Based on a carefully constructed sample of dwarf stars, a new optical-near infrared photometric calibration to estimate the metallicity of late-type K and early-to-mid-type M dwarfs is presented. The calibration sample has two parts; the first part i
ncludes 18 M dwarfs with metallicities determined by high-resolution spectroscopy and the second part contains 49 dwarfs with metallicities obtained through moderate-resolution spectra. By applying this calibration to a large sample of around 1.3 million M dwarfs from the Sloan Digital Sky Survey and the Two-Micron All Sky Survey, the metallicity distribution of this sample is determined and compared with those of previous studies. Using photometric parallaxes, the Galactic heights of M dwarfs in the large sample are also estimated. Our results show that stars farther from the Galactic plane, on average, have lower metallicity, which can be attributed to the age-metallicity relation. A scarcity of metal-poor dwarf stars in the metallicity distribution relative to the Simple Closed Box Model indicates the existence of the M dwarf problem, similar to the previously known G and K dwarf problems. Several more complicated Galactic chemical evolution models which have been proposed to resolve the G and K dwarf problems are tested and it is shown that these models could, to some extent, mitigate the M dwarf problem as well.
