Our main result shows that a continuous map
f
f
acting on a compact metric space
(
X
,
ρ
)
(X,\rho )
with a weaker form of specification property and with a pair of distal points is distributionally chaotic in a very strong sense. Strictly speaking, there is a distributionally scrambled set
S
S
dense in
X
X
which is the union of disjoint sets homeomorphic to Cantor sets so that, for any two distinct points
u
,
v
∈
S
u,v\in S
, the upper distribution function is identically 1 and the lower distribution function is zero at some
ε
>
0
\varepsilon >0
. As a consequence, we describe a class of maps with a scrambled set of full Lebesgue measure in the case when
X
X
is the
k
k
-dimensional cube
I
k
I^{k}
. If
X
=
I
X=I
, then we can even construct scrambled sets whose complements have zero Hausdorff dimension.