We assembled a sample of broad-line and narrow-line active galactic nuclei (AGNs) observed by ASCA, the excess variances of which have been determined. The central black hole masses in this sample can be obtained from the reverberation mapping method and the width of the Hβ emission line. Using the black hole masses and the bolometric luminosity, the Eddington ratio (the ratio of the ionizing luminosity to the Eddington luminosity) can also be obtained. We confirmed the strong anticorrelation between X-ray variability and the central black hole mass found by Lu & Yu. In addition, we found that narrow-line Seyfert 1 galaxies (NLS1s) also follow this relation, with a larger scatter, and that there is only a weak correlation between the X-ray variability and the Eddington ratio, which suggests that rapid variability and narrow lines in NLS1s are mainly due to small central black holes in NLS1s, not the difference of circumnuclear gas around NLS1s. A strong correlation was found between the hard X-ray photon index and the Eddington ratio. If the suggestion of two distinct accretion classes [namely advection-dominated accretion flow (ADAF) and thin disc accretion] in AGNs is correct, the strong correlation between the photon index and the Eddington ratio shows that there exists a kind of two-zone accretion disc, in which the outer zone is a thin disc and the inner zone is an ADAF disc. Otherwise, the accretion process is thin disc accretion and ADAF accretion is not required.