Figures

Figure 1: Evolution of p$_{max}$ with increasing number of population/sample fractions for a) a fixed number of measurements (k=60) and b) a fixed relevant fraction size (f=0.05). Solid lines represent worst- and dashed lines best-case scenarios. The shaded region on (a) marks the area where M$\geq$1/f and p is kept constant at p$_{max}$.

Figure 2: As shown in Figure 1, at most M$_{opt}$=6 bins are allowed in a sample histogram in order to reduce the chance that at least one fraction f$\geq$0.05 is missed of a perfectly uniform population (worst-case scenario) to less than p=20%. The black dots represent the 60 samples. In histogram a, M=20$\geq$M$_{opt}$. In this particular example, three bins are empty, amounting to a total fraction 0.15 of the population. When the data are grouped into 20 bins, this occurs 64% of the time. In histogram b, M=M$_{opt}$=6 and, for exactly the same sample, no fraction $\geq$ 0.05 is missed. Only 20% of the histograms with 6 bins will have an empty fraction in their worst-case scenario, which corresponds to five fractions of size 0.05 and one of size 0.75. While using M$_{opt}$ theoretically is a valid way to prevent relevant fractions of being missed, it will generally yield oversmoothed histograms and be of limited practical use. Instead, it is better to simply report f$_{act}$ and/or p$_{max}$ along with the data.

Figure 3: Graphs that allow a quick assessment of the number of grains (k) needed to push the chance of missing at least one fraction $\geq$f of a worst-case population below $p_{max}$, a) as a function of the number of grains (k) and b) the number of (relevant) fractions (M) or their size (f).

Table 1:
The adequate number of grains (k) as a function of the desired probability (p) of missing at least one fraction $\geq$f of a worst-case population.

Table 2
f$_{act}$, p$_{max}$ and M$_{opt}$ as a function of k.

Given a specified number of grains (k), this Table shows f$_{act}$ - the smallest population fraction that has not been missed with at least p% certainty - for four values of p; p$_{max}$ - the maximum probability of missing at least one fraction $\geq$f of a worst-case population - for four values of f; and M$_{opt}$ - the largest number of bins that are less than p% likely to miss at least one fraction $\geq$f of the worst-case population - for four values of f and p.

Figure 4: The random uniform numerical analogue to Figure 1, for k=60 and f=0.05. The solid black line gives the worst-case probability from Figure 1. The different symbols represent different levels of uniformity. The higher its "percentile", the closer a synthetic population is to a uniform distribution. For example, a "99 percentile" population is likely to be multimodal, while a "5 percentile" population would be nearly unimodal.

Figure 5: The numerical analogue to Figure 3, this plot allows a quick lookup of the maximum probability (p$_{max}$) of missing at least one fraction $\geq$f of a "95 percentile" synthetic population when k grains are dated.

Figure 6: a) probability density plot of the Nubian Sandstone [11] and b) the result of numerical resampling experiments for different numbers of grains (k), for f=0.05.

Figure 7: Same as Figure 6, but for the lunar impact spherules data [12].