  A set of permutations is called sign-balanced if the set contains the same
number of even permutations as odd permutations. Let $S_n(\sigma_1, \sigma_2,
\ldots, \sigma_r)$ be the set of permutations in the symmetric group $S_n$
which avoids patterns $\sigma_1, \sigma_2, \ldots, \sigma_r$. The aim of this
paper is to investigate when, for certain patterns $\sigma_1, \sigma_2, \ldots,
\sigma_r$, $S_n(\sigma_1, \sigma_2, \ldots, \sigma_r)$ is sign-balanced for
every integer $n>1$. We prove that for any $\{\sigma_1, \sigma_2, \ldots,
\sigma_r\}\subseteq S_3$, if $\{\sigma_1, \sigma_2, \ldots, \sigma_r\}$ is
sign-balanced except $\{132, 213, 231, 312\}$, then $S_n(\sigma_1, \sigma_2,
\ldots, \sigma_r)$ is sign-balanced for every integer $n>1$. In addition, we
give some results in the case of avoiding some patterns of length $4$.
