Individual-weighted mean group size

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.