A low false-positive rate means we are reasonably unlikely to be fooled into thinking correlations exist when they do not—i. A high specificity is ultimately the most important trait when investigating long-term human-environment interaction because spurious correlations abound in the real world and filtering out unlikely hypotheses is an important part of scientific research. On the other hand, the wide range of true-positive findings implies that we might miss important correlations because of chronological uncertainty, especially when the climate data are very noisy or the underlying correlation is weak.

This is clearly a problem that should be addressed with more methodological work, but for now the PEWMA method appears to be a good tool for testing hypotheses involving correlations between palaeoenvironmental records and archaeological count data. The third finding—that increasing the number of radiocarbon dates above five had no effect on the simulation results—is counterintuitive, though, and requires further thought. We initially expected that including more dates would markedly improve the true-positive rate and decrease the false positive-rate. That did not happen.

One possible explanation for the counterintuitive relationship between dates and true-positive rates is that chronological uncertainty is not relevant at all because using more dates seemed to have no impact on the results. This possibility, however, can be dismissed by looking at the results of a single bootstrap iteration. Recall that the simulation was broken down into experiments. Each experiment involved a combination of simulation parameters that was constant throughout a given experiment. Within each experiment, pairs of synthetic time-series were analyzed using the PEWMA algorithm—the top-level pairs.

Each top-level pair was subjected to a chronological bootstrap, which resulted in sub-pairs of time-series. Each sub-pair only differed from the others because different chronological anchors—i. So, if chronological uncertainty was irrelevant, we would expect the PEWMA analysis results to have been identical between sub-pairs.

What we saw instead was that each top-level result was a fraction ranging from zero to one, indicating the percentage of the sub-pairs for which the PEWMA method was able to identify the underlying correlation. Therefore, we can be sure that chronological uncertainty had an effect, which means that another explanation is required. A more likely explanation, we think, is that chronological uncertainty has an effect, but it is not as important as the other variables, namely the signal-to-noise ratio and the strength of the underlying correlation. So, large differences in the signal-to-noise ratio and the strength of the underlying correlation will mask the effect of chronological uncertainty to some degree.

Consequently, had we included chronological uncertainty in the archaeological time-series as well as the palaeoenvironmental time-series, we might have seen a greater effect. To some extent, therefore, these results should be considered relatively liberal, since archaeological time-series generally do contain chronological uncertainty.

In a similar vein, had we used an older portion of the calibration curve or wider radiocarbon dating errors for the individual dates, we would expect the utility of the model to decrease.

Still, since the effect we see in the simulation results is small, similar amounts of chronological uncertainty in the archaeological time-series, or small differences in other chronological uncertainties, should only slightly decrease the true-positive rate of the PEWMA method. These findings have implications for our previous research on climate change and Classic Maya conflict [ 18 ]. As we explained earlier, the present simulation study compliments our earlier use of the PEWMA method for testing the hypothesis that climate change drove Classic Maya conflict.

As part of our earlier research we performed sensitivity tests of the PEWMA method to account for various sources of bias. These tests indicated that our primary finding, that increases in temperature corresponded to increases in conflict at the centennial scale, was largely unaffected by temporal bias.

### Navigation menu

The present simulation looked specifically, and more completely, at the effect of chronological uncertainty in the palaeoenvironmental time-series by performing bootstraps to evaluate a very large number of what-if scenarios. The results suggest that the PEWMA method is robust to chronological uncertainty—in fact, chronological uncertainty appears to be the least important of the parameters we investigated. In addition, the portion of the calibration curve we used in the simulation is much older than the Classic Maya period, meaning it has greater chronological uncertainty associated with it.

Even so, the simulation results suggest that false positive findings are rare. Importantly, the false positive rate would decrease for time-series spanning more recent periods because the chronological uncertainty in the calibration curve is lower over more recent periods as well. Thus, we can be more confident that our findings in the Classic Maya case study were not the result of chronological uncertainty.

To appreciate the implications of our simulation results more generally, we can think in terms of conducting blind analyses—i. Our simulation suggests that having at least five to 10 radiocarbon dates per years for a given palaeoenvironmental series is sufficient as long as those dates are spread fairly evenly throughout the series. Spending resources on more dates would likely make little difference in the results. This means, for instance, that most of the palaeoenvironmental time-series that are readily available online have sufficient numbers of radiocarbon dates to create reliable PEWMA models.

