  This paper considers the query complexity of the functions in the family
F_{N,M} of N-variable Boolean functions with onset size M, i.e., the number of
inputs for which the function value is 1, where 1<= M <= 2^{N}/2 is assumed
without loss of generality because of the symmetry of function values, 0 and 1.
Our main results are as follows: (1) There is a super-linear gap between the
average-case and worst-case quantum query complexities over F_{N,M} for a
certain range of M. (2) There is no super-linear gap between the average-case
and worst-case randomized query complexities over F_{N,M} for every M. (3) For
every M bounded by a polynomial in N, any function in F_{N,M} has quantum query
complexity Theta (sqrt{N}). (4) For every M=O(2^{cN}) with an arbitrary large
constant c<1, any function in F_{N,M} has randomized query complexity Omega
(N).
