## Counting on the Census Numbers: Mathematical Approaches to the elep in Numbers

Aaron Adair MIT Department of Physics adairaar@gmail.com

Abstract : Censuses of the Israelites found in Numbers have been mulled over because of  both  their  large  values  and  the  particulars  of  those  values.  Efforts  to  fit  the  large numbers of Israelites escaping Egypt into something historically plausible or suggest a lexical development focus on reinterpreting the standard counter for 'thousands', elep ( אלף ). The most recent attempts use mathematical models, but the results have not been rigorously tested. A statistical evaluation finds the efforts wanting, and the underlying numbers are most likely artificial. The origin of the census values is then proposed, indicating source relationships within the Pentateuch rather than some archaic, pre-CBH features and sources. Mathematical tests also limit what esoteric meanings the census numbers might have.

Keywords : Book Numbers, Census, Source Criticism

## Measuring the Problem

In the traditional Exodus story, the number of Israelites fleeing Egypt is so large it gives pause to readers.  Exodus 12:37b says the number of fighting-age men as over 600,000, while Numbers 1 counts the soldiers from each tribe, totaling 603,550 (Num 1:46). The conservative assumption of one woman and two children per soldier gives a total population at over 2.4 million, on par with the entire population of ancient Egypt. Similarly large numbers in the Deuteronomistic History are at odds with the archaeological record, as the territory only began to hold millions of residences in the 20 th century.

While several approaches to this numerical issue have been  published, the most common way to maintain  the  historical  plausibility  of  these  numbers  is  suggesting  the  term  for  'thousand'  ( elep , אלף ) does not mean a numerical value but a unit, at least when used in these census counts. If one hypothesizes an older source for the census numbers, perhaps אלף would have had a different meaning, thus confusing a later editor and implying a massive army. As a working hypothesis, אלף may have a pre-CBH meaning that was lost with developments within the Hebrew language and conflated it with a numerical value. The time of this other meaning must be prior to all CBH outside of the Pentateuch and the Deuteronomistic History, since Amos 5:3 and Isa 7:23; 30:17; 37:36 use אלף as the numeric value of 1,000. 1

1  Cf. Mic 5:2 as the only non-Pentateuchal or Deuteronomistic CBH example of אלף as different from 'thousand'.

Though elep can mean a clan or tribe, and perhaps it could mean a military unit, the question is if this fits the context of the censuses in Numbers 1 and Numbers 26. The fullest mathematical treatment, following a suggestion going back to Flinders Petrie, 2 comes from Colin Humphreys, 3 wherein he argues that the alapim for each tribe in the Numbers censuses were small military units. When Reuben, for example, counted 46 elep and  500 men, this is not 46,500 soldiers, but 500 fighting men in 46 military units (see Table 1). This goes against the natural reading of the numbers as 46 elep plus 500 men (using a waw to indicate addition), so there must be textual issues to explain why we should read '46 אלף of 500 men' as if it were a construct state (cf. Num 10:36) instead of '46 אלף and ( ו ) 500 men' with the clear waw conjunction. The use of the singular elep instead of the plural alapim ( אלפים ) is also odd, if אלף were being used as a noun instead of as a number (cf. Num 31:14).

Currently, these grammar issues are not discussed, let alone explained, by advocates. While Petrie, Mendenhall, and Humphreys suppose there was another, similar word to elep (perhaps with a different vocalization) that meant 'contingent,' this would not avoid the need to explain the grammatical points. Instead, Denise Flanders provides a detailed analysis of the grammatical and lexical issues with the argument that elep could mean a military unit or contingent. 4 Her focus was primarily on the Deuteronomistic History, but the results apply to the Numbers censuses as well. However, she does not address the mathematical arguments of Humphreys, and his hypothesis might overcome the grammatical issues, though with ad hoc suppositions about the Priestly writer not understanding prior sources both lexically as well as grammatically. However, an additional problem for Humphreys is his requirement that the census values are very ancient (perhaps even from Moses), but not only are the census numbers from P, but a late stratum of P. 5 Humphreys does not engage in source criticism. Nonetheless, his hypothesis can and should be examined on mathematical grounds.

To date, several critiques of Humphreys exist, 6 but they do not carefully examine his modeling or perform tests with a statistical method, 7 and others seem to find no fault with the mathematics presented. 8 This may be in part because most published researchers in biblical studies do not have a strong mathematical background, while Humphreys is a physicist. To fill in this gap, the following analysis will critique the population modeling proposed by Humphreys, examining both its internal consistency as well as what statistical tests can say about the quality of the hypothesis. Moreover, an analytical approach to the numbers in the censuses will help indicate how they were generated, if they have a deeper purpose, and what might be the origins of those values given the sources of the Pentateuch.

2 W.M. Flinders Petrie, Researches in Sinai , 207-217. Similarly, George E. Mendenhall, 'The Census Lists of Numbers 1 and 26', 52-66.

3  Colin J. Humphreys, 'Number of People', 196-213; 'Numbers in the Exodus', 323-328.

4  Denise Flanders, 'A Thousand Times No, אלף Does not Mean 'Contingent' in the Deuteronomistic History', 484-506.

5 Suzanne Boorer, The Vision of the Priestly Narrative , 70-71.

6  Jacob Milgrom, 'On Decoding Very Large Numbers', 131-132; Mark M. McEntire, 'A Response to Colin

J. Humphreys's', 262-264; Rüdinger Heinzerling, 'On the Interpretation of the Census Lists', 250-252.

7 McEntire, 'A Response to Colin J. Humphreys's', 262 already found Humphreys' math to be 'torturous.' I fear what might be said about the present article.

8 McEntire, 'A Response to Colin J. Humphreys's,' 262 initially said the math was 'convincing' and 'internally consistent.' Gary A. Rendsburg, 'An Additional Note to Two Recent Articles on the Number of People in  the  Exodus  from  Egypt  and  the  Large  Numbers  in  Numbers  I  and  XXVI',  394  says  that  Humphreys' mathematics is 'most convincing.'

## Real or Artificial Numbers

Before addressing Humphreys' model, one should address how reliable are the numbers in the censuses in the first place. Humphreys, as well as Petrie before him, assumed the general historicity of the  narratives  that  contain  the  counts  of  Israelites,  and  they  made  observations  that  they  alleged supported accepting those numbers, though after reinterpreting the term elep .