The largest, and most popular, online source for palaeoenvironmental time-series is the NOAA website www. Perusal of their catalogue revealed that many of the time-series they curate come with more than five radiocarbon dates. Consequently, our hypothetical analysis could involve the existing palaeoenvironmental data, and if we need to gather a new dataset our chronometric costs would be low.

We could also be confident that our PEWMA analysis would be able to identify an important relationship if it existed, at least much of the time. Correlations with coefficients of 0. Thus, failing to find a relationship could suggest that there was no important relationship to find. If we hypothesized that rainfall variation, for instance, was strongly correlated to the rise and fall of Classic Maya socio-political complexity, then the PEWMA method should be able to identify such a relationship given a proxy time-series for past rainfall and one for socio-political complexity.

If it failed to identify a relationship, one possible reason is that the correlation is quite low, at least according to our simulation results. Failing to find such a correlation, then, might simply indicate that the underlying relationship is not very important, falsifying the hypothesis that a strong relationship existed.

A simple way to overcome this problem would be to test the hypothesis with additional time-series since that would increase the chances of finding a true-positive correlation.

- UNCERTAINTY IN RADIOCARBON DATING: A NUMERICAL APPROACH | Jamal Munshi - ovahiryripen.tk!
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- How reliable is radiocarbon dating?.

Therefore, with some replication we could be fairly confident in our findings. It is important to keep in mind, though, that our simulations also imply that one in ten positive results might be spurious. There are at least two obvious ways to control for false positive findings. One is to use a more stringent test for statistical significance. Since the PEWMA method we used relies on comparing AICs to determine when a significant relationship has been identified, we could change the baseline for significance from identifying AICs that are strictly lower than a benchmark AIC to a baseline that required AICs to be lower by some predetermined amount, giving a confidence buffer of sorts.

This is what we did in our previous analysis on climate change and Classic Maya conflict [ 18 ], and we strongly recommend it in general—though the specific size of the buffer is arbitrary and should be considered carefully for any specific case. The other way to control for false positives would be to conduct replication studies.

For the hypothetical blind analysis we would have to gather multiple archaeological and palaeoenvironmental time-series containing observations of the same underlying phenomena—e. Then, we would re-run the PEWMA analysis and make a decision about our hypothesis on the basis of multiple results taken together, instead of relying on a single comparison. Overall, though, a false positive error rate of 1 in 10 seems acceptable for archaeological research. Therefore, while we ought to make attempts to control for the false positive findings, our simulation results suggest that the PEWMA method is adequate for archaeological purposes.

Overall, our results indicate that the PEWMA method is a promising time-series analysis tool for archaeological and palaeoenvironmental research. The method is suitable for analysing any archaeological count time-series, which potentially includes a wide range of archaeological proxies for past human behaviour, and it performs well even with relatively few radiocarbon dates—only five dates for a time-series years long. Therefore, we can make use of many of the published palaeoenvironmental time-series readily available online and maintain low chronometric costs when gathering new data.

The method can also reliably find moderate to strong correlations between archaeological and palaeoenvironmental time-series when the latter have a strong signal. It should also be noted that leads and lags in a putative human-environment relationship could be tested for in the usual way—i. Thus, we think that the PEWMA method has the potential to contribute substantially to research on past human-environment interaction. There is one very important caveat to keep in mind, which is that the results yielded by applications of the PEWMA method to archaeological time-series are assumption dependent.

Like most statistical techniques, the PEWMA model was created with a specific class of problems in mind and therefore makes certain assumptions about the data. While it appears to be fairly robust to chronological uncertainty, it is best suited to cases where the count-based archaeological data represent a past process that 1 contained autocorrelation; 2 had temporal persistence that can be characterized adequately by exponential decay—e. The last of these traits is particularly important because the PEWMA model assumes a given process was the product of its past states, which includes the previous impacts of any relevant covariates.

So, the effect of covariates persists through time. If, in contrast, a process is suspected to have had covariates with only instantaneous impacts at any given time, then a PEWMA model may not be appropriate. It is, therefore, important to be aware of what one is attempting to model before using the PEWMA method. It would be wise to use the diagnostics outlined in Brandt et al. There are at least three avenues to explore in future research. One involves looking at the effect of calibrated radiocarbon date uncertainty on the dependent—i. We chose to focus on chronological uncertainty in the palaeoenvironmental data in order to limit the sources of error in the simulation and see the effects of chronological uncertainty as clearly as possible.

