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The statistical characterization of the measurement errors of a phasor measurement unit (PMU) is currently receiving considerable interest in the power systems community. This paper focuses on the characteristics of the errors in magnitude and angle measurements introduced only by the PMU device (called random errors in this paper), during ambient conditions, using a high-precision calibrator. The experimental results indicate that the random errors follow a non-Gaussian distribution. They also show that the M-class and P-class PMUs have distinct error characteristics. The results of this analysis will help researchers design algorithms that account for the non-Gaussian nature of the errors in synchrophasor measurements, thereby improving the practical utility of the said-algorithms in addition to building on precedence for using high-precision calibrators to perform accurate error tests.