# Statistical and Proactive Analysis of an Inter-Laboratory Comparison: The Radiocarbon Dating of the Shroud of Turin

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## Abstract

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## 1. Introduction

^{12}C and

^{13}C and the radioactive

^{14}C, in a ratio close to that of the Earth’s atmosphere. From the moment of death, if no external contamination occurs, the organism becomes a closed system in which the stable isotopes

^{12}C and

^{13}C maintain their concentration, while the amount of

^{14}C decreases at a known rate. By measuring how much

^{14}C an artifact contains, analysts determine the “radiocarbon age” of the sample. The true “calendar age” is obtained by combining the radiocarbon age with a calibration curve which takes into account the variable concentration of

^{14}C due to fluctuations of, e.g., solar wind and the Earth’s magnetic field, as well as to nuclear weapon testing and

^{14}CO

_{2}cycling between atmospheric, oceanic, and terrestrial carbon reservoirs. An overview of the metrological history of

^{14}C dating can be found e.g., in [9].

## 2. Radiocarbon Dating the Shroud: Context, Sampling, and Results

^{14}C dating techniques, namely, the proportional counter and the AMS.

_{5}distribution, that is, the t-statistic with 5 degrees of freedom. This was then made slightly broader to allow for the errors in the calibration curve before using the curve to give the calendar age. The details are on p. 614 of [8]. In this respect, we note that a consolidated and globally agreed approach/document on uncertainty evaluation was not available in 1988. In [8] uncertainty and errors are evaluated using a method no longer implemented by metrologists. In fact, during the 1980s, the approach to measurement uncertainty was based on random and systemic error statements. The Bureau international des Poids et Mesures at the end of the seventies launched a questionnaire on the evaluation of uncertainty. National Metrology Institutes highlighted the need for guidelines to tackle the uncertainty evaluation method discrepancies, merging the approaches and starting work for the first edition of the Guide for expression of Uncertainty in Measurements (GUM) in 1993, see Section 5.

## 3. Robust Statistical Analysis of the Official Data

- (i)
- The smaller one of the two Shroud samples given to the Arizona laboratory was not dated;
- (ii)
- The data published in [8] are heterogeneous and there is a linear spatial gradient of the 12 subsample ages. That is, the age of a piece at the top edge is systematically less than that of the adjacent piece. As a consequence, the subsample dating cannot be considered as repeated measurements of a single unknown quantity. Thus, the basic assumption of radiocarbon dating was not fulfilled.

^{14}C counts of four samples.

#### 3.1. Results

_{5}distribution), which is considered the estimated uncertainty of the individual measurements, as stated by [8].

- (1)
- Unweighted ANOVA: we ignore the estimated uncertainty of the individual measurements;
- (2)
- Weighted ANOVA: we use the scaled standard errors in Table 1 as weights;
- (3)
- Modified ANOVA: If the weights are correct, the observed standard errors should agree with those given by the weights. They do not for Arizona, with the weights giving standard errors that are appreciably smaller than the observed values. We use the observed standard errors detailed in [8] to scale up the individual weights for Arizona.

- No evidence of differences in variances among the three laboratories;
- Evidence of difference in means among the three laboratories.

- No evidence of differences in variances between the three laboratories.
- No evidence of a difference in means between the three laboratories.

- (1)
- The distribution of the t-statistic for the vertical coordinate is not centered around zero (as expected if the vertical coordinate did not play any role), but there is not enough statistical evidence to claim that it is significant. Given that the whole sample is rectangular with the long side horizontal, see the upper panel of Figure 2, we do not have enough information to detect significant variability along the vertical coordinate, which; however, cannot be ruled out.
- (2)
- The t-statistic for the horizontal component is always negative. It is not significant when Arizona is assumed to have dated both samples, but becomes significant if only the larger sample were dated, providing evidence of an anomalous and unexpected relationship, with a negative slope of about 50 years/cm, between age and horizontal position.
- (3)
- The configurations that assume Arizona dated both samples lead to regression models with outliers, which are absent under the assumption that Arizona dated just the larger sample. The latter allocations are those giving a significant slope [29].

