BIVARIATE STATISTICS MADE EASY

How Lone Classicist Aubrey Diller's Single-Shot Derringer
Beat MuffEstablishment MultiShooter Blunderbuss Army

DIO TO MUFFIA    —    IS 13-OUT-OF-13 ENUFFIA?
GetSmart Asks: Would-You-Belieeeve 14-OUT-OF-14?

Clear Crucial-Test Which Will Occam-Razor-Sharpen Our Curiosity:
How Does the Lock-Step History-of-Ancient-Astronomy Community
Differ from Herds of Projective Crank-Accusing Religious Cultists?

History-of-Ancient-Astronomy Pols'
TruthShun Priority SleepWalks On:
MUFFIA Devolves Into MORPHIA

[One of the characteristics that mark those at the top of their field — those whose self-confidence is founded on genuine total expertise and a realistically grounded sense of secure success — is a warm generosity towards the creativity of others, a gladness in the success of those who might superficially appear to be their competitors. E.g., Mahler towards Strauss. Or van der Waerden to even his most bigoted critics. It is those who inwardly know the infirmity of their views and worth who fear to rise beyond the amniotic warmth of clinging to the numerical & fiscal security-blankets provided by cults of their ilk.]

In recent years, certain deliberately-prestigious, captive, Muffia-servile journals have published effectively unrefereed history-of-astronomy research papers, that try to find the reliability of data-fits for 2-unknown Gaussian cases. E.g., Centaurus (Y.Maeyama 1984), the Journal for the History of Astronomy (A.Jones 2002 & B.Schaefer 2005), and the Journal for Astronomical History & Heritage (J.Brandt, P.Zimmer, & P.Jones)
[DIO has occasional experience in this area. Our non-metric photogrammetry on R.Peary's 1909 “North Pole” photos was an analytic least-squares simultaneous solution for 22 unknowns: DIO 1.1 [1991] ‡4 [p.29].]
Though DIO appreciates (e.g., DIO 6 [1996] ‡3 §B3 [p.37]) the valid contributions of these and other opposition scholars, it is a regrettable fact that none of the above-mentioned authors has understood how to solve Gaussian bivariate problems analytically — or, indeed, by any means besides monovariate testing, trial&error, or guesswork. Likewise for the 1975 attempt by ultimo-Muffioso O.Neugebauer to solve the Hipparchos-Strabo “klimata” data (also amateurishly investigated by the above-cited JHA 2002 paper) — a politically & ethically scandalous affair (persistent clique-robbery of credit from a great philologist). So we will (below) use the modern klimata controversy as a useful mathematical illustration of how to apply bivariate statistics with ease.
“Klimata” were used by ancient astrologers for house-division (DIO 14 [2008] ‡3 n.25 [p.35]). They defined a klima as a latitude L where the longest day M was some discrete value. Ancient klimata tables (e.g., Almajest 2.6) were usually computed at convenient intervals, such as every quarter-hour of M.
It is disappointing that — in addition to their pseudoreferee-undetected technical shortcomings — all three papers appear to be invalidly taking credit for others' discoveries.
This unfortunate history will be detailed in the paragraphs immediately following, but those interested in math more than in our brief passing diversion into academic-clique syciology, may simply CLICK their way past it. (Or to a politics-frei version of the present paper.)

Why Have Science's Skills & Attitude of Humble
OpenMindedness Atrophed in the Tight
History-of-Ancient-Astronomy Cult?

