  We used the updated [Fe/H] abundances of 168 F-G type dwarfs and calibrated
them to a third order polynomial in terms of reduced ultraviolet excess,
$\delta_{0.41}$ defined with $ugr$ data in the SDSS. We estimated the $M_g$
absolute magnitudes for the same stars via the re-reduced Hipparcos parallaxes
and calibrated the absolute magnitude offsets, $\Delta M_g$, relative to the
intrinsic sequence of Hyades to a third order polynomial in terms of
$\delta_{0.41}$. The ranges of the calibrations are $-2<$[Fe/H]$\leq$0.3 dex
and $4<M_g\leq6$ mag. The mean of the residuals and the corresponding standard
deviation for the metallicity calibration are 0 and 0.137 mag; while, for the
absolute magnitude calibration they are 0 and 0.179 mag, respectively. We
applied our procedures to 23,414 dwarf stars in the Galactic field with the
Galactic coordinates $85^{\circ}\leq b\leq90^{\circ}$, $0^{\circ}\leq
l\leq360^{\circ}$ and size 78 deg$^{2}$. We estimated absolute magnitude $M_g$
dependent vertical metallicity gradients as a function of vertical distance
$Z$. The gradients are deep in the range of $0<Z\leq5$ kpc, while they are very
small positive numbers beyond $Z=5$ kpc. All dwarfs with $5<M_g\leq6$ mag are
thin-disc stars and their distribution shows a mode at $(g-r)_0\approx 0.38$
mag, while the absolute magnitudes $4<M_g\leq5$ are dominated by thick disc and
halo stars, i.e. the apparently bright ones ($g_0\leq18$ mag) are thick-disc
stars with a mode at $(g-r)_0\sim0.38$ mag, while the halo population is
significant in the faint stars ($g_0>18$ mag).
