## Introduction

Secondary ion mass spectrometry (SIMS) is an important tool for understanding short-lived isotopic systems and for constraining early solar system chronology. The SIMS technique, like many other analytical techniques, is fundamentally a sampling experiment: a subsample of a parent population is measured to estimate certain parameters of the source population (e.g., an isotope ratio). It is typically assumed that these sampled ratios are unbiased estimates of the true isotope ratios in the object. When the number of counts of the denominator isotope is large, this assumption is generally safe, but when the number of counts is low, the expectation value of the ratio calculated from the measurement can be significantly higher than the true ratio in the object (see e.g., Pearson 1910). Count rates of the denominator isotope during SIMS measurements of short-lived radionuclide systems are often low, particularly when parent/daughter elemental ratios are high, making the isotope ratios susceptible to ratio bias. In this paper, we refer to the bias as the expectation value of the measured isotope ratio minus the true ratio in the object. Ogliore et al. (2011) discuss the issue of ratio bias as it applies to SIMS measurements. They show that positive bias in isotope ratios inferred from counting data can be significant and can result in incorrect inferences about the objects. The relative bias (the bias divided by the true ratio) in ratio estimation is approximately equal to the inverse of the number of counts in the denominator (assumed to be Poisson distributed). For example, if the number of total counts of the denominator isotope is 200, the bias of the estimated ratio will be 5‰. The more counts of the denominator isotope, the smaller the bias and the closer the estimated ratio is to the true ratio that the investigator seeks to measure.

Ratio bias is particularly insidious in SIMS measurements of short-lived radionuclide systems where statistical bias from low counts in the denominator isotope can produce a correlation similar to an isochron with a positive slope (Fig. 1a). The bias increases as the number of denominator counts decreases. In the example in Fig. 1a, the counts of ^{55}Mn have been held constant for each set of data, so the x-axis of the plot is effectively 1/^{52}Cr, producing a perfect correlation. In a real system, the ^{55}Mn counts will also vary, which weakens the correlation. But, for many natural systems, the parent/daughter element ratio is controlled primarily by variations in the daughter element abundance and the system approaches the modeled case. As we will show below, unrecognized ratio bias can easily be interpreted as evidence for the presence of a short-lived nuclide when the sample formed.

Averaging the ratios from many cycles of a measurement is especially prone to ratio bias, because the counts obtained over the measurement are divided up amongst the individual ratios, resulting in a lower number of counts for each ratio. For instance, if a measurement is partitioned into 100 cycles, the ratio calculated by the mean of these 100 ratios will have a relative positive bias about 100 times larger than the ratio determined from dividing the total counts of the numerator isotope by the total counts of the denominator isotope. Increasing the number of ratios (while maintaining the same number of counts per ratio) improves the statistical uncertainty, but it does not decrease the bias (Fig. 1b). If one averages 300 ratios with a mean of 200 counts in the denominator of each ratio, the estimated ratio will have an expectation value 5‰ greater than the true value and a statistical uncertainty of ±6.6‰ (the highest gray point in Fig. 1a). However, if one averages 900 ratios, the estimated ratio will still be 5‰ larger than the true ratio even though the uncertainty has decreased to ±3.8‰ (Fig. 1b). Totaling the counts before calculating the ratios will lower the bias. However, either method can be significantly biased if the counts in the denominator are low. The data for many of the published SIMS studies of short-lived radionuclide systems have been calculated using the mean of the ratios, implying that the published ratios may be significantly affected by statistical bias.

The most effective way to avoid ratio bias is to ensure that there are enough counts of the denominator isotope in each cycle of the measurement. In cases where this cannot be achieved, calculating ratios from the total counts will usually suffice to eliminate the effect of ratio bias. When the counts per cycle are very low, Beale’s estimator, a method that takes into account the total counts of the isotopes and the correlation between the numerator and denominator isotopes, will provide more accurate estimated ratios, as it is less biased (Ogliore et al. 2011). Coath and Steele (2013) proposed another ratio estimator with low bias. Determining the number of counts necessary to avoid biased ratios depends on the accuracy required to clearly observe the desired effects (e.g., excesses in radiogenic isotopes).

In this paper, we discuss the effect of ratio bias on isochron slopes, and we report recalculated results for the data published in Hsu et al. (1997) and Hsu (2005) on ^{53}Mn-^{53}Cr systematics of pallasites, in Tachibana and Huss (2003) and Guan et al. (2004) on ^{60}Fe-^{60}Ni systematics of sulfides from ordinary and enstatite chondrites, and in Tachibana et al. (2006) and subsequent abstracts from the University of Hawai’i on ^{60}Fe-^{60}Ni systematics of silicates from ordinary chondrites. We also recalculated some of the data reported in Mishra et al. (2010) on ^{60}Fe-^{60}Ni systematics of silicates from ordinary chondrites, and data for ^{10}Be-^{10}B systematics of CAIs from CV chondrites published by MacPherson et al. (2003). We corrected for possible biases in the published isotopic ratios by calculating the ratios using the total counts instead of averaging the ratios from each cycle. For the ^{10}Be-^{10}B data, we also used Beale’s estimator to calculate the ratios. The data reported here should be used in place of those reported in the original publications. We hope that re-evaluating these datasets will encourage other researchers to revisit their data. The new data will provide much needed clarification on the abundances of short-lived radionuclides in the solar system.