Petrie noted that the hundreds place in the census numbers was oddly distributed, with most values a 4 or 5, and no 0s, 1s, 8s, or 9s. 9 He suggested that the distribution indicates the reality of the numbers,  because  getting  such  a  distribution  by  chance  is  highly  improbable.  Petrie  does  not  specify what a natural  distribution  would  look  like,  or  why  an  artificial  selection  of  numbers  would  not have the features he noticed. Perhaps he assumed that fictitious numbers would follow a uniform distribution, where each digit is just as likely to appear; or if the numbers were poorly transmitted, they would randomize into a uniform distribution. Rather, the natural distribution of digits should follow Benford's law: a randomly generated number is most likely to start with a 1 (30.1% probability), then a 2 is the next-most likely (17.6% probability), and so on. 10 There are also expected probabilities for the next digits and combinations of digits. Especially after the first digit, so that there are orders of magnitude above and below to allow for variance for the numbers, Benford's law indicates there should be more 1s, not fewer, let alone none. This is most extreme when looking at the tens place; it is almost all 0s, except for a 5 and a 3, as if the writer generally rounded to the nearest hundreds place but included just a couple arbitrary examples of additional precision. Petrie's observations should make us doubt the values more than accept them, as the process that generated the numbers looks unnatural. 11

But there is no reason to only look at the hundreds place. Observing both the ten-thousands place and the thousands place (the first and second elep digits), the distributions do not suggest a natural distribution. The ten-thousands place distribution is similar to that of the hundreds place-mostly 4s and 5s, and no 0s, 1s, 8s, or 9s. That the first and third digits should have such similar distributions is highly unexpected with naturally occurring numbers. However, the thousands place does not follow the same pattern, as there are now 0s, 1s, and 9s, the mode is 2, the 4s and 5s are in the minority instead of the majority, and the general shape is closer to Benford's law. Still, 1 is underrepresented, while 2, 4, and 5 are overrepresented (see Figure 1). Measures of the divergences between the ex- Digit Counts

9 Petrie, Researches in Sinai , 210.

10 Mark J. Nigrini, Benford's Law .

11 Heinzerling, 'On the Interpretation of the Census Lists', 251 also noted other anomalous findings in the digits of the census counts.

8

7

5

21

1

0

Tens Place in elep Counts

Numbers 1 & 26

3

5

4

pected counts and the actual counts, such as either the Kullback-Leibler or the Jensen-Shannon Divergence, 12 indicate that the thousands place distribution is much better than the ten-thousands place when compared to Benford's law. The digit distribution suggests the leading digits for elep and for hundreds were artificial, while the next elep digit was closer to a natural distribution.

Digit Digit Digit

Figure 1 . Counts of digits in censuses in Numbers, plotted against expected counts following Benford's law. Note that the hundreds place was plotted against the first digit predictions of Benford's law because Petrie treated this as the first digit in the actual soldier counts. If instead the third digit predictions of Benford's law were used, there should be approximately a uniform distribution.

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However, given the small dataset, sweeping claims about the digits cannot be argued based on Benford's law alone. Instead, the problem is Petrie's vague logic going from the distribution of the hundreds place digits to historicity. Not only was Petrie unclear about why the distribution was an indicator for the reliability of the numbers, but the reality is the opposite. Similarly, when Humphreys says that it is 'statistically extremely unlikely that a set of fictitious invented numbers would display  such  curious  characteristics,' 13 this  is  without  support,  though  he  thinks  he  can  explain  the distribution of digits with his population model (see below). Petrie's logic is not sound, while Humphreys' discussion requires his modeling of the size of the tribes of Jacob.

Humphreys' analysis is based on several questionable assumptions. In particular, he assumes that the tribe of Levi is the about same size as any of the other tribes. 14 However, the straight-forward reading of Numbers 3 suggests that all one-month and older male Levites are 22 elep , 15 while other tribes  have  fighting  forces  several  times  that  (see  Table  1).  This  makes  Levi's  tribe  far  smaller. Their smaller size is also indicated by Deut 18:1, as the Levite were allotted no lands and were to live on sacrificial offerings. Special pleading is needed to suggest that the elep in Numbers 3 is different in size than the elep in Numbers 1 in such a way to make Levi the same size as Judah (more on this below). Additionally, not all these Levites can be in 'priestly troops' as Humphreys sug-

12 These measures are calculated using the implementations in the scipy library for the Python programming language.

13  Humphreys, 'Numbers in the Exodus', 327.

14  Humphreys, 'Number of People', 201-202.

15 Petrie, Researches in Sinai , 216-217 suggests the Levite numbers were a later addition, coming from perhaps as late as the monarchic period.

Ones Place in elep Counts

Numbers 1 & 26

Hundreds Place in Soldier Counts

Numbers 1 & 26

gests, 16 because priests had to be between the ages of 25 and 50 (Num 8:24-26), but the census counted any male at least one month old. The smaller number of priestly men is confirmed in Numbers 4, where 8 elep and 580 were of age and worked in the tent of meeting. Therefore, a Levite elep in Numbers 3 cannot be a priest troop, and the straight-forward reading of the Book of Numbers indicates Levi is a comparatively small tribe.

Additionally, the argument that all the tribes were about the same size at their inception and at the time of the censuses are problematized by modern studies and refuted by Humphreys' own counting. There is already the assumption that the tribal system existed in the mid-2 nd  millennium BCE and following the basic historicity of its origins as formed from the sons of Jacob. This is not supported by Humphreys but implicitly assumed while not addressing the scholarship questioning that assumption. 17 Nonetheless,  looking  at  the  biblical  numbers  from  the  various  tribes  through  time show large disparities. For example, First Chronicles 12 had the number of soldiers coming to David's call differ by an order of magnitude; Judah and Simeon could only muster several thousand apiece, Levi less than 5,000, while Zebulun provided 50,000. As for the time of the Exodus, the tribal sizes are found to vary significantly. 18  The Numbers 1 census has Dan more than double the size  of  Simeon  (assuming elep does  not  mean  1,000);  the  traditional  reading  of  Numbers  26  has Judah more than triple the size of Simeon. Moreover, while Humphreys suggests that a few centuries would not make one tribe significantly larger than another, the change in counts between Numbers  1  and  Numbers  26  shows Manasseh more than tripled in size (assuming the elep was  not  a count of people), while Simeon lost a third (or more than half, when reading elep traditionally), all in a matter of decades. If such fluctuations were real, then there are no grounds to say the tribes would be of the same order of magnitude in size centuries after their formation.

