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tokens: of magnetohydrodynamic (MHD) waves with observations of solar atmospheric waves and oscillations gives information on the physical parameters of medium (e.g., Nakariakov et al. 2016 ), including sunspots (e.g., see Snow et al. 2015 ; Sych 2016 for recent results). Moreels van Doorsselaere ( 2013 ) designed a method for deriving the temperature of plasma confined in photospheric magnetic flux tubes by using the intensity, velocity, and magnetic field strength fluctuation amplitudes. Reznikova et al. ( 2012 ) estimated the inclination angle of umbral magnetic field, based on the interpretation of umbral oscillation as a slow magnetoacoustic gravity (MAG) wave in an inclined magnetic field. Yuan et al. ( 2014 ) established that the magnetic inclination angle derived seismologically was 30%–40% larger than that given by the magnetic field extrapolation. Roberts ( 2004 ) reviewed the analytical expressions for the cutoff frequency as a function of observable parameters for different solar atmospheric models. Roberts ( 2006 ) provided an analytical form of angular cutoff frequency of a slow magnetoacoustic wave in a non-isothermally stratified atmosphere with a uniform vertical magnetic field. The cutoff frequency not only prescribes the lowest frequency of a propagating wave, but also acts as a natural frequency of a medium (e.g., Roberts 2004 ). It suggests that the atmosphere should be observed to oscillate at the frequency equal or higher than the cutoff frequency in response to an external broadband excitation (Botha et al. 2011 ; Chae Goode 2015 ; Kwak et al. 2016 ). Thus, the cutoff frequency is always below the detected frequency (e.g., Tziotziou et al. 2006 ; Yuan et al. 2014 ). Thus, the cutoff frequency involves the physical parameter of the medium, independently of the specific excitation mechanism of atmospheric oscillations. In this work, we determine the cutoff frequency of umbral oscillations by using the weighted frequency, , where P ( f ) is the power spectrum (Takahashi et al. 2015 ), for 478 sunspots observed in the continuum with the Heliospheric Magnetic Imager (HMI; Scherrer et al. 2012 ; Schou et al. 2012 ) on board the Solar Dynamics Observatory ( SDO ; Pesnell et al. 2012 ). Using the theory given by Roberts ( 2006 ), we determine the Alfvén speed, plasma-beta ( β ), and mass density inside umbrae. In Section 2 , we describe the method used to obtain the weighted frequency and its link with the Alfvén speed, plasma- β , and mass density. In Section 3 , we provide maps of those quantities as well as their average behavior for the 478 analyzed sunspots. Finally, we summarize and discuss our results. 2. Method: Seismology We derive the physical parameters of an umbra by using the seismology, developing the approach designed by Yuan et al. ( 2014 ), who estimated acoustic cutoff frequencies of umbral oscillations at the heights of 1700, 1600, and 304 Å line formation. The authors found that the acoustic cutoff frequency is