Many animals, including humans, tend to live in groups,

herd A herd is a social group of certain animals of the same species, either wild or domestic. The form of collective animal behavior associated with this is called ''herding''. These animals are known as gregarious animals. The term ''herd'' is ...

flocks Flocking is the behaviour exhibited when a group of birds, called a flock, are foraging or in flight. Computer simulations and mathematical models that have been developed to emulate the flocking behaviours of birds can also generally be applie ...

, bands, packs,

shoals In oceanography, geomorphology, and geoscience, a shoal is a natural submerged ridge, bank, or bar that consists of, or is covered by, sand or other unconsolidated material and rises from the bed of a body of water to near the surface. It o ...

, or

colonies In modern parlance, a colony is a territory subject to a form of foreign rule. Though dominated by the foreign colonizers, colonies remain separate from the administration of the original country of the colonizers, the '' metropolitan state'' ...

(hereafter: groups) of conspecific individuals. The size of these groups, as expressed by the number of people/etc in a group such as eight groups of nine people in each one, is an important aspect of their social environment. Group size tend to be highly variable even within the same species, thus we often need statistical measures to quantify group size and statistical tests to compare these measures between two or more samples. Group size measures are notoriously hard to handle statistically since groups sizes typically follow an aggregated (right-skewed) distribution: most groups are small, few are large, and a very few are very large. Statistical measures of group size roughly fall into two categories.

Outsiders' view of group size

* Group size is the number of individuals within a group; * Mean group size, the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...

of group sizes averaged over groups; *

Confidence interval In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...

for mean group size; * Median group size, the

median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...

of group sizes calculated over groups; *

for median group size.

Insiders' view of group size

As Jarman (1974) pointed out, average individuals live in groups larger than average. Therefore, when we wish to characterize a typical (average) individual’s social environment, we should apply non-parametric estimations of group size. Reiczigel et al. (2008) proposed the following measures: * Crowding is the size (the number of individuals) of a group that a particular individual lives in (equals to group size: one for a solitary individual, two for both individuals in a group of two, etc.). Practically, it describes the social environment of one particular individual. This was called ''Individual Group Size'' in Jovani & Mavor's (2011) paper.; * Mean crowding, i.e. the

of crowding measures averaged over individuals (this was called "Typical Group Size" according to Jarman's 1974 terminology); *

for mean crowding.

Example

Imagine a sample with three groups, where group sizes are one, two, and six individuals, respectively, then : mean group size (group sizes averaged over groups) equals

(1+2+6)/3=3

; : mean crowding (group sizes averaged over individuals) equals

(1+2+2+6+6+6+6+6+6)/9=4.555

. Generally speaking, given there are G groups with sizes n₁, n₂, ..., n_G, mean crowding can be calculated as: : mean crowding=

\sum_^G n^2_i/\sum_^G n_i

Statistical methods

Due to the aggregated (right-skewed) distribution of group members among groups, the application of

parametric statistics Parametric statistics is a branch of statistics which assumes that sample data comes from a population that can be adequately modeled by a probability distribution that has a fixed set of parameters. Conversely a non-parametric model does not as ...

would be misleading. Another problem arises when analyzing crowding values. Crowding data consist of non-independent values, or ties, which show multiple and simultaneous changes due to a single biological event. (Say, all group members' crowding values change simultaneously whenever an individual joins or leaves.) Reiczigel et al. (2008) discuss the statistical problems associated with group size measures (calculating

confidence interval In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...

s, two-sample tests, etc.) and offer a free statistical toolset (Flocker 1.1).

Literature

* Debout G 2003. Le corbeau freux (Corvus frugilegus) nicheur en Normandie: recensement 1999 & 2000. ''Cormoran,'' 13, 115–121. * Jarman PJ 1974. The social organisation of antelope in relation to their ecology. ''Behaviour'', 48, 215–268. * Jovani R, Mavor R 2011
Group size versus individual group size frequency distributions: a nontrivial distinction.
''Animal Behaviour'', 82, 1027–1036. * Lengyel S, Tar J, Rozsa L 2012
Flock size measures of migrating Lesser White-fronted Geese Anser erythropus
''Acta Zoologica Academiae Scientiarum Hungaricae'', 58, 297–303. (2012) * Reiczigel J, Lang Z, Rózsa L, Tóthmérész B 2008
Measures of sociality: two different views of group size.
''Animal Behaviour'', 75, 715–721. * Reiczigel J, Mejía Salazar MF, Bollinger TK, Rozsa L 2015
Comparing radio-tracking and visual detection methods to quantify group size measures.
''

European Journal of Ecology The ''European Journal of Ecology'' is an English-language, biannual, scientific journal founded in 2015. It publishes original, peer-reviewed papers (in categories like research articles, reviews, forum articles, policy directions) referring to a ...

'', 1(2), 1–4.

External links

Flocker 1.1 – a statistical toolset to analyze group size measures (with all the abovementioned calculations available)

Gallery

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flock {{collective animal behaviour Behavioural sciences Ethology Sociobiology Behavioral ecology Group processes Herding Statistical inference