However, most archaeological time-series will likely contain chronological uncertainty, usually from radiocarbon dating. While we suspect the effect of additional radiocarbon dating uncertainty in the response time-series to be small—since the overall effect of chronological uncertainty appears to be small—it would still be prudent to investigate it further. Future research should involve simulations that look at how the PEWMA method performs when both the response and predictor time-series are dated with radiocarbon.

The second avenue for future research involves estimating the magnitude of an underlying correlation in the presence of chronological uncertainty. Our experiment involved determining whether we could identify an underlying correlation. An obvious parameter to explore, therefore, was the strength of that correlation, which we varied between experiments. The bootstrap simulations resulted in a range of estimates of the magnitude of correlations between the synthetic archaeological and palaeoenvironmental series. Clearly, it would be useful to use the bootstrap estimates to produce a single estimate for the underlying magnitude.

That magnitude would indicate how important a given covariate was relative to other covariates, and it would also allow us to estimate effect sizes—i. However, combining the magnitude estimates from the chronological bootstrap is not straightforward and would have been an ad hoc exercise. In the future, we need to determine how best to combine the estimates while ensuring that the confidence intervals are calculated correctly.

This research will require statistical development and further simulation work. In the study reported here, we effectively used an annual resolution for the time-series, but often archaeological and palaeoenvironmental data have different resolutions. Many modern palaeoenvironmental records boast annual resolutions, for example, while most archaeological time-series will have much coarser resolutions.

Consequently, we have to change the resolution of one or both time-series in order to perform analyses. Future research, therefore, should explore the effect of changing the resolutions of the independent and dependent time-series to match each other. Exploring these two potential research avenues would help us to determine the limits of the PEWMA method, a method with considerable potential to deepen our insights into past human-environment interaction.

Time-series analysis has considerable potential to improve our understanding of past human-environment interaction. However, there is reason to think that its application could be undermined by the widespread reliance on calibrated radiocarbon dates for age-depth models. Calibrated radiocarbon dates have highly irregular uncertainties, as we mentioned earlier. These highly irregular uncertainties potentially pose a significant problem because they undermine the assumptions of standard statistical methods.

With this in mind, we conducted a large simulation study in which we explored the effect of calibrated radiocarbon date uncertainty on a potentially useful Poisson regression-based method for time-series regression, called PEWMA. To test the effect of calibrated radiocarbon date error on the PEWMA method, we simulated thousands of archaeological and palaeoenvironmental time-series with known correlations and then analysed them with the PEWMA algorithm. Our simulation experiments yielded three important findings. One is that the PEWMA method was able to identify true underlying correlations between the synthetic time-series much of the time.

Decreasing the noise levels and increasing the correlation coefficients to 0. While it is not surprising that stronger correlations in less-noisy data were easier to identify, it is important to be aware that the method might miss low correlation relationships. This is surprising because we were expecting the highly irregular chronological errors of radiocarbon dates to warp the time-series in ways that could cause many spurious correlations and therefore a high false positive rate.

The third, and perhaps most surprising finding, was that varying the number of radiocarbon dates used to date the time-series had no noticeable effect. The true-positive rates were largely consistent whether five, 10, or 15 radiocarbon dates were used. This was surprising because it seems like adding more dates should reduce chronological uncertainty by increasing the number of chronological anchors for the age-depth models. Thus, we expected that more dates would improve our ability to find underlying correlations. That increasing the number of dates above five had no substantial impact on the true- or false-positive rates indicates that the PEWMA method is fairly robust to chronological uncertainty.

Taken together, our findings indicate that the PEWMA method is a useful quantitative tool for testing hypotheses about past human-environment dynamics. It can be used to determine whether an underlying correlation exists between a calendrically-dated archaeological time-series and a radiocarbon-dated palaeoenvironmental time-series. Crucially, it has a low false-positive rate, a moderate-to-high true-positive rate, and it appears to be fairly robust to chronological uncertainty.

Methods with these traits are essential for analyzing archaeological and palaeoenvironmental time-series, which is a vital part of understanding past human-environment interaction. R script with top-level simulation function—called from S5 File. We thank our anonymous reviewers and academic editor for their advice.

The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. National Center for Biotechnology Information , U. Published online Jan Author information Article notes Copyright and License information Disclaimer. The authors have declared that no competing interests exist. Received Sep 25; Accepted Dec This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

R script for building age-depth models. R script for calibrating radiocarbon dates. R script containing low-level simulation functions. R script for running the simulation in parallel. The equation for the radioactive decay of 14 C is: During its life, a plant or animal is in equilibrium with its surroundings by exchanging carbon either with the atmosphere, or through its diet.