Instead of Humphreys' explanation, another hypothesis can explain the oddities in the digit distributions. As noted by Eryl Davies, 19 the earlier Yahwist (or simply non-P) source had a population of about 600,000 men leaving Egypt (Exod 12:37b; 20 Num 11:21). An even division of 600,000 gives 50,000 per tribe. Making half of the tribes larger than 50,000 and half smaller gives the abundance of 4s and 5s in the first digit (the ten-thousands place). As for the hundreds, when all added together this  total  is  close  to  6,000;  again,  taking  6,000  and  dividing  by  the  number  of  tribes,  and  there would be an additional 500 men per tribe. Adjusting the number up or down would similarly give an over-abundance of 4s and 5s. Also, Numbers 1 has half of the hundreds counts below 500, just like how half of the total counts were above and below 50,000. In other words, the same process of taking a number, evenly dividing it by 12, and then semi-randomly adjusting the numbers up and down produces a very artificial result. That semi-randomness in the adjustments especially effected the thousands place digits, which was noted earlier to more closely follow Benford's law. Overall, the evidence better fits the hypothesis that the census numbers are artificial creations.

16  Humphreys, 'Number of People', 205.

17  See Andrew Tobolowsky, The Sons of Jacob and the Sons of Herakles , for summary of modern research and further analysis.

18 McEntire, 'A Response to Colin J. Humphreys's', 262-263 for a similar critique.

19  Eryl W. Davies, 'A Mathematical Conundrum', 466.

20 R. S. Driver, The Book of Exodus , 100-101; John Van Seters, The Life of Moses , 229. Joel S. Baden, The Composition of the Pentateuch , 199 argues Exod 12:37 was from P, but his argument only applies to 12:37a, in agreement with Driver.

However, these confounds only undermine Humphreys' explanation for the digit distributions, his population estimate (c. 20,000 Israelites), and the reliability of the numbers themselves. His major premise is not yet addressed: the number of men per tribe were divided up into the specified number of alapim . This can be tested in three ways: correlations between counts of elep and soldiers, mathematical consistency, and narrative consistency.

## Correlations and Testing the Null Hypothesis

Given the hypothesis that the numbers are soldiers in elep -units, there ought to be a correlation between the number of alapim from a given tribe and their number of soldiers. If there are more soldiers, there ought to be more military units/ elep -groups, following a simple linear model. A very simple linear model would say as the number of soldiers doubled, the number of units would double. The basic linear model (Figure 2) for this census hypothesis is expressed by equation (1), where y is  the  count  of  soldiers, x is  the elep count, m is  the  number  of  men per elep ,  and y0 is  the y -intercept (discussed below).

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To see if this model fits the data well, 21 indicating a relationship between soldier and elep counts, I calculate  the  (Pearson)  correlation  coefficient, 22 R ,  in  four  places:  the  counts  in  Numbers  1,  in Numbers 26, the combination of Numbers 1 and Numbers 26, and the changes between Numbers 1 and Numbers 26. If there is a strong correlation between the count of soldiers and the count of elep -groups in each tribe, such that more soldiers means more elep -groups, then R should be close to 1. Because the counts of soldiers are round numbers and no more precise than at the tens place, then the correlation will not be perfect. Nonetheless, a value greater than 0.7 is expected for the model to explain most of the variance, 23 while a value close to 0 indicates no relationship at all.

21 The standard fitting method is known as ordinary least squares (OLS), which draws a straight line through the data points that minimizes the vertical distance between the data points and the line. Calculations were done with the statsmodels library for the Python programming language.

22 The Pearson correlation is the ratio of the covariance between two variables to the product of each variable's standard deviation. The covariance is the average of how much the product of each value's difference from the variable's mean, while the standard deviation is basically the average difference between a value and the mean of all values. Basically, it is a ratio of the correlated variance to the amount of variance in the data. A correlation of 0 implies no relationship between two variables (a change in x has no predicted change in y ), while a correlation of 1 implies a direct, positive relationship (a positive change in x will have an exact, positive change in y ), and a correlation of -1 implies a direct, negative relationship. Normally, the correlation is between -1 and 1, as most relationships are not direct or sufficient to explain all the variance in one variable in terms of another.

23 Variance explained is given by R 2 . If R = 0.7, then R 2  0.5 = 50%.

700 -

600-

500-

400-

300-

200-

20

Yo Scatterplot of elep and Soldier Counts Numbers 1&26

m

....

=-

y=mx+yo

·

·

·

Figure 2. Equation of a line (red). The coordinates are on an x-y plane, the slope (m) is the change in y (Δy) given a change in x (Δx), the green dot is the y-intercept (y0) where the red line intersects the y-axis (where x = 0), and the black dot is the origin of the coordinate system (x = 0, y = 0).

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The results should also be statically significant, meaning they should not be plausibly achieved by randomness. The common rule-of-thumb in statistics is that the probability, p , of having such a correlation R or better when there is no actual correlation (the null hypothesis) should be less than 1 in 20 ( p < 0.05); only then is a result considered significant. With the small amount of data involved in this analysis, the less strict p < 0.10 threshold may be acceptable. 24 Having the criteria of a minimum R as well as a minimum p will balance weak statistical power (that is, few data points) with minimizing Type II errors (inadvertently excluding true positives). Now, we can test Humphreys' hypothesis and see if R > 0.7 while also being statistically significant in any of the four cases. The results are in Table 2. Additionally, scatter plots of these counts are displayed to see if there are any patterns.

Figure 3. Scatterplot of the elep counts versus the hundreds counts. The source of the numbers, either Numbers 1 or Numbers 26, are color-coded.

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Looking at the counts of men and elep -groups in Numbers 1, there is only a weak correlation without  statistical  significance.  As  for  the  counts  from  Numbers  26,  the  correlation  is  almost  nonexistent  and  not  statistically  significant.  Combining  these  counts  together  provides  more  reliable statistics, as the statistical test is evaluating more data and will be more powerful. Unfortunately, the correlation is weak and not significant. A scatterplot of these counts indicates no pattern (Figure 3).

24 For a correlation of R = 0.7, 12 samples, and an 80% chance of correctly rejecting the null hypothesis, then p  0.10. See Stephen B. Hulley et al., Designing Clinical Research , 79.

