The sky rms is the simplest model-independent characterization of a cosmological anisotropy signal. We show that the rms temperature fluctuations determined from the COBE Differential Microwave Radiometer (DMR) 4 yr sky maps are frequency independent, consistent with the Planckian spectrum expected for the cosmic microwave background signal and therefore with the hypothesis that they are cosmological in origin. The typical rms amplitude is ~35 +/- 2 μK at 7° and ~29 +/- 1 μK at 10°. An analysis of the rms anisotropy determined from the data in both Galactic and ecliptic coordinates is used to determine the rms quadrupole normalization, Qrms-PS, for a scale-invariant Harrison-Zeldovich power-law model. Corrections are applied for small biases observed in the likelihood analysis. While there are variations depending on the data selection, all results are consistent with a Qrms-PS normalization of ~18 +/- 2 μK. This is also shown to be true for a ``standard'' cold dark matter model of cosmological anisotropy. The difference in the normalization amplitudes derived when the quadrupole is either included or excluded from the analysis is attributable to contamination of the observed sky quadrupole by foreground Galactic emission.NASA/GSFC is responsible for the design, development, and operation of the Cosmic Background Explorer (COBE). Scientific guidance is provided by the COBE Science Working Group. GSFC is also responsible for the development of the analysis software and for the production of the mission data sets.