# Analysis of Radiocarbon Dating Statistics in Reference to the Voynich Manuscript

The purpose of this article is to identify potential errors in radiocarbon dating with the view to evaluating the Voynich manuscript 14C data. Anyone reading this article should have a basic understanding of statistics.

- Type of sample, i.e. protein vs. carbohydrate vs. carbonate.
- Preparation of sample, i.e. different solvents used for removing surface dirt.
- Errors from delta 13C isotope dilution measurements.
- Accelerator Mass Spectrometry background 14C contamination.
- Counting errors.
- Terrestrial radiocarbon age calibration curve.

The U. of Arizona corrects their measurements for Delta 13C isotope dilution and for 14C background contamination, Donahue et.al. (*1990 Radiocarbon, vol 32 No. 2, p 135-142.*)

Laboratory | Offset | σ1 | σ2 | k | No.samples |
---|---|---|---|---|---|

Belfast Irish Oak 1986 | -6 +/- 1 | 21.5 | 29.4 | 1.37 | 407 |

Belfast Irish Oak 2002, 2004 | 4 +/- 2 | 21.3 | 27.6 | 1.21* | 124 |

Waikato | 10 +/- 2 | 22.0 | 22.9 | 1.04* | 100 |

- The mean of the values, weighted, according to the individual error estimates,
**Mw**and unweighted**Mu**. **n**the number of values.**σi**is the estimated SD for an individual sample measurement.**σ1**is the expected standard deviation based on the counting statistics, the average standard deviation of the individual samples**σi****σ2**is the standard deviation in the 14C age of replicate samples.**σw**an estimate SD for the weighted mean,**Mw**, using the individual sample estimates in the weighting.**σw2 = 1/sumσi-2**for i = 1 to n**σu**an estimated SD for the population of sample measurements where the individual estimates are ignored. It is usually referred to as an estimate of the standard error of the mean and may be used as an approximation for the total error.**σu2 = σ22/n****Chi square**a statistic used to estimate the uncertainties in data without making the additional measurements.

- Failure to detect systematic errors, this affects the accuracy of the results.
- Nonstatistical fluctuations in the instruments or the measurements.
- Carelessness.

Vinland map | Shroud of Turin | Sample 2 | Sample 3 | Sample 4 | |
---|---|---|---|---|---|

14C BP yr σi | 338 ± 116 | 591 ± 30 | 922 ± 48 | 1838 ± 47 | 724 ± 42 |

14C BP yr σi | 406 ± 30 | 690 ± 35 | 986 ± 56 | 2041 ± 43 | 778 ± 88 |

14C BP yr σi | 537 ± 51 | 606 ± 41 | 829 ± 50 | 1960 ± 55 | 764 ± 45 |

14C BP yr σi | 486 ± 26 | 701 ± 33 | 996 ± 38 | 1983 ± 37 | 602 ± 38 |

14C BP yr σi | 574 ± 69 | 894 ± 37 | 2137 ± 46 | 825 ± 44 | |

Range | 236 | 110 | 167 | 299 | 223 |

Chi sq | 9 | 9 | 9 | 22 | 17 |

df | 4 | 3 | 4 | 4 | 4 |

Significance | 5% | 4% | 5% | 1% | 2% |

Mu ± σ2 | 468 ± 96 | 646 ± 57 | 925 ± 69 | 1992 ± 110 | 739 ± 84 |

Mu ± σu | 468 ± 43 | 646 ± 29 | 925 ± 31 | 1992 ± 49 | 739 ± 38 |

Mw ± σ1 | 468 ± 58 | 646 ± 35 | 927 ± 46 | 1995 ± 46 | 721 ± 51 |

Mw ± σw | 468 ± 17 | 646 ± 17 | 927 ± 20 | 1995 ± 20 | 721 ± 20 |

U of A | 468 ± 27 | 646 ± 31 | 925 ± 32 | 1995 ± 46 | 722 ± 43 |

k = σ2/σ1 | 1.6 | 1.7 | 1.5 | 2.4 | 1.6 |

ksher ~ σu/σw | 2.5 | 1.7 | 1.6 | 2.5 | 1.9 |

For the Shroud of Turin and controls the radiocarbon ages were calculated using the procedure of Stuiver and Polach,(*1977 Radiocarbon, Vol.19, No. 3, p. 355-363.*) The errors include the statistical (counting) error, the scatter of results for standards and blanks, and the uncertainty in the delta13C determination. The Oxford Radiocarbon laboratory rounded errors in their measurements of the Shroud of Turin and the controls that were below 40 to 40 BP years.

A chi square was calculated for the five data sets in order to determine whether the errors from the individual measurements adequately represented the total error. Table 2 shows that in all cases a chi square of 5% or less was obtained, making it unlikely that the counting errors represent the total error in the radiocarbon dating of the above five samples.

