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tokens: obs , and I obs were then, for the i th value, and“Observed” values of p ′ and were calculated using Equations ( 2 ) and ( 8 ), respectively, for a given value of . We repeated this process 10 4 times. Figure 2 shows a single realization of our Monte Carlo simulations for three cases: α = 1, 0.5, and 0 (constant polarization fraction). In all cases, p 100 mJy beam –1 = 0.025. The results are shown without any debiasing of the polarization fraction and colored according to their S/N in I and p ′. Figure 2. Monte Carlo simulations of the observed polarization fraction p ′ as a function of S/N I / σ QU for three cases: α = 0 (top), 0.5 (middle), and 1 (bottom). In all cases, p 100 mJy beam –1 = 0.025. Results are shown without debiasing. The polarization fraction p ′ is given as an absolute value (not as a percentage), so any value p ′ > 1 (marked in bold on plot axes) is unphysical. Points are colored according to their S/N in I and p ′; gray points meet none of the criteria shown in the legend. The solid black line shows p ′ = μ p , the mean of the Rice distribution, for the given values of α and . The dashed black line shows the true polarization fraction p = ( I / σ QU ) − α . The dotted line shows the null-hypothesis/low-S/N relation . In the cases where α 1, the dotted–dashed line marks the critical value . Figure 2 shows that where I / σ QU ( I / σ QU ) crit , the distribution shows an identical p ′ ∝ I −1 behavior in all cases, and that the underlying power-law behavior does not dominate over the I −1 dependence until I / σ QU ≫ ( I / σ QU ) crit . Thus, unless a reasonable number of data points have S/Ns significantly greater than the critical value, the true power-law behavior is not recoverable. The overall behavior of the recovered polarization fraction is well described by the mean of the Rice distribution. 4.1. Uncertainty on Polarization Fraction In keeping with standard practice in observational polarimetry and the default behavior of the POL-2 pipeline (e.g., Kwon et al. 2018 ), we estimated the uncertainty on the polarization fraction using the relation(e.g., Wardle Kronberg 1974 ). We note that this result is derived using classical error propagation and so assumes that δQ , δU , and δI are small and uncorrelated. We here take and . As discussed above, measurement errors δp calculated using Equation ( 19 ) can only be treated as representative of a Gaussian distribution around p ′ when p ′/ δp is large. 4.2. Fitting Methods We investigated the results of fitting two models to our Monte Carlo simulations. Fitting was performed using the scipy routine curve _ fit . In all cases, we attempted to recover the input values of and α , assuming that σ QU is a fixed and directly measurable property of the data set. We supplied our calculated δp values to curve _ fit as 1 σ uncertainties on p ′. Although this is technically valid only at high S/N, the default behavior of this fitting routine is to consider only the relative magnitudes of