It will therefore have the same proportion of 14 C as the atmosphere, or in the case of marine animals or plants, with the ocean. Once it dies, it ceases to acquire 14 C , but the 14 C within its biological material at that time will continue to decay, and so the ratio of 14 C to 12 C in its remains will gradually decrease.

The equation governing the decay of a radioactive isotope is: Measurement of N , the number of 14 C atoms currently in the sample, allows the calculation of t , the age of the sample, using the equation above. The above calculations make several assumptions, such as that the level of 14 C in the atmosphere has remained constant over time. The calculations involve several steps and include an intermediate value called the "radiocarbon age", which is the age in "radiocarbon years" of the sample: Calculating radiocarbon ages also requires the value of the half-life for 14 C.

Radiocarbon ages are still calculated using this half-life, and are known as "Conventional Radiocarbon Age". Since the calibration curve IntCal also reports past atmospheric 14 C concentration using this conventional age, any conventional ages calibrated against the IntCal curve will produce a correct calibrated age. When a date is quoted, the reader should be aware that if it is an uncalibrated date a term used for dates given in radiocarbon years it may differ substantially from the best estimate of the actual calendar date, both because it uses the wrong value for the half-life of 14 C , and because no correction calibration has been applied for the historical variation of 14 C in the atmosphere over time.

Carbon is distributed throughout the atmosphere, the biosphere, and the oceans; these are referred to collectively as the carbon exchange reservoir, [32] and each component is also referred to individually as a carbon exchange reservoir. The different elements of the carbon exchange reservoir vary in how much carbon they store, and in how long it takes for the 14 C generated by cosmic rays to fully mix with them.

This affects the ratio of 14 C to 12 C in the different reservoirs, and hence the radiocarbon ages of samples that originated in each reservoir. There are several other possible sources of error that need to be considered. The errors are of four general types:. To verify the accuracy of the method, several artefacts that were datable by other techniques were tested; the results of the testing were in reasonable agreement with the true ages of the objects.

Over time, however, discrepancies began to appear between the known chronology for the oldest Egyptian dynasties and the radiocarbon dates of Egyptian artefacts. The question was resolved by the study of tree rings: Coal and oil began to be burned in large quantities during the 19th century. Dating an object from the early 20th century hence gives an apparent date older than the true date. For the same reason, 14 C concentrations in the neighbourhood of large cities are lower than the atmospheric average. This fossil fuel effect also known as the Suess effect, after Hans Suess, who first reported it in would only amount to a reduction of 0.

A much larger effect comes from above-ground nuclear testing, which released large numbers of neutrons and created 14 C.

## Radiocarbon dating

From about until , when atmospheric nuclear testing was banned, it is estimated that several tonnes of 14 C were created. The level has since dropped, as this bomb pulse or "bomb carbon" as it is sometimes called percolates into the rest of the reservoir. Photosynthesis is the primary process by which carbon moves from the atmosphere into living things. In photosynthetic pathways 12 C is absorbed slightly more easily than 13 C , which in turn is more easily absorbed than 14 C.

This effect is known as isotopic fractionation. At higher temperatures, CO 2 has poor solubility in water, which means there is less CO 2 available for the photosynthetic reactions. The enrichment of bone 13 C also implies that excreted material is depleted in 13 C relative to the diet. The carbon exchange between atmospheric CO 2 and carbonate at the ocean surface is also subject to fractionation, with 14 C in the atmosphere more likely than 12 C to dissolve in the ocean.

This increase in 14 C concentration almost exactly cancels out the decrease caused by the upwelling of water containing old, and hence 14 C depleted, carbon from the deep ocean, so that direct measurements of 14 C radiation are similar to measurements for the rest of the biosphere.

## Radiocarbon dating - Wikipedia

Correcting for isotopic fractionation, as is done for all radiocarbon dates to allow comparison between results from different parts of the biosphere, gives an apparent age of about years for ocean surface water. The CO 2 in the atmosphere transfers to the ocean by dissolving in the surface water as carbonate and bicarbonate ions; at the same time the carbonate ions in the water are returning to the air as CO 2.

The deepest parts of the ocean mix very slowly with the surface waters, and the mixing is uneven. The main mechanism that brings deep water to the surface is upwelling, which is more common in regions closer to the equator. Upwelling is also influenced by factors such as the topography of the local ocean bottom and coastlines, the climate, and wind patterns. Overall, the mixing of deep and surface waters takes far longer than the mixing of atmospheric CO 2 with the surface waters, and as a result water from some deep ocean areas has an apparent radiocarbon age of several thousand years.

