Origins 10(1):58 (1983).
Re: Hayward & Casebolt: Genesis 5 & 11 Statistical Study (ORIGINS
9:7581)
Related page —  REACTION 
This interesting study has hardly said the last word. In fact,
probably the best explanation for last digit nonrandomness has been overlooked (or, at
least, thoroughly obscured under the heading of "digit preferences").
That explanation is rounding, to a nearby multiple either of 10 or of
5. What could be more likely, when nearly half the last digits are 0 (19 of 40, per Tables
5 & 6) and another 20% are 5 (8 of 40)? Furthermore, the total absence of 1 and 6 as
last digits hints at a rounding down to 0 and 5, respectively.
If the numbers in Table 4 had occurred in a modern document (as, for
example, figures on an income tax return), rounding would doubtless have been the first
explanation to come to mind for last digit nonrandomness. But critical scholarship is ever
pushing us to view the early chapters of Genesis as irrational and arbitrary and to prefer
explanations which match that presupposition.
In reality, rounding was even more likely in Bible times than now.
"The people of Bible times, as in many parts of the East today, thought more in terms
of round numbers than we do, and did not demand mathematical exactitude."^{1}
It is extremely unlikely that Moses would even attempt, like the modern demographer,
"to handle fractions of years consistently" by using "the age of the
individual at his most recent birthday." (p. 77)
If rounding alone is sufficient to account for the nonrandom
distribution of last digits, then nonrandomness of last digits would not imply
nonrandomness of the absolute values of the ages themselves. In that case,
Hayward and Casebolt have not actually demonstrated that the "age data
form a significantly nonrandom distribution." (p. 75)
In more precise terms, the study of Andrews University age data showed
that "If age data is nonrounded and random, then last digits are
random" ([p Ù q ] Þ r). This
is logically equivalent to its contrapositive, "If last digits are nonrandom, then
age data is either rounded or nonrandom" (~ r = [~p Ú ~q]). Thus, if rounding has occurred, nonrandomness of age data is
not necessarily implied.
Rounding to a multiple of 10 or of 5 is obviously tied to the base 10
number system. Thus one way of testing whether last digit nonrandomness could be due
solely to (base 10) rounding is to change all data to base 11 and see whether last digits
then become random. They do — See Tables A and B. (To simplify matters, only
pregenerative years are discussed here, since only they are really pertinent to
chronology. Results are even more favorable — from my point of view — for
postgenerative years.)
TABLE A: Pregenerative years of 20 patriarchs in Gen. 5 & 11 (MT)
Base 10 Base 11 Base 10 Base 11 Base 10 Base 11 130 109^{2} 187 160 30 28 105 96 182 156 32 2T 90 82 500 415 30 28 70 64 100 91 29 27 65 5T^{3} 35 32 70 64 162 138 30 28 100 91 65 5T 34 31
TABLE B: Frequency distribution of base 11 last digits from Table A
Last
digitObserved
frequencyExpected frequency = 20/11. 0 1 1 3 c^{2} = 7.5, d.f. = 10, P » 0.05:
Observed and expected frequencies are not significantly different. More exactly, P = 0.67±, that is, by chance alone, distributions in which observed values differ even more widely from the expected will occur in 2 cases out of 3.
2 2 3 0 4 2 5 1 6 2 7 1 8 4 9 1 T 3
A more direct, but theoretically more suspect, way of testing whether the nonrandomness of last digits can be explained solely by rounding is simply to "unround" the numbers by likely guessing. Two possible prerounding distributions are found in columns (a) and (b) of Table C.
TABLE C. Frequency distribution of (base 10) last digits from Table A after unrounding by conjecture.
Last digit Observed
frequency
after rounding(a) Likely "observed"
frequency before
rounding(b) Minimally unrounded
"observed" frequency
before rounding0 10 4 5 1 0 3 3 2 3 3 3 ü 3 0 0 0 ï 4 1 2 1 ï 5 4 2 4 ý (unchanged) 6 0 1 0 ï 7 1 1 1 ï 8 0 1 0 þ 9 1 3 3 (Expected frequency = 2; d.f. 9.)
Column (a) assumes that one 6 and one 4 had been rounded to the
nearest 5, and that three 1's, two 9's, and one 8 had been rounded to the nearest multiple
of 10. c^{2} = 7.0, P » 0.05: Observed and expected
frequencies are not significantly different. More exactly, P = 0.64±, that is,
by chance alone about 2 distributions out of 3 will have observed values that differ even
more widely from the expected.
Column (b) represents minimal unrounding to avoid a significant
difference. It assumes only that three 1's and two 9's had been rounded to the nearest
multiple of 10. c^{2} = 15.0, P > 0.05: Observed and
expected frequencies are not significantly different.