And yet, even these lackluster correlations are overestimates, because the y -intercept ( y0 ) was not constrained. The y -intercept indicates the value of y when x is zero. For the current hypothesis, the y -intercept is how many soldiers there are when there are no elep -groups. In some cases, there were significant results where y0 was well above zero, indicating that even without a single elep ,  there would be hundreds of soldiers. This is not possible if soldiers were all assigned to an elep . Worse, these statistically significant intercept values are much larger than any elep ,  which ranged in size between 5 and 17 soldiers; y0 , on the other hand, was over 340 for the combined dataset, when there were only about 460 soldiers per tribe. It is as if most (~75%) of the troops cannot be accounted for. The fact that the y0 results were ever statistically significantly different than zero is unexpected given the hypothesis.

Instead, the hypothesis of alapim related to the soldier counts requires this constraint: zero soldiers implies zero elep -groups and vice versa, so the y -intercept ought to be set to zero. Therefore, equation (1) should have y0 = 0, and equation (2) ought to be fitted to the data.

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This model, which more accurately conforms to Humphreys' hypothesis, provides uninspiring coefficients ( R ( y0 = 0) Pearson in Table 2). 25 Some are not real numbers ( NaN ), suggesting that a horizontal line with no trend is better than the fitted model. In other words, in several cases there was no relationship between elep counts and soldier counts at all. With such small or non-existent correlation values and large p -values, almost all the variance is either explained by other variables or just random noise.

As another approach, instead of the Pearson correlation, the Spearman correlation (ρ) looks at the rank of the values. 26 This  tests  the  hypothesis:  the  largest elep counts  go  with  the  largest  soldier counts, and the converse. This does away with the restrictions of the linear model, wherein the values  must  always  proportionally  increases  together  rather  than  just  generally  increase  together. Spearman also does away with the y -intercept, removing an additional modeling constraint. Thus, the Spearman correlation may work better for the sake of Humphreys' hypothesis. However, this results  in  similarly  bad  correlations  ( ρ  Spearman in  Table  2).  The  negative  correlation  for  the Numbers 26 counts suggests more alapim means fewer troops, the opposite of the expected trend. No results were statistically significant.

25 R is calculated differently when there is a set y -intercept as in equation (2), so these coefficients are not directly comparable with that calculated using equation (1). The p -values also cannot be calculated. Nonetheless, small and non-existent R -values are informative.

26 Spearman correlation uses the rank of each value (smallest to largest) from 1 to n in a set with n distinct values. Then the rank values are used to perform the same calculations as in the Pearson correlation, using covariance and standard deviation. If all the values are different, then an easier calculation of the coefficient uses the differences in rank ( di ) for each set of points ( xi , yi ) and the number of data points ( n ). If all xi and yi values have the same rank, then the correlation will be 1. If the ranks are random and the variables have no relationship, we expect the correlation to be close to 0.

Change in number of Soldiers

600

400 -

200 -

0

-200 +

-400 +

-40

"ГГ

Scatterplot of elep and Soldier Counts

Changes between Numbers 26 & 1

20-

The failure to reject the null hypothesis ought to be sufficient to no longer pursue the elep -group hypothesis. However, for completeness, suppose that another variable is each tribe had its own rules for organizing a military unit. While the hypothesis is ad hoc , it is already required to explain why one tribe had as few as 5 men in an average elep , while another had as many as 17. Perhaps then these divergent-sized alapim could explain the weak correlations. One way to account for this is by looking  at  proportional  changes  in  the  counts  of elep -groups  and  men  between  Numbers  1  and Numbers 26. In that case, looking at the correlation between proportional changes in elep counts and in soldier counts within a tribe will avoid the possible confound of inter-tribal differences. At first, there appears to be a better correlation ( 26 - 1prop. in Table 2), but it is almost completely driven by outlier values, particularly Manasseh skewing the results with high statistical leverage-that is, the value is so different from the rest of the data points that it has an undue influence on the fitted line. If removing one data point notably changes the overall trend, then it is likely an outlier. Without Manasseh, the result is no better than before ( 26 - 1prop.+clean in Table 2). The scatterplots have no pattern, and Manasseh was clearly an outlier (Figure 4), driven by the near tripling in size of Manasseh in 40 years-a sure sign that there is something amiss with the value, given Humphreys' model. There is simply no statistical support for the hypothesis.

Figure 4. Absolute and proportional changes to elep and hundreds of soldiers counts between Numbers 26 and Numbers 1.

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In fact, several tribal values show the folly in the approach advocated by Humphreys. Both Ephraim and Naphtali had 8 fewer elep in Numbers 26 than in Numbers 1, but their soldier counts were unchanged. More starkly, Asher gained 12 elep -groups, a 29% increase, but their soldier count went down by 100, a 20% decrease. Absurdly, Dan lost nearly half its population, but it gained 2 more elep . In about half of the tribes, the change in elep count went in the opposite direction as the soldier  count.  There  is  no  discernable  trend,  indicating  there  is  no  underlying  relationship  between elep -counts and additional soldier counts.

Additionally, the hypothesis is directly refuted when looking at the Levite numbers. If the interpretation that only the hundreds count is the personnel count, while the elep is a group of arbitrary size,

Scatterplot of elep and Soldier Counts

Proportional Changes between Numbers 26 & 1

then there is the absurd situation wherein Kohath had only 300 males 27 (Num 3:27-28) yet sent 750 men into priestly services (Num 4:34-37). This impossibility falsifies Humphreys' reading, while the traditional reading of 8,300 males with 2,750 in service is perfectly consistent. The same problems exist for Gershon (500 total males, 630 in priestly service) and Merari (200 total males, 200 in priestly service).

This last impossibility was noted by Jerry Waite, 28 but he speculated that the next number after elep was originally a tens instead of a hundreds. The Priestly writer then assumed that elep was 1,000, and thus the next number was a hundreds. Not only is there no evidence of this, but it requires either the editor to have read a value like 50 ( חמשים ) as חמשׁ מאות ,  or  an  earlier  writer  wrote חמשׁ for חמשים , and then the editor imputed מאות . The first scenario is not plausible, while the latter requires mistakes on both the part of the original writer and the later editor, which is improbable. The fix Waite supposes also contradicts Humphreys' model that has the Levite tribe of similar size as other tribes; now, there were only 100 male Levites, confounding the population model in the first place. But these are all the least of the problems with Waite's ad hoc hypothesis. The Levite census counts in  Numbers  4  makes  the  reading  of  tens  values  as  hundreds  impossible,  because  Numbers  4  includes the tens along with the hundreds, making the alleged misreading unviable. But even forcing the factor of ten change on Numbers 4, this would mean there are 158 men in the priesthood (75 from Kohath, 63 from Gershon, 20 from Merari)-this is still more men in the priesthood than total Levite males.

