  Schools with the highest average student performance are often the smallest
schools; localities with the highest rates of some cancers are frequently small
and the effects observed in clinical trials are likely to be largest for the
smallest numbers of subjects. Informal explanations of this "small-schools
phenomenon" point to the fact that the sample means of smaller samples have
higher variances. But this cannot be a complete explanation: If we draw two
samples from a diffuse distribution that is symmetric about some point, then
the chance that the smaller sample has larger mean is 50\%. A particular
consequence of results proved below is that if one draws three or more samples
of different sizes from the same normal distribution, then the sample mean of
the smallest sample is most likely to be highest, the sample mean of the second
smallest sample is second most likely to be highest, and so on; this is true
even though for any pair of samples, each one of the pair is equally likely to
have the larger sample mean.
  Our conclusions are relevant to certain stochastic choice models including
the following generalization of Thurstone's Law of Comparative Judgment. There
are $n$ items. Item $i$ is preferred to item $j$ if $Z_i < Z_j$, where $Z$ is a
random $n$-vector of preference scores. Suppose $\mathbb{P}\{Z_i = Z_j\} = 0$
for $i \ne j$, so there are no ties. Item $k$ is the favorite if $Z_k <
\min_{i\ne k} Z_i$. Let $p_i$ denote the chance that item $i$ is the favorite.
We characterize a large class of distributions for $Z$ for which $p_1 > p_2 >
\cdots > p_n$. Our results are most surprising when $\mathbb{P}\{Z_i < Z_j\} =
\mathbb{P}\{Z_i > Z_j\} = \frac{1}{2}$ for $i \ne j$, so neither of any two
items is likely to be preferred over the other in a pairwise comparison.