### Associated Data

Upwelling mixes this "old" water with the surface water, giving the surface water an apparent age of about several hundred years after correcting for fractionation. The northern and southern hemispheres have atmospheric circulation systems that are sufficiently independent of each other that there is a noticeable time lag in mixing between the two. Since the surface ocean is depleted in 14 C because of the marine effect, 14 C is removed from the southern atmosphere more quickly than in the north.

For example, rivers that pass over limestone , which is mostly composed of calcium carbonate , will acquire carbonate ions. Similarly, groundwater can contain carbon derived from the rocks through which it has passed. Volcanic eruptions eject large amounts of carbon into the air. Dormant volcanoes can also emit aged carbon. Any addition of carbon to a sample of a different age will cause the measured date to be inaccurate. Contamination with modern carbon causes a sample to appear to be younger than it really is: Samples for dating need to be converted into a form suitable for measuring the 14 C content; this can mean conversion to gaseous, liquid, or solid form, depending on the measurement technique to be used.

Before this can be done, the sample must be treated to remove any contamination and any unwanted constituents. Particularly for older samples, it may be useful to enrich the amount of 14 C in the sample before testing. This can be done with a thermal diffusion column.

Once contamination has been removed, samples must be converted to a form suitable for the measuring technology to be used. For accelerator mass spectrometry , solid graphite targets are the most common, although gaseous CO 2 can also be used. The quantity of material needed for testing depends on the sample type and the technology being used.

There are two types of testing technology: For beta counters, a sample weighing at least 10 grams 0. For decades after Libby performed the first radiocarbon dating experiments, the only way to measure the 14 C in a sample was to detect the radioactive decay of individual carbon atoms. Libby's first detector was a Geiger counter of his own design. He converted the carbon in his sample to lamp black soot and coated the inner surface of a cylinder with it. This cylinder was inserted into the counter in such a way that the counting wire was inside the sample cylinder, in order that there should be no material between the sample and the wire.

Libby's method was soon superseded by gas proportional counters , which were less affected by bomb carbon the additional 14 C created by nuclear weapons testing. These counters record bursts of ionization caused by the beta particles emitted by the decaying 14 C atoms; the bursts are proportional to the energy of the particle, so other sources of ionization, such as background radiation, can be identified and ignored. The counters are surrounded by lead or steel shielding, to eliminate background radiation and to reduce the incidence of cosmic rays. In addition, anticoincidence detectors are used; these record events outside the counter, and any event recorded simultaneously both inside and outside the counter is regarded as an extraneous event and ignored.

The other common technology used for measuring 14 C activity is liquid scintillation counting, which was invented in , but which had to wait until the early s, when efficient methods of benzene synthesis were developed, to become competitive with gas counting; after liquid counters became the more common technology choice for newly constructed dating laboratories. The counters work by detecting flashes of light caused by the beta particles emitted by 14 C as they interact with a fluorescing agent added to the benzene.

Like gas counters, liquid scintillation counters require shielding and anticoincidence counters. For both the gas proportional counter and liquid scintillation counter, what is measured is the number of beta particles detected in a given time period. This provides a value for the background radiation, which must be subtracted from the measured activity of the sample being dated to get the activity attributable solely to that sample's 14 C. In addition, a sample with a standard activity is measured, to provide a baseline for comparison.

The ions are accelerated and passed through a stripper, which removes several electrons so that the ions emerge with a positive charge. A particle detector then records the number of ions detected in the 14 C stream, but since the volume of 12 C and 13 C , needed for calibration is too great for individual ion detection, counts are determined by measuring the electric current created in a Faraday cup.

Any 14 C signal from the machine background blank is likely to be caused either by beams of ions that have not followed the expected path inside the detector, or by carbon hydrides such as 12 CH 2 or 13 CH. A 14 C signal from the process blank measures the amount of contamination introduced during the preparation of the sample.

These measurements are used in the subsequent calculation of the age of the sample. The calculations to be performed on the measurements taken depend on the technology used, since beta counters measure the sample's radioactivity whereas AMS determines the ratio of the three different carbon isotopes in the sample.

To determine the age of a sample whose activity has been measured by beta counting, the ratio of its activity to the activity of the standard must be found. To determine this, a blank sample of old, or dead, carbon is measured, and a sample of known activity is measured.