Finally, the very first equation used by Humphreys builds on a false premise, given his hypothesis, that leads to nonsensical results. Noting Num 3:46 that there were 273 more first-born males among the  non-Levite  tribes  than  there  were  male  Levites,  Humphreys  builds  his  model  on  this  difference. 29 He creates equation (3) (his equation (1)) where If is the number of first-born non-Levitical Israelites and L is the number of male Levites.

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However, the  count  of  first-born  Israelites  was  given  as  22 elep and  273  (Num  3:43).  If elep is treated as 1,000, then there were more first-born males than all the Israelites as estimated by Humphreys. This also forces the number of Levite males to also exceed 20,000, and there are no grounds for following the elep -group hypothesis for any of the census data. If instead elep is  treated  as  a group counter, as was done with in the elep -military unit hypothesis, then there were only 273 firstborn Israelites ( If ) in 22 elep -groups (about 12 first-borns per elep ). Following equation (3), then the number of Levite males ( L )  must be zero! This is not only an absurd result, but Humphreys also notes that his reading of Numbers 3 and his model indicate there were 1,000 Levite males. Using that as L ,  then  the  number of first-born Israelites would be 1,273, which corresponds to no other values in the censuses, particularly Num 3:43.

27  MT says 600, but the total (Num 3:39) makes this impossible, suggesting Greek manuscripts more accurately preserved the number.

28 Jerry Waite, 'The Census of Israelite Men after their Exodus from Egypt', 489-490. Also, Heinzerling, 'On the Interpretation of the Census Lists', 250-251.

29  Humphreys, 'Number of People', 201.

There is only one last way to save this estimate, which is to assume that originally Num 3:43 referred not to 22 elep of 273 first-born males, but 21 elep -groups, and then the number of first-borns is  1 elep +  273  = 1,243.  In other words, the hypothetical source had to use elep in  two  different ways next to each other (i.e., עשׂרים ואחת אלף אלף ) in an unhelpful case of antanaclasis. Why the Priestly writer looked at this and neither understood two different meanings of elep nor treated it as a  million  (i.e.,  1  Chron  21:5)  is  yet  another  grammatical  issue  Humphreys'  hypothesis  requires. Nonetheless, this opens the possibility that this same sort of issue would exist in all the other census counts for each tribe. For example, Reuben may have not had 46 elep -groups, but perhaps one or more of those alapim is actually 1,000. This means we would not know if the number of soldiers from Reuben was 500, 1,500, 2,500, and so on up to 46,500. Equations (4) and (5) expresses all the possibilities, where n and m are whole numbers ( ℕ0 ) that have to add up to the total number of elep of either kind.

<!-- formula-not-decoded -->

Similarly, this uncertainty exists for all other tribal counts, so each tribal census requires its own version of equation (5) with different values of n and m . The mixture of elep -unit and elep -thousand is also needed to fix the problem with the Levite counts between Numbers 3 and 4, as noted earlier.

To solve for such counts analytically is impossible; each tribal count has two unknowns (how many elep -groups and how many 1,000s; n and m , respectively), and each tribal count is independent of the other. This requires solving for 60 unknown variables with 30 sets of equations and constraints (15 for each tribe or sub-tribe, repeated twice for two censuses, and each having two unknowns). At best, one can look at every possible combination of elep -counts in these 30 equation-sets and see which combinations seem to work. With literally thousands of possible combinations (on average, about 50 possibilities per equation), this is a daunting prospect, perhaps requiring numerical simulations and some objective function to see what results are best.

However, we can already find a solution that is far simpler: set n = 0 in all cases, and then elep = 1,000 in all cases.

## Simply 1,000

Simple statistical tests falsify a relationship between the less-than-one-thousand-troop counts and the counted elep from a tribe, basic math disproves the hypothesis, and the underlying model is built from a demonstrably false premise. All ways to save the model require making it uninformative. Conversely, basic algebra shows that the writer of Numbers 1 interpreted elep as 1,000. Note that the totals provided in Num 1:46 are 603 elep and 550 men; summing the individual tribal counts, and there are 598 elep and 5550 men. Since these should be equal, as described by equation (6), then solving for the value of the elep is as follows:

<!-- formula-not-decoded -->

<!-- formula-not-decoded -->

The same sort of algebra applied to Numbers 3, 4, and 26 give the same results for elep .

When looking at the excess number of first-born non-Levitical Israelites in Num 3:43 (22 elep + 273), the count of Levite males in Num 3:39 (22 elep ),  and  equation (3) that was based on Num 3:46, we can prove the Israelite elep and the Levite elep are identical. If the Israelite elep is designated as elepI and the Levite elep unit is elepL ,  without assuming these alapim are the same, then equation (7) again proves elep must have the same size in both cases.

<!-- formula-not-decoded -->

As the elep of the Israelite soldiers was shown to be 1,000, then the Levite elep must be 1,000. This also refutes Humphreys' estimation that the average military elep was about 9.3 soldiers, while the Levitical elep was about 48 priests. 30

Independently, the same outcome is derived when examining the sum of half-shekels provided by each soldier (Exod 38:25-26): 100 talents and 1,775 shekels. As 1 talent was 3,000 shekels, 31 this gives 301,775 shekels. Doubling this to account for each soldier giving a half-shekel, and this gives 603,550 soldiers, exactly as reported by Exod 38:26 and Num 1:46, but only if elep is 1,000. If this shekel-count is not well-sourced, then it indicates the writer of Exodus 38 (usually believed to be P) is inventing numbers based on their own calculations. If some numbers are invented, then the probability that other numbers are invented increases necessarily. Conversely, if the shekel-count is historically reliable, then elep must be 1,000 and the soldier count was huge.

The Deuteronomistic History also contradicts the smaller elep model. In Gideon's fight against the Midianites (Judges 6-7), he first claimed his elep [family/clan] is the smallest in Manasseh. Nonetheless, his army started with 32 elep , but under instructions from God to reduce his army's size, it became 10 elep , and then finally 300 soldiers. If an elep were 9.3 soldiers as estimated, then Gideon started with 298 soldiers then reduced his army to 93; and yet it was reduced to 300-an impossibility. This example also undermines the argument that an editor would confuse the different possible meanings of elep , since the writer of Gideon's story fluently went between its possible valences.