#### 3.2. Implications of the Evidence of a Spatial Trend

## 4. Statistical Analysis of the Raw Data

#### 4.1. Results

^{14}C among the dated subsamples is rejected. The evidence is:

- (a)
- The heterogeneity of the raw data;
- (b)
- The consistent ages of the control samples.

## 5. The Correct Sampling Design

^{2}potential experimental units. A set of n units is then chosen for experimentation. The spatial cover is achieved by choosing the units such that there is one in each row and column. The units can be chosen at random, and any seemingly unsatisfactory pattern, such as one that is spatially too regular, rejected. Alternatively, sampling can be only from a set of units which have some desirable spatial property. [39] prefers the latter. Once the units for experimentation have been chosen, the samples need to be assigned to the three laboratories in a suitably randomized way to, e.g., avoid all samples assigned to a given laboratory coming from one end or side of the material, as actually happened with the Arizona laboratory. The design of spatial experiments is given in [40].

## 6. Specific Problems of Textile Dating

^{14}C from that originally present, and the overall

^{14}C count adds the original radiocarbon to the ‘new’ one, thus skewing the dating result. As a consequence, the accuracy in the radiocarbon dating of a textile depends on its age, handling, and exposure to contaminants during its history. This problem was particularly serious in the 1980s, when contamination was a main contributory cause of the unsatisfactory results (three outliers out of 18 radiocarbon ages of three textile samples) obtained in inter-comparison work between four AMS and two gas counter laboratories [14]. The authors indeed ascribed the presence of outliers to inadequate cleaning procedures.

#### 6.1. Clean Samples of Polluted Cloth

- (a)
- (b)
- The presence of spurious cotton detected by microscopy [47] and by analytical spectroscopy in [48] inside threads from the Raes sample (which is adjacent to the radiocarbon sample, see Figure 2) might reveal an attempt to repair and reweave one of the portions historically removed from the Shroud by the Savoy family over the centuries for various reasons, as reported by [49];
- (c)
- According to [30], the thickness of the remnant of the Arizona sample is 0.25 mm, which is considerably less than the average 0.39 mm thickness of the Riserva (minimum 0.34 mm, maximum 0.43 mm) measured by the textile expert [50], thus supporting the hypothesis of the presence of a foreign fabric in the Arizona sample.

## 7. Discussion and Proposal

- (i)
- The anomalous relationship between subsample age and its position revealed by the robust statistical analysis (see Section 3.1) might suggest the presence of contaminants that escaped cleaning pretreatments. These contaminants could have biased the radiocarbon age;
- (ii)
- The use of additives or preservatives during the history of the Shroud is highly probable albeit poorly documented. We know that thymol was left in the wooden reliquary during the whole day of the 21 April 1988 when the Shroud was taken down for the sampling operation [53]. As a consequence, the Shroud was exposed to thymol vapors released by the wood from 1988 to 1998, when the Shroud was moved into an especially designed exposition reliquary.

^{14}C measurement of the Shroud could still result in an inaccurate age determination due to thymol exposure. To get valuable information in advance of any possible new dating, we propose a strategy to determine to what extent the influence of contaminants may skew the radiocarbon age of the Shroud, as detailed below.

#### 7.1. Charred Material

^{14}C/

^{12}C ratio, possibly via isotopic exchange between the cloth and some volatile combustion products, as discussed by [19] and [54]. Since it is likely that any effect would have been proportional to the temperature of the fabric during the fire, a comparison between the radiocarbon ages of charred material and of unaffected Shroud threads should provide valuable information on this issue. Moreover, the burned threads come from several areas of the Shroud and are less porous than intact threads, thus less prone to being contaminated. Then, dating charred threads could give information on the spatial distribution of contaminants attached to the cloth before the 1532 fire.

#### 7.2. Holland Cloth

#### 7.3. Shroud Fibers and Raes Sample

## 8. Conclusions

- (a)
- The single-site Shroud sampling procedure (see Figure 1), which does not protect against the possibility that the sample is not representative of the whole, as discussed in Section 2 and Section 5, as well as chemical and FTIR spectroscopic data for the radiocarbon sample, which showed the area of the radiocarbon test was atypical and not representative of the rest of the Shroud [20,23,24];
- (b)
- The difficult cleaning of textiles whose handling and exposure to contaminant sources during their history is unknown, as discussed in Section 6, and
- (c)
- The results of statistical analyses performed on both official and raw data described in Section 3.1 and Section 4.1. These reveal the reasons for the lack of homogeneity of the Shroud data and identify the systematic spatial gradient of the ages as the source of the heterogeneity in means that was not detected by [8], it appears that the conclusion in [8] “The results provide conclusive evidence that the linen of the Shroud of Turin is medieval” needs to be reconsidered.

