## Calculating an age from multiple datesEdit

A significant proportion of age constraints for Neoproterozoic strata are derived from U-Pb dates on zircons from volcanic rocks. The final reported date and associated uncertainty are often weighted mean dates derived from a number (n) of individual dates on different zircons (or zircon sub-domains). This is the case for data acquired using both ID-TIMS and microbeam techniques. The weighted mean weights each individual analyses (such as a single SIMS spot or single grain ID-TIMS analyses) according to its precision so analyses with a low uncertainty contribute more to the weighted mean than those with high uncertainty. Importantly, the use of a weighted mean algorithm (or other averaging) is underpinned by the expectation of a single population with normally distributed errors (i.e., there is no correlation between precision of analyses and age). If the errors on the individual analyses are approximately equal (as is typical for microbeam U/Pb data) then the weighted mean uncertainty is proportional to 1/√n, therefore high-n datasets can be used to reduce the overall age uncertainty for data collected on a single population with normally distributed errors. If the uncertainties on the individual analyses are variable then the weighted mean uncertainty is controlled by the most precise analyses making the weighted mean date less proportional to n. High-n datasets are critical for assessing analytical vs geological scatter, however the real limit on the precision is the analytical uncertainty of single (spot or grain) analyses as this controls our ability to resolve real variation within a series of analyses due to ‘open-system’ behaviour.

A common measure of the “coherence” of a data set is a statistical parameter called the MSWD (mean square of the weighted deviates; (York, 1966, 1967)). A value of approximately 1 indicates that the scatter in the data can be explained by analytical uncertainties alone, values much less than 1 indicates that analytical uncertainties have been overestimated, and values greater than 1 can indicate either that the uncertainties have been underestimated or that another source of scatter, often called “geological” scatter is present. Furthermore, the actual MSWD value for which the scatter of the data can be considered due to analytical factors alone, is not restricted to a value of one but in fact varies according to the number of data points in the calculation (Wendt and Carl, 1991). So, to be 95% confident that the scatter of the data is due to the analysis when n=5, an acceptable MSWD range would be 0.2 - 2.2 but for n = 25 this would be 0.6-1.5 (Wendt and Carl, 1991). Although not often explicitly stated, an MSWD of 1 does not necessarily mean there is a single (age) population. Rather, it indicates that if real (age) variation is present, it cannot be resolved within the precision of the individual analyses.

## Uncertainties as a result of geologic complexityEdit

Uncertainty as a result of geologic complexity is the most difficult to quantify. The most common cause of excess scatter is open system behaviour resulted from either inheritance of older zircon and/or Pb loss. For U-Pb zircon analyses reduced errors on single analyses often exposes fine-scale variability that may reflect protracted or punctuated crystallisation of zircon crystals in a magma chamber or the effects of very subtle open system behaviour thus that high-precision analyses do not always transform into reduced uncertainties in calculated weighted mean dates.

Complex U-Pb zircon systematics

In the past decade errors associated with ID-TIMS analyses have been reduced by almost an order of magnitude. These reduced errors offer unprecedented precision but also expose geological complexity at the <0.1 % level sometimes resulting in scatter that exceeds analytical uncertainties. It is now common for a geochronologist to be faced with a population of zircon analyses that do not form a coherent cluster and the crucial question is how to interpret the data to arrive at a depositional age. The advent of CA-TIMS pre-treatment for the elimination of Pb loss has been extremely important as it gives one confidence that in many cases Pb loss need not be considered as a cause of excess scatter. Furthermore, for Neoproterozoic rocks, the concordia curve has a shallow enough slope, and the 207Pb/235U dates measured precisely enough to be able to evaluate discordance at the per mil level, however this is not the case for microbeam U/Pb dates. As outlined above, microbeam U/Pb dates on volcanic rocks rely upon the averaging of a relatively high-n dataset (10-20) of relatively imprecise (ca. 2 to 4%) U/Pb determinations to get a weighted mean date with precision ca. 1% or less. Underpinning these lower uncertainties is the assumption of a single population with normally distributed errors. However, it is the low precision of each analysis combined with variability of the standard analyses that bracket unknowns that often precludes the detection of subtle amounts of Pb loss or inheritance. Stated another way, if the amount of Pb-loss or inheritance is less than the precision of a single spot analyses then it cannot be detected via normal statistical proxies (such as the MSWD) therefore the assumption of a normal distribution maybe be invalid (see Fig. 4). If Pb-loss is the main source of open-system behaviour, this will have the effect of lowering the 206Pb/238U date on some analyses as well as the weighted mean 206Pb/238U date.