Logistically, the Numbers narrative assumes much larger numbers of soldiers than estimated otherwise. For example, the war against the Midianites (Numbers 31) involved 12 Israelite alapim (one elep per tribe), and Israel won, killed five kings, and brought back hundreds of thousands of heads of livestock. Those 'thousands' of animals are confirmed by arithmetic results given in the narrative, wherein one five hundredth of the spoils are given to God, and the count of items sacrificed are given and perfectly match those values multiplied by 500 (i.e., 675 sheep for God, given 337,500

30  Humphreys, 'Number of People', 205.

31 Carlo Zaccagnini, 'Note on the Weight System at Alalah VII', 472-475.

(675  500) as spoils for the solders). As for the victorious Israelite army, given an elep -group of 9.3 soldiers, then all of this was accomplished by about 110 men. Each soldier had to shepherd over 6000  sheep,  as  well  as  650  cattle  and  550  donkeys.  Also,  these  soldiers  captured  32,000  virgin women, half of which would be spoils to the soldiers who participated (Num 31:27). If there were only 110 soldiers, then each of them received more than 140 sex slaves, a Solomonic proportion. Conversely, if there were 12,000 troops, then each soldier took one or two virgin women, and their shepherding duties would have been within the realm of plausibility. The tinyelep model makes the story strange in a way the author likely did not intend.

The  plagues  on  Israel  also  demonstrate  the  numbers  must  be  larger  than  Humphreys'  estimated population of 20,000. For example, Num 25:9 says that 24 elep were killed by the plague. Here an elep cannot be either a military or priestly unit, since this is about how many Israelites in general were killed by the plague. There is also no plausibility that elep means clan or family, as this would suggest the plague killed 24 families entirely and all the others lost not a single member. In addition,  an  earlier  plague  killed  14,700  Israelites  (Num  16:49),  and  similarly  none  of the  arguments about other meanings of elep can consistently apply here. If it were 14 families of 700 people, then, applying that reading to Num 25:9, the 24 families killed had zero people; if this were 14 families and an additional 700 people, then this reading is contrary to how the same was done for the military and priestly units. When every hypothetical reading of elep must be exceptional, then the hypothesis is not informative. Instead, when reading the text straight-forwardly, the plagues are killing more than Humphrey's estimated population.

Clearly the writer(s) or redactor of Numbers interpreted the elep in the census as 1,000. Suggesting a misreading of one elep for  'military units' and another elep for  'thousands' early on would not explain the shekel count, nor the counts from battles against the Midianites, nor the plague deaths, nor does it help with the mathematical problems identified above. Whatever one thinks of the numbers, they cannot be minimized in a mathematically sound way, nor can it be done without doing violence to the narrative. As the hypothetical reading of elep as 'military unit' or 'contingent' also introduces grammatical anomalies, while reading elep as 1,000 is grammatically sound and mathematically consistent, all the evidence points away from the hypothesis.

## Origins of the 600,000

As noted earlier, the P-sourced size of the tribes of Israel as all around 50,000 fighting men was plausibly derivate of the approximate 600,000 men that left Egypt according to the non-P source. This leaves the question as to how J/non-P derived a number like 600,000. One plausible reason is that J is writing after the creation of the Deuteronomistic History. While Deuteronomy has no census of Exodus-era soldiers, the Book of Joshua begins to indicate the size of Israel's fighting force during  the  conquest,  as  Joshua  had  around  40,000  soldiers  laying  siege  to  Jericho  (Josh  4:13). 32

32  Liane M. Feldman, The Consuming Fire , 22-23, 241 suggests this verse is from P, but the only evidence is the mention of Jericho and the claim that only P mentions Jericho (which is contrary to most proponents of the  Documentary  Hypothesis;  i.e.,  Deut  34:3  is  non-P).  If  the  Deuteronomistic  history  is  older,  mentions Jericho, and then influences P, then the same evidence is explained just as well, so Feldman's argument is

Even larger numbers are found in Judges 20 as 400,000 soldiers were arrayed for battle against the Benjaminites. Later, Saul gathered 330,000 men to his cause (1 Sam 11:8), and then David's own census found 800,000 fighting men from Israel and another 500,000 from Judah (2 Sam 24:9). Later still,  when  Rehoboam was planning to attack Israel, the tribes of Judah  and Benjamin alone had 180,000 capable fighting men (1 Kgs 12:21).

A reader of these books would assume that the population of ancient Israel was huge, much like the Chronicler who put the size of David's army at over one million (a thousand thousand; 1 Chron 21:5). Perhaps then it was almost a requirement that a similarly large number left Egypt in the first place so that there could be these large armies to fight for Joshua, Saul, and David. A magnitude of tens of thousands would be too small to explain how there were enough Israelites to become the hundreds of thousands of soldiers that fought in Judges 20.

An additional indication of the size of the fleeing Israelite population could be gleaned from the size of the army Joshua used to seize Jericho. Only three of the twelve tribes (Reuben, Gad, and Manasseh) were involved when the siege was first described (Josh 4:12), which may suggest a quarter of Israel's fighting force was prepared for war. If a reader looked at this situation as similar to the story  of  Gideon  (Judg  7:3-8),  who  started  with  32,000  men,  then  cutting  down  by  about  a  third  to 10,000, and then cutting down by about a third again to 300, and he further divided his armies into thirds (Judg 7:16), then the same logic of a third of a third could be applied to Israel's fighting forces in Joshua. If only a quarter of the available tribes went to Jericho, and if only a quarter of each of those tribes participated, then the approximate 40,000 were a quarter of a quarter of all the fighting forces, which would be about 4  4  40,000 = 640,000. The approximate 160,000 that Reuben, Gad, and Manasseh had, given this hypothesis (4  40,000), is close to the total of soldiers given in Numbers 26 (136,930), so at least P may have made this deduction.

A value of about 600,000 may have been desirable to J/non-P if he already deduced a value of close to that from reading Joshua 4, but this rounded value would also have had a nice mathematical feature. As a number easily divisible by 12, this would give a round number of soldiers to each of the tribes. This would not have been true for 500,000 or 700,000, and the easy division of 600,000 by 12 seems to have been utilized by P.

