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The wide-area imaging surveys with the {\it Herschel} Space Observatory atsub-mm wavelengths have now resulted in catalogs of order one hundred thousanddusty, star-burst galaxies. We make a statistical estimate of $N(z)$ using aclustering analysis of sub-mm galaxies detected at each of 250, 350 and 500$\mu$m from the Herschel Multi-tiered Extragalactic Survey (HerMES) centered onthe Bo\"{o}tes field. We cross-correlate {\it Herschel} galaxies against galaxysamples at optical and near-IR wavelengths from the Sloan Digital Sky Survey(SDSS), the NOAO Deep Wide Field Survey (NDWFS) and the Spitzer Deep Wide FieldSurvey (SDWFS). We create optical and near-IR galaxy samples based on theirphotometric or spectroscopic redshift distributions and test the accuracy ofthose redshift distributions with similar galaxy samples defined with catalogsof the Cosmological Evolution Survey (COSMOS), as the COSMOS field has superiorspectroscopy coverage. We model-fit the clustering auto and cross-correlationsof {\it Herschel} and optical/IR galaxy samples to estimate $N(z)$ andclustering bias factors. The $S_{350} > 20$ mJy galaxies have a bias factorvarying with redshift as $b(z)=1.0^{+1.0}_{-0.5}(1+z)^{1.2^{+0.3}_{-0.7}}$.This bias and the redshift dependence is broadly in agreement with galaxiesthat occupy dark matter halos of mass in the range of 10$^{12}$ to 10$^{13}$M$_{\sun}$. We find that the redshift distribution peaks around $z \sim 0.5$ to1 for galaxies selected at 250 $\mu$m with an average redshift of $< z > = 1.8\pm 0.2$. For 350 and 500 $\mu$m-selected SPIRE samples the peak shifts tohigher redshift, but the average redshift remains the same with a value of $1.9\pm 0.2$.

HerMES: A Statistical Measurement of the Redshift Distribution of Herschel-SPIRE Sources Using the Cross-correlation Technique