What implications would rounding have for the traditional practice of
adding the pregenerative age data to calculate the time since creation? If rounding up
(e.g., 9 to 10) and rounding down (e.g., 1 to 0) occurred on a roughly equal scale, it
makes practically no difference. The rounding per columns (a) or (b) of Table C makes
no more than a year's difference in the time from creation to the birth of Isaac. In the
light of the much larger differences in the time periods given by the various ancient
texts (LXX, MT, etc.), even a few years difference caused by rounding would be hardly
worth mentioning.
I conclude that:
Mitchell P. Nicholaides
Camden, South Carolina
P.S. Since Seth was not Adam's "firstborn son," the second sentence of Hayward and Casebolt's article contains its own sort of "rounding."
————————
^{1} SEVENTHDAY ADVENTIST BIBLE DICTIONARY, revised edition (1979), p.
208.
^{2} ( = 1×11^{2} + 0×11 + 9×1 )
^{3} ( = 5×11 + 10×1)
J. L. Hayward and D. E. Casebolt reply:
We commend Mitchell Nicholaides for his careful statistical analyses
and lucidly stated opinions. We heartily agree that our paper "has hardly said the
last word" on the topic. As stated in our introduction, "We do not attempt to
completely resolve the genealogy/chronology problem." However, for several reasons we
disagree with Nicholaides' a priori assertion that "Rounding is a totally sufficient
explanation and the most likely explanation for the observed nonrandomness" in the
age values, and his implicit conclusion that a meaningful chronology can be constructed
from the genealogical data.
First, if rounding of real age values occurred as Nicholaides posits,
it did not occur consistently. While 27 of the 40 independent age values are multiples of
5 or 10, 13 are not. Nicholaides' analysis implies that many digits one number removed
from 5 and 10 are rounded. For example, in constructing his Column (a) of Table C he
assumes that one 6, one 4, two 9s and even one 8 have been rounded to either 5 or 10. Yet,
to choose only two examples (without considering the numbers ending in 2s or 7s) we find
that Eber's and Nahor's pregenerative ages of 34 and 29, respectively, were not rounded.
Second, deviation from an expected distribution of values in the age
data is particularly apparent when multiples of 100 are considered. Again, assuming a
random distribution of frequencies, the expected frequency of multiples of 100 in a series
of 40 numbers is less than 1 (40/100 = 0.4). However, examination of our Table 3 reveals
that eight (20%) of the 40 independent age values are multiples of 100. (A c^{2} comparison between the observed and expected values is
not too meaningful due to an expected value of less than 1. However, the difference
between observed and expected values is so large that statistical testing is unnecessary
to evaluate these data.) Those who defend the rounding hypothesis must consider the
possibility that not only is a large percentage of the numbers rounded to the nearest 10,
but an unnaturally high proportion seem rounded to the nearest 100 as well.
Third, several factors suggest that the writer of the Genesis
genealogies was more concerned with style than with chronology. For example, Enoch, the seventh
from Adam "walked with God" and was therefore "taken" by God (Genesis
5:2224). (Also, when he was "taken" he was 365 years old, corresponding to the
365 days of a complete year.) Noah, the tenth from Adam "walked with
God" (Genesis 6:9), "won the Lord's favour" (Genesis 6:8), and during the
Flood "only Noah and his company survived" (Genesis 7:23). Abraham, the tenth
postFlood patriarch, was made the father of "a great nation" (Genesis 12:2)
with the prediction that "All the families on earth will pray to be blessed as you
are blessed" (Genesis 12:3). (All the above quotes are from the NEB.) One searches in
vain for mention of comparable benefits accruing to patriarchs of numerically
nonsignificant generations. (Of course, other numbers like 3 and 12 held special
significance to Hebrew writers, though the patriarchs of corresponding generations did not
receive extraordinary blessings. But we think it more than coincidental that each of the
patriarchs receiving exceptional recognition by God were members of numerically
significant generations.)
Also, the genealogies are neatly organized into two groups of ten
patriarchs each, the first group containing antediluvians and the second group containing
postdiluvians to the time of Abraham, the father of the Hebrew people. Interestingly, the
last patriarch to bear children in each group of ten bore three children, presumably as
triplets: Noah fathered Shem, Ham, and Japheth during his 500th year (Genesis 5:32), and
Terah begat Abraham, Nahor, and Haran during his 70th year (Genesis 11:26).
Finally, the average pregenerative age of the first nine patriarchs was
116 years, while Noah's pregenerative age was 500 years, over four times the average. If
Noah had had sons at the average preFlood age, and if these sons had fathered children at
the same average age, and if these children produced descendants of their own at the same
average age, and these descendants the same, there would have been an additional three
generations of patriarchs alive during the Flood besides Noah and his sons. But, of
course, this would have altered the balance the writer achieved by placing ten patriarchs
before and ten after the Flood.
The Genesis genealogies would suit well a Hebrew writer's intent to
show ancestral continuity between the Creator God and the Hebrew people, albeit in a
somewhat stylistic fashion. But to force Genesis 5 and 11 to assume the role of
"chronogenealogies" demands more of scripture than we believe was intended by
the inspired writers or is warranted by the evidence.
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