The bias in periodogram spectral estimators is computed as a function of the sample size N by assuming a model power spectrum that decays like f-α at high frequencies. For α = 2, it is shown that when the aliasing of the measured power spectrum is properly taken into account the bias in the "raw" periodogram is nearly independent of frequency for large N. For the range of values 1.7 $lesssim$ α <2, an upper bound on the bias is provided by the case α = 2. Theoretical calculations of the maximum absolute bias as a function of N are used to determine when the periodogram is approximately unbiased and when the bias is significant enough to require the use of a modified periodogram which incorporates data tapering, also called data windowing. For solar wind velocity data acquired by the ACE spacecraft and a chosen low frequency cutoff of 10-7 Hz, the bias in periodogram spectral estimators is found to be less than 4% for sample sizes N greater than 216 = 65536. This corresponds to a 49 day record of 64 s data.