Maeyama 1984's correct but clumsily induced date for Aristyllos (c.260 BC) was actually discovered analytically and prominently published earlier by uncited Rawlins Isis 73 pp.259-265 (1982) p.263. (Some amusing statistical problems with the 1984 paper are described at DIO 1.2 [1991] n.126 [p.125].)
Schaefer's by-now-notorious 2005 paper was — ere mirthful April Fool's Day collapse — touted as the 1st (convincing-to-archons) proof of Hipparchos' authorship of the Ancient Star Catalog; this, despite the previous existence (since Tycho) of multiple reliable proofs of the point, to the near-unanimous satisfaction of 4 centuries of astronomers.
Aubrey Diller's greatest discovery was establishing 2nd century BC use of sph trig, by showing a sph trig function's neat fit to a given set of (now) THIRTEEN (Hipparchos-Strabo) klimata data. Jones 2002 says nothing in favor of said discovery; but he seems to believe (p.17) — on the basis of a non-jelling (DIO 11.1 [2002] [p.26]) and literally baseless (DIO 5 [2009] n.25 [p.9]) fit to ONE utterly-misunderstood datum — that he has established it himself. (Full details on baselessness will appear if necessary in upcoming DIO 16 [2009] §F.) And the DIO 4.2 [1994] (p.56 Table 1) proof that including normal ancient 5' rounding ensures Diller's victory — producing a nearly (now fully, as of 2009/4/1) PERFECT fit to all 13 data — is not even cited. H.Thurston & DR were shocked at this — but direct 2002 contact proved fruitless. Nonetheless, hoping for Jones' refereeing-input and his responsiveness to evidence, DIO promptly informed him by email that DR's 2009/4/1 fresh look at Strabo 2.5.36 had found that one of the only 2 reasonable interpretations of this Strabo passage had brought the sole once-apparently-nonfitting datum into precise accord with Diller-DR. (DIO's email included the text of DIO 5 [2009] n.25 [p.9] — entire issue still revisable since it and vol.16 were not yet mailed out even as of 2009 early November.) Evidently DIO's attitude of welcoming skeptical checkups (by, e.g., A.Jones & D.Duke) of DR's work — and our thankfulness for the resultant corrections of our (very occasional) errors — is not exactly contagious in opposition quarters. (See, e.g., our grateful, congratulatory front-cover DIO 11.2 [2003] delight, welcoming the latest wonderful Jones-Duke discovery, this one undoing an unhistorical DR view.) Instead, in response to the 2009 April email, Jones kept denigrating Diller's find and Strabo's numerical accuracy, somehow missing an unmissably-glaring major advance here, namely, that Strabo's accuracy is now spectacularly & 13-fold-perfectly redeemed. The solution has been revealed not by guess but by mathematical analysis. (Similarly: the DIO math-test that discovered the rocks in JHA Assoc.Ed. J.Evans' head.) And this in a case whose perfect solution is far, far more fragile than hitherto understood. See DIO 16 [2009], e.g., Fig.1 & caption [p.24]. (Published at end of 2009.)
To set all this in perspective: Jones 2002 p.17 explains-away the non-fit of his attempted derivation of the origin of obliquity ε = 23°51'20" as due to trigonometric “imprecisions”, evidently seeing no significance — not to mention humor — in such patent dodgeball, especially when competing with a SIMPLER theory that scores over a DOZEN on-the-nose klimata-fits without any recourse to such convenient excuses for non-match.
An odd false-appearances scenario (of a sort we realize is required in more familiar theologies). It only looks like DIO is right and JHA wrong; but that's just an illusion — something akin to the devil's work. A tightly-focused maleficent force must have sullied what would-surely-have-been a perfect match for Jones' theory — and (to spoof Jones 2002 n.9) sly satan even more fiendishly polluted the Diller-Neugebauer-DR klimata-data-base in just such a way (by adding a secret 100 stade constant onto all klimata) as to obscure the now-buried truth (which 7y of subsequent research has failed to recover, despite the stimulus right along of a $1000 DIO van der Waerden Award for anyone who finds a better solution here than Diller-DR's for the klimata) while creating a false data-set that happens to hit the bulls-eye: a target MUCH more microscopic than JHA's archons knew when gleefully publishing the 2002 fantasy. [Ever heard of an elephant challenging a mouse to a pistol-duel? — or, better, a dueling elephant who imagined his mouse-opponent was as big as an elephant?] Thus creating-by-pure-wild-flukish-luck a PERFECT fit to a sph trig model based upon [i] satan's constant, and [ii] the only accurate obliquity we (by independent verification: DIO 5 [2009] §D3 [pp.8-9]) know was anciently used and was Hipparchan.
Let's be clear about this: there is no harm and much benefit in attacking by looking for an alternate theory, especially when entertaining in-passing such an imaginatively exploratory one as we see in Jones 2002 (though its untenability should have been obvious long before publication). Indeed, Jones & Duke assisted DR magnificently, by finding a valid alternate theory, displacing an invalid 1987 DR theory. The problem here is the failure to own up when it is one's own attack that is invalid — and in this case assists in postponing immortal credit to a highly eminent fellow philologist.
Since Jones 2002 could show only helter-skelter predictive hits on klimata (vs all klimata for Diller), what was the point of publishing it? The paper too closely resembles classic Muffia try-anything sand-in-eyes harassment (of non-Muffia scholarship) to permit our avoidance of such an otherwise-uncomfortable question.
NB. Jones' over-the-shoulder fleeing of engagement on the klimata issue is not the only evidence that he is consciously engaging in academic misbehavior:
[a] Why is he the only one of the four participants (Diller, Neugebauer, DR, Jones) in the klimata debate who has not published a table of his theory's fit to the data? — which would of course immediately reveal its invalidity.
[b] He has repeatedly non-cited the superior-fit DR edition of Diller's table: originally at DIO 4.2 [1994] p.56; now perfect-fit at DIO 16 [2009] ‡3 Tables 1&2 [pp.20-21] and Fig.2 [p.24].]