While 600,000 can be represented in a sexagesimal number system (2:46:40:0), such as that of the Babylonians, there are no strong indications that the numerical values in the censuses were based on such a system, let alone complex numerical values for planetary cycles has had been suggested previously. 33 Furthermore, the distribution of census values is suggestive of there being no meaning to the particular numbers given to the tribes. As seen in Figure 5, the distribution of values closely follows a normal distribution (also known as a bell curve). A normal distribution is expected if there is one source for a measurement, along with random variation around the mean value of that distribution. Tests suggest that the distribution is surprisingly close to a well-shaped normal distribution;

not persuasive. Additionally, the approximate number of soldiers mentioned in Josh 4 clashes with the exacting numbers from P.

33  For discussion, see Davies, 'A Mathematical Conundrum', 457-458.

8

6-

5

3-

2-

1

20000

Histogram of Census Counts

the main test for normality (the Shapiro-Wilk test) confirms this (the null hypothesis is consistent with the data, p > 0.95).

30000

40000

50000

60000

Census Tribe Sizes

Figure 5. Histogram of Numbers 1 and 26 Censuses. A fitted normal distribution curve (red) is added to the bin counts.

<!-- image -->

The fact that the distribution is not multimodal (it has one peak) is suggestive that there is only one underlying cause for the values of the tribal populations. With a mean value of about 50,000 and more than half of the counts between 40,000 and 60,000, this distribution implies that the underlying cause is having tribes with about 50,000 soldiers with a bit of adjustment up or down (the standard deviation is about 13,000). If there were other causes for the particular numbers in each of the tribes, then such a clean distribution would have been unlikely. Values based on gematria, 34 for example, would not so nicely conform to a normal distribution, as each value would have a different cause,  and this would have introduced additional modes (or 'bumps') in the overall distribution. Instead, the deviations around the mean value are consistent with randomness around a central value.

The existence of a central value in a normal distribution is also more expected if P is using J/non-P rather than non-P using P and rounding P's numbers. If each tribe's size were randomly guessed by P,  there  is  no  reason  to  expect  such  a  distribution  (a  uniform  distribution  seems  most  likely),  let alone most tribes having sizes of nearly 50,000. The abundance of 4s and 5s in the first digit is also expected on the division of 600,000 by 12, while random generating numbers would be closer to Benford's law (discussed above). Instead, this is more expected if P had some total from which it started and then divided that value up nearly evenly between all the tribes. P is also not likely deriving his numbers directly from D, since the logic of getting a total of 600,000 from 40,000 at Jericho required rounding 40,000  4  4 down. If P directly used 40,000  4  4, then the census would have had a total closer to 640,000, and the average tribe would have had 53,000 fighting men. Instead, P is using the rounded-down value found in non-P, and P would not likely have rounded down, given the propensity for more exact numbers compared to D and non-P. This provides some evidence for P's knowledge of non-P.

34  Davies, 'A Mathematical Conundrum', 452-453.

Additionally, there is not a consistent size of the tribes between Numbers 1 and 26. Of the six tribes that had less than 50,000 soldiers in the first census, barely more than half (four of the six) continued to  be  so  small;  similar  results  exist  for  the  tribes  originally  larger  than  50,000.  Also,  barely more than half of the tribes (7 of 12) had their numbers increase, yet the total number of Israelites decreased, as if there were no trend but random occurrences of tribes getting larger or smaller. The changes in tribe sizes appear to be random and symmetric with a mean change close to zero. There is  also  only  a  weak and  non-statistically significant correlation between the early and late census counts ( R = 0.45, p = 0.14). As both sets of census numbers have an average of 50,000 soldiers per tribe, the counts in either census were largely independently generated from dividing up the 600,000 into twelve tribes. At most, a couple of values were made particularly large or small, but a larger project of making some tribes larger or smaller based on a criterion cannot be supported. Looking for meaning in exact values is unlikely to be fruitful. 35

Of the sizes that may be meaningful, there are at most three examples that have some evidence. One is that Judah was always recorded as the largest tribe, and Judah was likely the tribe of special importance to the Priestly writer. 36 Second is the massive drop in Simeon, from 59,300 to 22,200. This may relate to the P-plague story in Numbers 25; Zimri of Simeon was the only person mentioned from a particular tribe who was killed because he was with a Midianite woman (Num 25:14). 37 But the motivation for the small size of Simeon, as well as Levi, may go back to Gen 49:5-7 wherein Jacob curses these two tribes who would be scattered. This could be another example of J/non-P influencing the numbers in P, and this would explain why Simeon and Levi had approximately the same counts in the censuses (roughly 22,000). Third, Benjamin is the smallest tribe in Num 1 (accounting for Ephraim and Manasseh as half-tribes from Joseph), and this may reflect 1 Sam 9:21 where Saul says that Benjamin was the smallest of all the tribes. However, Benjamin's numbers grew  significantly  between  Numbers  1  and  26,  so  the  random  adjustments  to  the  tribal  sizes  in Numbers 26 created an inconsistency in the final results. The case for Benjamin's initial size being made small is therefore only tentative.

## Conclusions

The analysis performed above suggests both that the data in the censuses in Numbers are not likely to be historical or based on a pre-CBH relic, and instead they appear to be artificially generated, especially by P. The efforts to avoid the very large numbers in the Numbers censuses by finding another definition for elep were found to be wanting, either rejected on statistical grounds or refuted by simple arithmetic considerations.

Some evidence was found for the census numbers as a development between Pentateuchal sources. The P source was most likely working from a starting number from J/non-P, who was influenced by the numbers in the Deuteronomistic History. The trend of D influencing J, then influencing P, follows trends others have found with the relationship between the Pentateuchal sources, such as the anti-Egyptian plague traditions. 38 This assessment adds to the growing consensus that P knew and used non-P. 39 However, P was not likely giving particular values for the size of tribal armies for some deeper meaning. Instead, P was producing counts around a set mean value of 50,000 and generally had random changes to that mean value.

35 Statistical inferences from the Levite numbers are even less likely to be fruitful, because there are so few data points.

