= 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 $n_1,\ldots,n_k$, the probability of selecting someone in group $i$ is $n_i/\sum_jn_j$, giving the <weighted mean>
$$
G=\frac{\sum_i n_i^2}{\sum_i n_i}.
$$
The 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 therefore
$$
G=\mu+\frac{\sigma^2}{\mu}\geq\mu.
$$
Equality 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.<Effects of domestication on neophobia: A comparison between the domesticated Bengalese finch and its wild ancestor, the white-rumped munia by Kenta Suzuki and others>[Birds used in this study: group membership counts]{ref} Choosing the appropriate average depends on whether the scientific question concerns typical groups or the social setting experienced by typical individuals.
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