Though having written his above-cited prominent 2002 paper on the Diller matter, Jones now (following DIO's email) made no attempt to converse with DR, and just (2009/4/9) brushed off the whole issue as minor. A particularly ironic contention:
[a] It is fruitless to gratuitously insult the import of Diller's obviously major contribution to the history of mathematics.
(Has DR's involvement become the insuperable impediment? Has once-seemingly neutral Jones adopted the path mapped out at DIO 2.3 [1992] ‡6 n.14 [p.93]? See also ibid §E1 [pp.93-94].)
[b] The issue was unminor enough for Jones to have volunteered as the sole scholar in 1/3 of a century (1975-2009) publicly to attack Diller's theory, on what he (incredibly) regarded as substantial grounds, rushing this into JHA print at the very moment when Isis was seeking refereeing (from, e.g., hist.astron mogul and DR's indefatigable denigrator O.Gingerich of JHA) for the eventual 1st-ever hist.sci-journal citation (H.Thurston Isis 93.1:58-69 [2002] p.67 n.18) of the very DIO 4.2 [1994] Table 1 [p.56] which the Muffia cult had hitherto successfully pretended didn't exist. Jones' paper continued this shun-sham anyway, during its failed try at finding the grail of a plausible Alternate Explanation to Diller-DR's.
As Thurston's Isis paper was circulating in 2001, several scholars finally began (note parallel to a prior similarly-timed Muffia tactical switch from omertà to counter-attack: DIO 2.3 [1992] §E3 [pp.93-94]; and DIO 4.3 [1994] ‡15 §§D6&F4 [pp.126-127&130]) trying to-the-max to puncture every one of the numerous DR discoveries cited in the Isis paper. These probes (which should have been forthcoming collegially years earlier — not instead as a seething last-ditch effort to squelch hated heresy) fortunately resulted in circulated attacks on DR's proposed solutions for the Almajest Mars-Jup mean motions and for the Hipparchos klimata. The former attack (Jones-Duke) was valid, the latter invalid. (Note that the fact that DR's many other discoveries survived all 2002 attacks hasn't since enlightened or un-seethed a single Muffioso. I.e., if a math-check were negative, that would be reported. If positive, nothing is said. Needless to say, that is not DIO's approach to its opposites.)]

Desperate-Institutional Ethics Where the Shun Don't Shine:
The Jones 2002 paper under discussion appeared in the JHA of pol's-pol Gingerich, whose genius is such that, even as he nears age 80y, he can super-psychically, with absolute certainty (eternally self-circumscribed by inescapable shunner-improvidence: DIO 6 [1996] ‡1 §J3 [p.25]) perceive that every single contribution in DR's lifetime of discovery is worthless — since each must BETTER have an Alternate Explanation.
Including even all future DR discoveries. Sound incredibly weird? Yes. But consider: how else can a shunning be made to stick? Especially when the repulsive art of shunning is so deeply dark that its chief designers modestly will not even (publicly) take credit for its martially-strict maintenance.
[A creditable exception should be noted: in 2005 (Archimedes, “Wrong for All the Right Reasons”), Jones fairly cited and supported DR's already-universally accepted and British Museum-displayed surprise discovery that cuneiform text BM55555's yearlength was based on Greek observations, thus providing the hour (dawn) of Hipparchos' 135 BC solstice.]

