Social group size

Social group size is the number of individuals assigned to a social group under a stated membership rule and observation period. A headcount at one moment, a count of regular members and the number present during an experiment can differ. Report which quantity was measured before comparing groups.

The way groups are sampled changes their apparent typical size. Consider two groups containing 2 and 8 animals. Their arithmetic mean size is $(2+8)/2=5$. If an animal is instead sampled uniformly from the ten individuals, the expected size of its group isMore animals experience the larger group. For group sizes $n_1,\dots ,n_k$, this individual-weighted mean group size is ${\sum }_i{n}_i^2/{\sum }_i{n}_i$, rather than ${\sum }_i{n}_i/k$. These are original illustrative calculations, not measured animal data.

Social group size also affects the interpretation of replication. A finch experiment used 70 Bengalese finches in seven groups and 58 white-rumped munias in six groups. The groups contained ten birds each except one eight-bird munia group. Those totals describe birds and groups, not interchangeable counts of independent treatment assignments.^[ref] The experimental unit must follow the design; a larger headcount alone does not solve pseudoreplication.

The individual-weighted mean group size is the expected size of the social group containing an individual chosen uniformly from all individuals in the groups being studied. Larger groups receive more weight because they contain more potential sampled individuals. For positive group sizes $n_1,\dots ,n_k$, the probability of selecting someone in group $i$ is $n_i/{\sum }_j{n}_j$, giving the weighted meanThe focal individual is included; its expected number of group companions is $G-1$.

For an original example, groups of sizes $1,1,4$ contain six individuals. Their group-based arithmetic mean is 2, but $G=(1+1+16)/6=3$. Four of the six individuals experience a group of four, so an individual-based observation gives the larger group more influence.

Write $\mu =k^{-1}{\sum }_i{n}_i$ and ${\sigma }^2=k^{-1}{\sum }_i(n_i-\mu {)}^2$, using a divisor of $k$ for this finite collection. Expanding the squares gives ${\sum }_i{n}_i^2=k({\sigma }^2+{\mu }^2)$ and thereforeEquality holds precisely when all groups have equal size. The difference is determined by group-size variance, rather than a contradiction between two legitimate averages.

An experiment reporting numbers of animals and numbers of groups supplies both levels of description. For instance, a finch study used 70 Bengalese finches in seven groups and 58 white-rumped munias in six groups.^[ref] Choosing the appropriate average depends on whether the scientific question concerns typical groups or the social setting experienced by typical individuals.