^{14}C measurement. In Section 7 we proposed a set of samples to be radiocarbon dated in order to quantify the bias introduced by contaminants not completely removed, or by other sources of heterogeneity such as, for example, mending added to repair the Shroud mentioned in Section 6.1. In detail, we recommended dating the charred yarns of the Shroud, the Holland cloth, the Raes sample, and yarns outside the image from the middle of the Shroud. Taken alone, none of the dating results of these samples could be expected to be the individual source of a reliable estimate of the age of the Shroud. However, the comparison of their ages, in a multivariate analysis, could give selective information on the skewing influence of many of the possible contaminants, as well as their spatial distribution across the cloth.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**(

**Top**): schematic of the Shroud sample to be dated and its initial partition. In the first drawing, the shadowed parts are those trimmed. The second drawing is the part used for the partition. The third drawing shows the retained part, called “Riserva”, on the left and the part to be dated on the right [17]. (

**Bottom**): photo of the bottom left-hand edge of the frontal image of the Shroud framed in 1978. (Credit: 1978 Barrie M. Schwortz Collection, STERA Inc., Florissant, CO, USA). We added the subdivisions of the sample and their relative position. The sample removed by Raes in 1973 and the part retained as Riserva are also shown.

**Table 1.**Estimated radiocarbon BP years of the individual subsamples with scaled standard errors from t

_{5}distribution. Those for Arizona exclude one source of error (see text).

Arizona | RC Dating | 591 | 690 | 606 | 701 | |

Scaled standard error | 30 | 35 | 41 | 33 | ||

Oxford | RC dating | 795 | 730 | 745 | ||

Scaled standard error | 65 | 45 | 55 | |||

Zurich | RC dating | 733 | 722 | 635 | 639 | 679 |

Scaled standard error | 61 | 56 | 57 | 45 | 51 |

**Table 2.**Significance levels of tests of homogeneity of variances and means of the three laboratories for unweighted and weighted analyses.

Unweighted | Original Weights | Modified Weights | |
---|---|---|---|

Variance Homogeneity | 0.787 | 0.354 | 0.700 |

Difference in Means | 0.0400 | 0.0043 | 0.0497 |

Day | Subsample | Years BP |
---|---|---|

6 May | A1D(2) | 606 ± 41 |

A1D(2)’ | 574 ± 45 | |

12 May | A1D(1) | 753 ± 51 |

A1D(1)’ | 632 ± 49 | |

24 May | A1C(1) | 676 ± 59 |

A1C(1)’ | 540 ± 57 | |

2 June | A1C(2) | 701 ± 47 |

A1C(2)’ | 701 ± 47 |

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**MDPI and ACS Style**

Di Lazzaro, P.; Atkinson, A.C.; Iacomussi, P.; Riani, M.; Ricci, M.; Wadhams, P.
Statistical and Proactive Analysis of an Inter-Laboratory Comparison: The Radiocarbon Dating of the Shroud of Turin. *Entropy* **2020**, *22*, 926.
https://doi.org/10.3390/e22090926

**AMA Style**

Di Lazzaro P, Atkinson AC, Iacomussi P, Riani M, Ricci M, Wadhams P.
Statistical and Proactive Analysis of an Inter-Laboratory Comparison: The Radiocarbon Dating of the Shroud of Turin. *Entropy*. 2020; 22(9):926.
https://doi.org/10.3390/e22090926

**Chicago/Turabian Style**

Di Lazzaro, Paolo, Anthony C. Atkinson, Paola Iacomussi, Marco Riani, Marco Ricci, and Peter Wadhams.
2020. "Statistical and Proactive Analysis of an Inter-Laboratory Comparison: The Radiocarbon Dating of the Shroud of Turin" *Entropy* 22, no. 9: 926.
https://doi.org/10.3390/e22090926