The ground-based magnetometer network has long been a powerful tool for monitoring and observing the variations of the currents flowing in the magnetosphere-ionosphere (M-I) system. The time series of magnetograms are nonstationary and their frequency behavior changes over time. They are therefore not amenable to traditional time domain or spectral (Fourier) analysis. In recent years, various new mathematical techniques have been developed to analyze magnetometer data and the wavelet technique has stood out as being particularly relevant. In order to correctly make statistical inferences based on wavelet analysis, the wavelet coefficient distributions of magnetograms must be examined. In this work, we apply the discrete wavelet transform to the 1-min magnetometer records and then use several statistical techniques to analyze the probability distributions of the wavelet coefficients. It is found that the distributions of these coefficients for both storm and quiet times are highly nonnormal and can be classified as being heavy tailed. This finding suggests that when applying statistical techniques to the wavelet coefficients of the magnetograms, one must make sure that these techniques are robust to large departures from Gaussianity manifested by the presence of heavy probability tails. It is also found that the tail indexes for storm times are on average smaller than those of quiet times, which reflects the stronger impulsive and nonstationary features in magnetometer data during storm times, and the shifts are most significant for the wavelet coefficients corresponding to physical scales of 4--8 min.