  We have measured the root-mean-square (RMS) amplitude of intensity
fluctuations, $\Delta I$, in plume and interplume regions of a polar coronal
hole. These intensity fluctuations correspond to density fluctuations. Using
data from the \rev{Sun Watcher using Active Pixel System detector and Image
Processing} (SWAP) on \textit{Project for Onboard Autonomy (Proba2)}, our
results extend up to a height of about 1.35~$R_{\sun}$. One advantage of the
RMS analysis is that it does not rely on a detailed evaluation of the power
spectrum, which is limited by noise levels to low heights in the corona. The
RMS approach can be performed up to larger heights where the noise level is
greater, provided that the noise itself can be quantified. At low heights, both
the absolute $\Delta I$, and the amplitude relative to the mean intensity,
$\Delta I/I$, decrease with height. However, starting at about 1.2~$R_{\sun}$
$\Delta I/I$ increases, reaching 20--40\% by 1.35~$R_{\sun}$. This corresponds
to density fluctuations of $\Delta n_{\mathrm{e}}/n_{\mathrm{e}}
\approx$~10--20\%. The increasing relative amplitude implies that the density
fluctuations are generated in the corona itself. One possibility is that the
density fluctuations are generated by an instability of Alfv\'en waves. This
generation mechanism is consistent with some theoretical models and with
observations of Alfv\'en wave amplitudes in coronal holes. Although we find
that the energy of the observed density fluctuations is small, these
fluctuations are likely to play an important indirect role in coronal heating
by promoting the reflection of Alfv\'en waves and driving turbulence.
