A systematic time lag of approximate to 20 ms between the 25-50 keV and 50-100 keV hard X-ray (HXR) emission has been recently discovered in solar flares. This was interpreted in terms of electron time-of-flight differences (Aschwanden, Schwartz, & Alt 1995c). Here we attempt to determine the accuracy and uncertainties of such energy-dependent time delay measurements using burst-trigger data from DISCSC/BATSE on the CGRO spacecraft, recorded with a time resolution of 64 ms. We evaluate the time delays by cross-correlating entire flare time profiles at different energies and evaluate the statistical uncertainty of a delay measurement with a Monte Carlo method, in which random noise is added to the raw data. We examine also uncertainties resulting from aliasing, incomplete sampling, and pulse pileup. We measure the time delays tau = t(25 keV) - t(50 keV) in 622 flares, with a statistical uncertainty of u less than or equal to 32 ms in 29% of the events, or u less than or equal to 64 ms in 65% of the events. The distribution f(tau) of the time delays from all flares can be characterized with three components: (1) a Gaussian peak at tau = 23.2 +/- 1.2 ms with a standard deviation of sigma(t) = 25.5 ms, (2) a power-law tail with a slope of -2.0 for large positive delays (tau = 0.1-4 s), and (3) a power-law tail with a slope of -1.3 for large negative delays (\-tau\ = 0.1-7.7 s). The percentages of flares in these three regimes are 15%, 69%, and 16%. Flares with short delays (\tau\ less than or equal to 0.1 s) exhibit rapid fluctuations with subsecond pulses. Such rapid fluctuations are almost absent in flares with longer delays. We find also a systematic trend of softer spectra in flares with large positive delays. We develop a simple physical model that combines electron time-of-flight differences in the thin-target and thick-target model. We are able to reproduce the observed time delay distribution in the range of \tau\ less than or similar to 0.1 s, requiring a distribution of electron densities in the range of n(e) < 3.0 x 10(12) cm(-3) and flare loop heights in the range of h less than or equal to 35,000 km. Large negative delays (tau greater than or similar to -0.1 s) can be produced in low-density loops with efficient magnetic trapping. Large positive delays (tau greater than or similar to 0.1 s) occur in flares with a strong thermal component due to the convolution of the injection profile with the heating and cooling function. This study demonstrates that energy-dependent HXR time delays can be used as a diagnostic and discriminator of flare models.