The additional samples allow errors such as background radiation and systematic errors in the laboratory setup to be detected and corrected for. In fact it would be difficult to describe this distribution as a mathematical equation. The slices we used are shown in Figure 3 and the computed probabilities and cumulative probabilities implied by the slices are listed in Table 1. Calibration of LB3 age distribution Figure 3: Numerical probability distribution of dates for LB3 4. A screenshot of the worksheet is shown in Figure 4.

It shows that the sample mean age us The ten-year slices of the histogram of dates are shown in Figure 6. A numerical probability distribution of dates implied by the areas of the slices is tabulated in Table 2 which serves as our tool for describing the MI1 sample. The data and computational details for these results are included in the data archive for this paper Munshi, Radiocarbon paper data archive, Calibration of MI1 age distribution Figure 6: Numerical probability distribution of dates for MI1 4.

The combination of the age data are carried out in the spreadsheet called Late Iron-I in the data archive for this paper Munshi, Radiocarbon paper data archive, The screenshot in Figure 7 shows a sample mean age of The calibration of these values is shown in Figure 8 and the slicing of the histogram of dates for image processing appears in Figure 9. Table 3 summarizes the results of the image processor in terms of a tabulated probability distribution of dates. Calibration of LI1 age distribution Figure 9: Sample probability distribution of dates for LI-I 4. A bootstrap resampling procedure is used to address the additional uncertainty imposed by calibration.

The sample probability distributions shown in Tables 1, 2, and 3 are resampled slice by slice with frequencies corresponding to the size of the slices. A screenshot of the relevant Excel worksheet for LB3 is shown in Figure The spreadsheet itself is included in the data archive for this paper Munshi, Radiocarbon paper data archive, These numbers are used in conjunction with the cumulative probability values in column B to pick random numbers from the distribution of dates A large number of dates are generated and placed in column D.

The sample means are recorded in column E. Columns F and G show from which rows of column D the samples are taken. For example, the first sample is taken from rows 2 through 17 of column D and the second sample is taken from rows 18 through 33 of column D and so on. The same procedure is used to resample the MI1 and LI1 distributions with the important differences being the probability distribution in column B and the sample size The cumulative probability distributions of the samples and the sampling distributions are compared in Figures 11 through The symmetry, normality, and greater precision of the sampling distributions are contrasted with the skewed, irregular, and wider overlapping distributions of the samples.

The results are summarized in Table 4. The values for the ranges and the numerical confidence intervals in Table 4 are derived directly from probability distributions shown in Figures 11 through 16 without any distributional assumption. The critical values of the t-statistic used in these computations are noted in the table. All of the intervals derived from the bootstrapped sampling distributions show a clear distinction between the archaeological ages being compared.

We find that the radiocarbon dating procedure is sufficiently precise in the date range under consideration to clearly distinguish between the three archaeological times. The discrimination power of radiocarbon dating among adjacent archaeological ages thus established offers greater confidence in the use of this tool for relating the relative chronology of stratigraphy to absolute time in calendar years.

This relationship is the essential link between archaeology and history. By way of contrast one notes the overlapping sample distributions highlighted in red in Table 4. Also by comparing the intervals and their midpoints for the samples it is easy to see that the sample distributions are skewed and may not be assumed to be symmetrical or normal. The distributions of sample means appear to be much closer to normal by virtue of the Central Limit Theorem The data are presented only to demonstrate the use of numerical methods for the analysis of radiocarbon data and not necessarily to draw any archaeological conclusions from them.

The results indicate that radiocarbon dating is sufficiently precise to serve as a dating tool for archaeology Summary of results 5. Distributional and simplifying assumptions as well as assumptions of convenience are unnecessary in the numerical approach. The calculations are not only more accurate and precise but they are also easier to use and easier to understand than analytical equations. Access to Microsoft Excel or Matlab is all that is necessary. The Microsoft Excel spreadsheets used in this demonstration are available for download in the data archive for this paper Munshi, Radiocarbon paper data archive, The data analysis provided here is not meant to draw any archaeological or historical conclusions but rather to show that computational statistics may be applied to radiocarbon data.

The results appear to indicate that the use of a bootstrap sampling distribution demonstrated here can increase the precision of radiocarbon dating in archaeology. Imrpoved radiocarbon age estimation using the bootstrap. Retrieved , from Harpers: Correlation with Juvenility Index. Annals of Statistics, 7: Radiocarbon dating the iron age in the Levant. Retrieved , from antiquity.