  The second moment of the H$\beta$ emission line is calculated for 329 narrow
line Seyfert 1 galaxies (NLS1s) selected from the Sloan Digital Sky Survey
(SDSS), which is used to calculate the central supermassive black hole (SMBHs)
mass of each. We find that the second moment depends strongly on the broader
component of the H$\beta$ line profile. We find that for the NLS1s requiring
two Gaussians to fit the H$\beta$ line the mean value of the SMBH mass from the
H$\beta$ second moment is larger by about 0.50 dex than that from the full
width at half maximum (FWHM). Using the gas velocity dispersion of the
core/narrow component of \oiii $\lambda$ 5007 to estimate the stellar velocity
dispersion, $\sigma_{*}$, the new mass makes NLS1s fall very close to the $\mbh
- \sigma_{*}$ relation for normal AGNs. By using $\sigma_{*}$ measured directly
from SDSS spectra with a simple stellar population synthesis method, we find
that for NLS1s with mass lower than $10^7 \msun$, they fall only marginally
below the $\mbh - \sigma_{*}$ relation considering the large scatter in the
mass calculation.
