Modern Free-Electron Laser facilities produce X-ray pulses as short as hundreds of attoseconds for use in ultra-fast science experiments. These experiments ideally require accurate characterization of the attosecond pulse duration; however, direct time-resolved diagnostic techniques are far from being routine procedure. In the meantime, the most experimentally accessible information about the X-ray pulses is available in their spectra and pulse energies. We show that the statistics of these quantities from a large dataset are sufficient to determine the average "lasing" length of the electron bunches, after which FEL theory determines the statistics of the x-ray pulses. These predicted statistics can be compared to those obtained from direct time-resolved measurements. A strategy is proposed to systematically reconstruct the average FEL parameter and electron bunch length from the measured pulse spectra and energies. On the way, we extend the analytical frameworks put in place by Bonifacio and Saldin, generalizing to finite length, non-uniform bunches with startup from shot noise.