Randomized and deterministic query complexity of symmetric functions
Ok, I found the answer in this survey: http://homepages.cwi.nl/~rdewolf/publ/qc/dectree.pdf

The sensitivity $s(f)$ of a (nonconstant) symmetric function $f$ is $s(f) \geq \lceil\frac{n+1}{2}\rceil$. However, $D(f) \geq s(f)$ and $R_{1/4}(f) \geq \frac{s(f)}{3}$...

There are other interesting results concerning symmetric functions in this survey (see Section 6.1).