36  Davies, 'A Mathematical Conundrum', 466 n. 44.

37  Humphreys, 'Number of People', 208-209.

38 J. Van Seters, 'A Contest of Magicians?', 569-580.

39 Boorer, The Vision of the Priestly Narrative , 90-110.

## Bibliography

- Baden, Joel S. 2012. The Composition of the Pentateuch: Renewing the Documentary Hypothesis . New Haven: Yale University Press. https://doi.org/10.12987/9780300152647.
- Boorer, Suzanne. 2016. The Vision of the Priestly Narrative: Its Genre and Hermeneutics of Time . Atlanta: SBL Press. https://doi.org/10.2307/j.ctt1gxxq00.
- Davies, Eryl W. 1995. 'A Mathematical Conundrum: The Problem of the Large Numbers in Numbers I and XXVI'. Vetus Testamentum 45 (4): 449-469. https://doi.org/10.1163/1568533952662351.
- Driver, R. S. 1953. The Book of Exodus: In the Revised Version with Introduction and Notes . Cambridge: Cambridge University Press.
- Feldman, Liane M. 2018. The Consuming Fire: The Complete Priestly Source, from Creation to the Promised Land . Oakland, CA: University of California Press. https://doi.org/10.1525/9780520383661.
- Flanders, Denise. 2018. 'A Thousand Times No, אלף Does not Mean 'Contingent' in the Deuteronomistic History'. Biblica 99 (4): 484-506. https://doi.org/10.2143/bib.99.4.3285660.
- Heinzerling, Rüdinger. 2000. 'On the Interpretation of the Census Lists by C. J. Humphreys and G. E. Mendenhall'. Vetus Testamentum 50 (2): 250-252. https://doi.org/10.1163/156853300506341.
- Hulley, Stephen B. 2013. Steven R. Cummings, Warren S. Browner, Deborah G. Grady, and Thomas B. Newman. Designing Clinical Research: An Epidemiologic Approach . Philadelphia: Lippincott Williams & Wilkins.
- Humphreys, Colin J. 1998. 'The Number of People in the Exodus from Egypt: Decoding Mathematically the Very Large Numbers in Numbers I and XXVI'. Vetus Testamentum 48 (2): 196-213. https://doi.org/10.1163/1568533982721550.
- ---. 2000. 'The Numbers in the Exodus from Egypt: A Further Appraisal'. Vetus Testamentum 50 (3): 323-328. https://doi.org/10.1163/156853300506396.
- McEntire, Mark M. 1999. 'A Response to Colin J. Humphreys's 'The Number of People in the Exodus from Egypt: Decoding Mathematically the Very Large Numbers in Numbers I and XXVI''. Vetus Testamentum 49 (2): 262-264. https://doi.org/10.1163/156853399774229019.
- Mendenhall, George E. 1958. 'The Census Lists of Numbers 1 and 26'. Journal of Biblical Literature 77: 52-66. https://doi.org/10.2307/3264330.
- Milgrom, Jacob. 1999. 'On Decoding Very Large Numbers'. Vetus Testamentum 49 (1): 131-132. https://doi.org/10.1163/156853399774228588.
- Nigrini, Mark J. 2012. Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection . Hoboken, NJ: Wiley. https://doi.org/10.1002/9781119203094.
- Petrie, W. M. Flinders. 1906. Researches in Sinai . London: John Murray.
- Rendsburg, Gary A. 2001. 'An Additional Note to Two Recent Articles on the Number of People in the Exodus from Egypt and the Large Numbers in Numbers I and XXVI'. Vetus Testamentum 51 (3): 392396. https://doi.org/10.1163/15685330152913666.
- Tobolowsky, Andrew. 2017. The Sons of Jacob and the Sons of Herakles : The History of the Tribal System and the Organization of Biblical Identity . Tübingen: Mohr Siebeck. https://doi.org/10.1628/978-316-155597-8.
- Van Seters, John. 1994. The Life of Moses: The Yahwist as Historian in Exodus-Numbers . Louisville, KY: Westminster John Knox Press.
- ---. 1995. 'A Contest of Magicians? The Plague Stories in P'. In Pomegranates and Golden Bells : Studies in Biblical, Jewish, and Near Eastern Ritual, Law, and Literature in Honor of Jacob Milgrom , edited by David P. Wright, David Noel Freedman, and Avi Hurvitz. Winona Lake, IN: Eisenbrauns: 569-580.

Waite, Jerry. 2010. 'The Census of Israelite Men after their Exodus from Egypt'. Vetus Testamentum 60 (3): 487-491. https://doi.org/10.1163/156853310X499853.

Zaccagnini, Carlo. 1979. 'Note on the Weight System at Alalah VII'. Orientalia 48 (4): 472-475.

Table 1. Counts of elep and Additional Soldiers per Tribe.

|              | Numbers 1   | Numbers 1   | Numbers 26   | Numbers 26   |
|--------------|-------------|-------------|--------------|--------------|
| Tribe        | elep        | +Soldiers   | elep         | +Soldiers    |
| Reuben       | 46          | 500         | 43           | 730          |
| Simeon       | 59          | 300         | 22           | 200          |
| Gad          | 45          | 650         | 40           | 500          |
| Judah        | 74          | 600         | 76           | 500          |
| Issachar     | 54          | 400         | 64           | 300          |
| Zebulun      | 57          | 400         | 60           | 500          |
| Ephraim      | 40          | 500         | 32           | 500          |
| Manasseh     | 32          | 200         | 52           | 700          |
| Benjamin     | 35          | 400         | 45           | 600          |
| Dan          | 62          | 700         | 64           | 400          |
| Asher        | 41          | 500         | 53           | 400          |
| Naphtali     | 53          | 400         | 45           | 400          |
| TOTAL        | 598         | 5550        | 596          | 5730         |
| Stated Total | 603         | 550         | 601          | 730          |

Table 2. Correlations and significance tests for elep and soldier counts. Numbers 1 and 26 censuses are treated individually, combined (1 & 26), and changes between the censuses (26 - 1). Statistically significant results (p < 0.10) have an asterisk (*).

| Numbers Chapter    | R Pearson   |   p R | y 0   |   R ( y 0 = 0) Pearson |   ρ Spearman |   p ρ |
|--------------------|-------------|-------|-------|------------------------|--------------|-------|
| 1                  | 0.40        |  0.20 | 230   |                   0.11 |         0.30 |  0.34 |
| 26                 | 0.10        |  0.75 | 426*  |                    NaN |        -0.10 |  0.76 |
| 1 &26              | 0.23        |  0.28 | 346*  |                    NaN |         0.08 |  0.70 |
| 26 - 1 counts      | 0.39        |  0.22 | 16    |                   0.38 |         0.27 |  0.40 |
| 26 - 1 prop.       | 0.64*       |  0.02 | 0.13  |                   0.62 |         0.34 |  0.27 |
| 26 - 1 prop.+clean | 0.28        |  0.41 | -0.03 |                   0.29 |         0.15 |  0.66 |