A Chimeral Castle-in-the-Air & a Trip From Sane to Zane:
J.Evans' 1987 JHA monster-length double-Pb papers provided yet another link in the decades-long chain of an ever-subsidized tradition of shots at making-real that alluring castle-of-eliminations-of-every-DR-result that has hovered tantalizingly for 1/3 of a century in the shunny air between frustrated OG's ears — a project which Jones alone has ever been able to lay a single actual history-of-astronomy brick of.
Two Evans-vs-Jones distinctions worth noting:
[i] Encouraged in 1994 by non-cultists (the highly able scholars Curtis Wilson & Hugh Thurston), Jones formerly went outside Muffia bounds in assenting to a few DR discoveries; but, as those benign influences gave way to Muffia ones, he appears to have become priority-hypnotized out of the former passing phase.
[ii] Jones 2002 delusionally-aggressively tried (p.17) to replace sane theory with zane theory. (This in an instance in which the two theories were not even mutually exclusive: see 2002-2003 printing of DIO 4.2 [1994] inside front cover n.2 [p.54].) Said folly is beyond even Gingerich's slightly less obvious standard-antiDR gambit, which is merely to create Alternate-Theory doubt (the only recourse, when as always DR's math maddeningly can't be overturned) — not to defy common sense by prosecuting his shaky Alternates as exclusively true. (An example from a different controversy: by simply proposing that all sorts of credibility-stretching theories [other than DR's obvious one] might-might-might be true, Evans 1987 followed his mentor Gingerich's recipe by taking care not to come down in favor of one of his joke Alternates, thus he never committed himself by going so unjustifiably far-out as Jones 2002 has.)
[Another distinction: when Britton (c.2000) told DR he was siding with Neugebauer's authority vs Diller's math it was at least another's authority. Jones' 2009/4/9 brush-off of Diller-DR is based on Jones' idea of the ultimo Authority: Alexander Jones. (Jones 2002 p.17, or DIO 16 [2009] ‡3 §G3 [p.31]: “I believe we have to regard the shadow ratio [(41 4/5)/120] as the more trustworthy datum”.) Jones' self-confidence is sometimes a plus. And DIO is the better for that. But this is not one of those instances.)]

Fast-forwarding from 2002: cultism's then-8y of nonciting Diller-DR's 1994 table has by 2016 become more than doubled (while heading for trebled [at 24y by 2018].)
And nowadays, we are fortunate to be able to announce a miracle of cultist freewill-death&transfiguration: thanks to the JHA's inimitable archonal trio, the seemingly-aging Muffia has grafted-devolved into a hybrid hydra — as mentally-deadicated as ever to its unthinkingly rigid idea-killing exilings — which (with due respect to its somnambulistically obedient immortal-zombiedum) we may dub double-entrendrily:
THE MORPHIA.
[We will continue using the older term in most cases following the present document, for several reasons too obvious to belabor here.]
The very reason why the above-cited 1994 table has been Muffia-Morphia-shunned [for 24y] IS its embarrassing UNminorness for the history-of-ancient-astronomy clique; the table's virtually (now exactly) perfect fit showed more simply, dramatically, embarrassingly, & consistently than any other case (though Ptolemy's Venus double-date Experimentum Crookis comes close: DIO 11.3 [2002] p.70) how densely and-or politically evidence-immune said clique has been. As noted during the 1994 announcement of Diller's vindication through accounting for ancient-standard 5'-rounding (see DIO 4.2 pp.55-57, Table 1), even young children perceive his case's coherence. (Their secret? Genuine neutrality. Which, revealingly, turns out to be much more vital than Muffia semi-numeracy for discerning the truth of the issue.)
From DIO 11.3 [2002] ‡6 n.12 [p.73]: the Muffia cult keeps insisting (ibid n.13 [p.73] & DIO 1.1 [1991] ‡3 §D [p.20]) “that the [Ptolemy Controversy]'s losers are the true experts, and the winners are mere cranks … all this, while themselves convincingly imitating the key feature of cranks: clinging to long-held opinions despite avalanche after avalanche of evidence against them.”

