdus assimilis 1.01 T. migratorius 1.03 T. grayi 1.04 T. plebejus 1.05 Regulus calendula 1.15 Polioptila caerula 1.13 Phylloscopus horealis 1.04 Cisticola juncidis 1.06 Orthotomus atrogularis 1.07 Megalurus palustris 1.24 Hypothymis azurca 1.14 H. coclestsis 1.06 Rhipidura javanica 1.20 Nectarinia jugularis 1.21 Acthopyga pulcherrima 1.29 Arachnothera clarae 1.23 Dicaeum pygmaeum 1.10 D. hypoleucum 1.17 D. trigonostigma 1.00 D. australe 1.01 Vireo osburni 1.46 V. altiloquus h 1.22 V. altiloquus 1.43 V. altiloquus h? 1.25 V. altiloquus 1.69 V. gilvus 1.05 V. flavifrons 1.10 V. olivaceus 1.12 V. gilvus 1.06 V. solitarius 1.06 V. leucophyrs 1.10 V. flavoviridis 1.31 Hylophilus ochraceiceps 1.11 Dendroica petechis 1.17 D. pharetra 1.07 D. petechia 1.07 D. petechia 1.07 Microligea palustris 1.03 Dendroica tigrina 1.00 D. virens 1.01 D. fusca 1.00 D. castanea 1.10 Seiurus motacilla 1.08 Wilsonia canadensis 1.21 Vermivora peregrina 1.05 V. chrysoptera 1.06 Myioborus torquatus 1.04 Basileuterus delatri 1.04 B. melanogenys 1.08 B. tristriatus 1.00 Agelaius phoeniceus 1.32 Icterus galbula 1.10 Agelaius tricolor 1.04 Icterus Cucullatus 1.13 I. graduacauda 1.14 I. wagleri 1.09 I. mesomelas 1.12 I. pectoralis 1.08 I. gularis 1.22 I. wagleri 1.07 I. chrysater 1.04 Caclcus cela 1.30 Cassidix mexianus 1.33 Piranga rubra 1.15 Tanagra affinis 1.03 T. minura 1.00 T. luteiacapilla 1.07 T. anneae 1.07 T. lauta 1.06 T. icterocephala 1.01 T. nigrocincta 1.00 T. lavinia 1.02 T. chrysophyrs 1.07 T. gyrola 1.03 Thraupis palmarum 1.07 Ramphocelus passerini 1.02 R. icteronatus 1.07 Tachyphonus delatri 1.08 T. rufus 1.26 Habia fuscicauda 1.04 Piranga bidentata 1.30 P. flava 1.12 Dacnis cayana 1.05 Cyanerpes lucidus 1.07 Lonchura leucogastra 1.02 Tiaris bicolor 1.02 T. olivacea 1.09 Acanthus flammea 1.09 Spizella passerina 1.03 Melospiza lincolni 1.03 M. melodia 1.12 Zonotrichia atricapilla 1.14 Loxia curvirostra 1.06 Amphispiza bilineate 1.00 Carpodacus pupureus 1.13 C. cassini 1.07 Aimophila ruficeps 1.14 Sporophila nigricollis 1.10 S. aurita 1.06 S. schistacea 1.05 Spinus atriceps 1.10 S. pinus 1.00 S. notatus 1.05 Spinus psaltria 1.00 Passerina leclancheri 1.10 P. rositae 1.29 Aimophila notosticta 1.05 A. botteri 1.00 A. mystacalis 1.02 A. rufescens 1.30 A. ruficauda 1.05 A. quinquestriata 1.04 A. ruficauda 1.11 A. rufescens 1.21 Pipilo fuscus 1.15 Arremonops conirostris 1.17 Atlapetes albinucha 1.26 A. brunnei-nucha 1.05 Melozone leucotis 1.10 A. brunnei-nucha 1.07 Pipilo fuscus 1.01 P. albicollis 1.07 P. ocai 1.02 Oryzoborus nuttingi 1.36 Cyanocompsa cyanoides 1.70 Saltator maximus 1.03 S. airiceps 1.28 Hesperiphona abeillei 1.03 Pheucticus chrysopeplus 1.41

 Bin Frequency 1-1.05 100 1.05-1.1 88 1.1-1.15 56 1.15-1.2 35 1.2-1.25 31 1.25-1.3 28 1.3-1.35 21 1.35-1.4 10 1.4-1.45 13 1.45-1.5 12 1.5-1.55 5 1.55-1.6 2 1.6-1.65 1 1.65-1.7 2 1.7-1.75 3 1.75-1.8 1 1.8-1.85 1 1.85-1.9 1

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