  The stellar velocity dispersion, $\sigma$, is a quantity of crucial
importance for spiral galaxies, where it enters fundamental dynamical processes
such as gravitational instability and disc heating. Here we analyse a sample of
34 nearby spirals from the Calar Alto Legacy Integral Field Area (CALIFA)
spectroscopic survey, deproject the line-of-sight $\sigma$ to $\sigma_{R}$ and
present reliable radial profiles of $\sigma_{R}$ as well as accurate
measurements of $\langle\sigma_{R}\rangle$, the radial average of $\sigma_{R}$
over one effective (half-light) radius. We show that there is a trend for
$\sigma_{R}$ to increase with decreasing $R$, that $\langle\sigma_{R}\rangle$
correlates with stellar mass ($M_{\star}$) and tested correlations with other
galaxy properties. The most significant and strongest correlation is the one
with $M_{\star}$: $\langle\sigma_{R}\rangle \propto M_{\star}^{0.5}$. This
tight scaling relation is applicable to spiral galaxies of type Sa $\mbox{--}$
Sd and stellar mass $M_{\star}\approx10^{9.5}\mbox{--}10^{11.5}\
\mbox{M}_{\odot}$. Simple models that relate $\sigma_{R}$ to the stellar
surface density and disc scale length roughly reproduce that scaling, but
overestimate $\langle\sigma_{R}\rangle$ significantly.
