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 $4<M_gleq6$ 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 $85^{circ}leq bleq90^{circ}$, $0^{circ}leq lleq360^{circ}$ 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 $0<Zleq5$ kpc, while they are very small positive numbers beyond $Z=5$ kpc. All dwarfs with $5<M_gleq6$ mag are thin-disc stars and their distribution shows a mode at $(g-r)_0approx 0.38$ mag, while the absolute magnitudes $4<M_gleq5$ are dominated by thick disc and halo stars, i.e. the apparently bright ones ($g_0leq18$ mag) are thick-disc stars with a mode at $(g-r)_0sim0.38$ mag, while the halo population is significant in the faint stars ($g_0>18$ mag).
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