Table 2 also shows that **σw** is considerably less than σu. and that an error **multiplier factor, k,** should be applied to each data set. This table also shows that **σu**, the estimate of the standard error of the mean, agrees reasonably well with the estimates the U of Arizona reported for the total errors for the Shroud of Turin and the three controls, but not for the Vinland map. It appears that an additional **error multiplier factor** should be applied to the stated error for the Vinland map. The **k** calculated according to the procedure of Reimer et.al. (*2004, Radiocarbon, vol 46 No. 3, p 1034-1036* emphasize the contribution of the largest error measurement to **σ1**, thereby making the corresponding **k** value smaller. The method I use sensibly de-emphasizes the larger errors and in turn makes the **ksher** value larger.

Table 3 shows a comparison of the mean 14C BP years reported by U of Arizona for the four Shroud of Turin samples with those reported by the two other laboratories of equal prestige.

Sample | Shroud of Turin | Sample 2 | Sample 3 | Sample 4 |
---|---|---|---|---|

Arizona | 646 ± 31 | 927 ± 32 | 1,995 ± 46 | 722 ± 43 |

Oxford | 750 ± 30 | 940 ± 30 | 1,980 ± 35 | 755 ± 30 |

Zurich | 676 ± 24 | 941 ± 23 | 1,940 ± 30 | 685 ± 34 |

Chi sq (2df) | 6.4 | 0.1 | 1.3 | 2.4 |

Significance % | 5 | 90 | 50 | 30 |

The U of Arizona and the U of Oxford show a difference of 104 years in their reported mean 14C BP years for the Shroud of Turin. The chi square (2 degrees of freedom) calculated for this data is 6.4 with a level of significance of 5%. The probability that random error alone is responsible for the scatter between the results reported by three labs is less the 5%. This indicates the possibility of the presence of systematic errors in the radiocarbon measurements for the Shroud of Turin, perhaps due to different sample preparation methods.

The need to provide the best possible data for converting radiocarbon ages into calendar ages resulted in an update in the atmospheric decadal tree ring data set in 2004, Reimer et.al. (*Radiocarbon, Vol. 44, No. 3, p. 1029-1056*). The Supplemental Data on which this curve is based may be found at http://www.radiocarbon.org/IntCal04.htm. I have extracted a portion of the data used to construct the terrestrial radiocarbon age calibration curve for the years 1400 – 1500 A.D. from: http://www.radiocarbon.org/IntCal04%20files/intcal04.14c.

Calendar age | BP age | 14C BP age | error yr 14C BP age |
---|---|---|---|

1380 | 570 | 646 | 12.0 |

1385 | 565 | 628 | 12.0 |

1390 | 560 | 604 | 13.0 |

1395 | 555 | 583 | 12.0 |

1400 | 550 | 572.0 | 12.0 |

1405 | 545 | 560.0 | 11.0 |

1410 | 540 | 543.0 | 12.0 |

1415 | 535 | 526.0 | 12.0 |

1420 | 530 | 513.0 | 13.0 |

1425 | 525 | 502.0 | 12.0 |

1430 | 520 | 491.0 | 13.0 |

1435 | 515 | 477.0 | 12.0 |

1440 | 510 | 459.0 | 13.0 |

1445 | 505 | 438.0 | 12.0 |

1450 | 500 | 415.0 | 12.0 |

1455 | 495 | 398.0 | 11.0 |

1460 | 490 | 393.0 | 12.0 |

1465 | 485 | 389.0 | 12.0 |

1470 | 480 | 381.0 | 13.0 |

1475 | 475 | 374.0 | 12.0 |

1480 | 470 | 372.0 | 13.0 |

1485 | 465 | 368.0 | 12.0 |

1490 | 460 | 359.0 | 12.0 |

1495 | 455 | 352.0 | 11.0 |

1500 | 450 | 350.0 | 12.0 |

- Using Table 4, I extrapolated the 14C BP year values of 565 and 473, for the calendar years 1404 and 1438 respectively.
- Total σ = (565 –473)/4 (1SD)
- Total σ = [(sample σ)2 + (curve σ) 2]1/2 (
*Stuiver and Becker 1993, Radiocarbon, Vol.35, No.1, p. 39.*)

[ 242 = (sample σ)2 + 122 ] - Sample σ = 21 BP yrs.
- Mean 14C BP year = 514 +/- 21 (1SD)

If the individual 14C BP year values for the four pages of the Voynich Manuscript are available, it is possible from the variance in their ages to calculate an estimate of the standard error of the mean. This estimate would indicate whether an error multiplier factor should be applied to the 1SD error of 21.

