  $f(Q)$ gravity is an extension of the symmetric teleparallel equivalent to
general relativity (STEGR). This work shows that based on the
scalar-nonmetricity formulation, a scalar mode in $f(Q)$ gravity has a negative
kinetic energy. This conclusion holds regardless of the coincident gauge
frequently used in STEGR and $f(Q)$ gravity. To study the scalar mode, we
further consider the covariant $f(Q)$ gravity as a special class in
higher-order scalar tensor (HOST) theory and rewrite the four scalar fields,
which play a role of the St\"{u}eckelberg fields associated with the
diffeomorphism, by vector fields. Applying the standard Arnowitt-Deser-Misner
(ADM) formulation to the new formulation of the $f(Q)$ gravity, we demonstrate
that the ghost scalar mode can be eliminated by the second-class constraints,
thus ensuring that $f(Q)$ gravity is a healthy theory.
