  A sufficient and necessary condition ensuring that the backward shift
operator on the K\"{o}the sequence space admits an invariant distributionally
$\varepsilon$-scrambled set for some $\varepsilon>0$ is obtained, improving the
main results in [F. Mart\'{\i}nez-Gim\'{e}nez, P. Oprocha, A. Peris, J. Math.
Anal. Appl., {\bf 351} (2009), 607--615].