Table 5 shows the raw data used to obtain Table 4. http://www.radiocarbon.org/IntCal04%20files/IntCal04_rawdata.csv.

The raw data was converted to the atmospheric decadal tree ring data set using a random walk model (Buck and Blackwell, *2004, Radiocarbon, Vol.46, No.3, p 1093-1102*)

Cal age | BP age | 14C BP age | 14C BP age error | N | Laboratory |
---|---|---|---|---|---|

1389 | 561 | 637.6 | 18.8 | 946 | Seattle |

1389 | 561 | 627.9 | 18.9 | 947 | Seattle |

1389 | 561 | 651 | 23 | 108 | Belfast 2002 |

1389 | 561 | 623 | 18.7 | 57 | Waikato |

1399 | 551 | 574.7 | 18.6 | 948 | Seattle |

1399 | 551 | 574 | 23 | 107 | Belfast 2002 |

1399 | 551 | 599 | 46.6 | 175 | Belfast 1986 |

1399 | 551 | 588 | 18.7 | 56 | Waikato |

1409 | 541 | 564.5 | 17.3 | 949 | Seattle |

1409 | 541 | 577 | 14.5 | 106 | Belfast 2002 |

1409 | 541 | 556 | 18.7 | 55 | Waikato |

1419 | 531 | 520.3 | 18.1 | 950 | Seattle |

1419 | 531 | 539 | 21.8 | 105 | Belfast 2002 |

1419 | 531 | 512 | 19.2 | 174 | Belfast 1986 |

1419 | 531 | 536 | 18.7 | 54 | Waikato |

1429 | 521 | 499.5 | 17.4 | 951 | Seattle |

1429 | 521 | 511 | 23 | 104 | Belfast 2002 |

1429 | 521 | 499 | 18.7 | 53 | Waikato |

1439 | 511 | 468.2 | 18.6 | 952 | Seattle |

1439 | 511 | 484 | 20.6 | 103 | Belfast 2002 |

1439 | 511 | 500 | 26 | 173 | Belfast 1986 |

1439 | 511 | 488 | 17.7 | 52 | Waikato |

1449 | 501 | 417.7 | 15.3 | 953 | Seattle |

1449 | 501 | 457 | 20.8 | 51 | Belfast 2002 |

1459 | 491 | 388.5 | 18.2 | 954 | Seattle |

1459 | 491 | 389.5 | 17.8 | 955 | Seattle |

1459 | 491 | 415 | 25.4 | 101 | Belfast 2002 |

1459 | 491 | 458 | 23.3 | 172 | Belfast 1986 |

1459 | 491 | 377 | 17.7 | 50 | Waikato |

1469 | 481 | 395 | 18.7 | 956 | Seattle |

1469 | 481 | 397 | 21.8 | 100 | Belfast 2002 |

1469 | 481 | 406 | 13.7 | 171 | Belfast 1986 |

1469 | 481 | 375 | 17.7 | 49 | Waikato |

Table 5 shows that the Seattle raw data age estimates tend to be younger (positive offset) than the other data sets (Reimer et.al. 2004, Radiocarbon, Vol.46 No.3, p.10345). This indicates the presence of a systematic error between data from Seattle and data from Belfast and Waikato. I have not been able to find any information indicating what the offset might be in 14C dating between U of Arizona and the 2004 atmospheric decadal tree ring data set.

It should be remembered that 14C dating measures sample activity not sample age and that the conversion from 14C BP age to calendar age is dependent on the use of the atmospheric decadal tree calibration curve with its own set of limitations with respect to accuracy and precision. I am concerned by the large variation between the decadal 14C data from the three labs for the years from 1419 to 1459 in Table 5, years critical to the dating of the Voynich Manuscript. The three labs made a large number of measurements on samples of Belfast Irish oak of known age and this data was used to produce the 2004 atmospheric decadal tree ring data set.

In conclusion, until a better method becomes available, radiocarbon dating is the best method for determining the approximate age of small samples of organic material. The Oxford Radiocarbon laboratory seems to believe that a realistic estimate of the S.D. of the 14C BP age should not be less than 40 years http://www.shroud.com/nature.htm and additionally they do not accept responsibility for any financial loss as a result of an erroneous report: http://www.shroud.com/vanhelst.htm. Having reviewed the available data and taking into account the variety of possible errors in 14C dating, I have come to my personal conclusion that the animal(s) whose skins were used to make the parchment for the Voynich Manuscript were probably killed some time during the first half of the 15th century.

Error is a normal part of science, no method is immune, results should be subjected to a critical examination and control experiments performed to determine the accuracy of the measurements. Finally it never hurts to review the literature, this should always be the initial step in any endeavor.

The references quoted in this paper can be found on the Radiocarbon web site http://www.radiocarbon.org/.