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tokens: h is the Planck constant, k is the Boltzmann constant, T d is effective dust temperature, and β is the emissivity index, which determines the steepness of the Rayleigh–Jeans slope of the spectrum. The python implementation emcee (Foreman-Mackey et al. 2013 ) was used to build a Markov Chain Monte Carlo (MCMC) analysis of the fitted SED for each data set, allowing the dust temperature and normalization as free parameters to sample the posterior probability distribution of the model. Where possible, we also leave β as a free parameter in the model, allowing for a test of the range of dust emissivities that are consistent with the data. However, fitting for β requires at least four data points to constrain the peak and Rayleigh–Jeans slope of the modified blackbody function. For many sources, our Herschel /SPIRE data are the only measurement in the FIR–submillimetre regime. Thus, we assume a value of β = 1.5 for these sources, as is frequently applied in the literature (e.g. Magnelli et al. 2012 ). Various combinations of T d and β have been found for samples of high-redshift quasars. For example, Priddey McMahon ( 2001 ) find an average of T d = 41 ± 5 K and β = 1.95 ± 0.3, whereas Beelen et al. ( 2006 ) find an average of T d = 47 ± 3 K and β = 1.6 ± 0.1 for their sample. For the sources that were not detected, or had only one detection to constrain the fit, we assume the median fitted dust temperature of the sample (38 K, see Fig. 6 ) and β = 1.5, and fit only for the normalization. For these sources with only one detection, we fit the 16th and 84th percentile values of the median fitted temperatures to estimate our errors on the FIR luminosity. As β is highly correlated with T d , errors on the derived properties of sources without β-fitting may be underestimated. However, the FIR luminosity is not strongly affected by our assumptions due to the joint dependency of T d , β and normalization, so this will not have a significant effect on the inferred values of L FIR or SFR. This is unsurprising, as the luminosity is derived by the integral of the fit defined by the data points. Figure 6. Histogram of effective dust temperatures for 53 quasars in the sample with temperature fitting, excluding those without a known redshift and synchrotron-dominated sources. The median dust temperature is K. Where two dust temperatures are fit, only the colder component is included here. For the purpose of spectral fitting, the ten sources without a known redshift are assumed to have z = 1.8, equivalent to the median redshift of the sample. The choice of redshift significantly affects the luminosity distance; thus, these objects are not included in the overall statistics. For three sources (APM 08279+5255, H1413+117, and iras F10214+4724), there are sufficient data in the mid-IR (MIR) to motivate fitting a two-temperature dust model. Table 3 shows the number of sources fitted with each combination of spectral parameters. Table 3. Number of sources fitted with