On 2009/9/30 DR left two messages on Jones' phone-mail, asking that we chat, emphasizing hopes of producing amicable agreement on the klimata issue and wishing that he would consider at last publicly acknowledging the merit and strength of Diller's original discovery. (The fruit of a scholar as creative as himself. Given Jones' [formerly] deserved eminence, his recognition of this would [outside the eternally ineducable: Old Guard Muffiosi & aHoly Trinity atop the JHA] end the Diller-Strabo pseudo-controversy at a stroke — and [for this history] an atypically amiable stroke.)
Jones was simultaneously informed of the uploading of an early version of the present document (the linking of which was postponed until 2009/11/3), which delineates the statistical situation of Diller-vs-Jones.
Jones emailed DIO that he'll be busy for some weeks.
Disappointing so far. Anyone can make mistakes. (DR certainly does!) But persistent failure to acknowledge them goes beyond mere error (especially in the context of ALL associates' like rigidity). It ultimately disallows blaming others (pseudo-refs in this case) and concentrates attention on the perp's character — as regards, e.g., humility, self-confidence, & etc. I.e., it is questionable whether the 2002 paper can much longer merit even a tenuous defense painting it as just a mistake.
(Was it ever credibly thus evaluable? See DIO 11.3 [2002] ‡6 n.20 [p.75]. Or did it constitute a made-man cult-initiation requirement? Does academic advancement in some fields actually require going counter to academic principles?)

Simplifying the Math Required to Evaluate
Jones' Gratuitous Attack on Diller

Given the cited journals' problems with bivariate statistical analysis, it may be a mercy to lay out a fresh, direct method that requires no knowledge of how to compute a least-squares problem analytically. And, some good news for the uninitiated: computing a two-unknown probability P is in some respects easier than for any other number of unknowns.
[It helps that when computing probability P from probability density pd for even numbers of unknowns, we do not need to use the difficult error-function. Here, throughout, we compute P as the cumulative probability exterior to the locus of points with the same pd as the point of interest — exactly as is routinely done for 1-dimensional statistical analysis.]

Most textbooks give a formula for the bivariate probability density pd that is not only messy but requires specialized analytic investigation to determine the inputs: the two unknowns' standard deviations and their correlation. So some (especially historians & classicists with non-extensive math backgrounds) may find it a pleasant surprise to learn of a clear, accurate, & efficient method for finding P while avoiding all that bother & clutter.

We have a set of N values of a variable j for specified values of an independent variable t, and there is a proposed function f(x,y;t) that is intended to fit the values of j as closely as possible — for the best choices of two unknowns: x&y.

We compute function f 's value corresponding to each of the N values of t for which values of j are available. Each difference between a datum j and the computed function f(x,y;t) [which is supposed ideally to equal j] is called a “residual” Δ, with sign-convention such that Δ = j − f. The probability of any given pair of values of x&y is measured by the sum S of the residuals-squared:

At the x&y where S is minimized (values which could be found by trial without sophisticated analysis, as already noted), a subscript m is appended to x&y and to S.

Our quick&undirty exact solution-for-amateurs will start with a trivial calculation, finding the unknowns' normalized deviations from their best fitted values, via the relative Difference D between the sum S at the point of interest (x,y) vs that (S_m) at S's minimum (best-fit):

Then our solution for P is simply:

where (again) N is just the number of data. That's all. It's that simple. And it's as accurate as any other method.

[It should be noted that all standard procedures must find the foregoing quantities (S & its minimum, etc) anyway, en route to their more elaborate solutions — e.g., finding standard deviations σ, correlation(s), pd, and so on. So if finding P is the goal (and it is obviously the main one), our equation makes P easily accessible at the outset, obviating any necessity for getting into complexities beyond. (The main advantage of sophisticated least-squares analysis is that, without trial&error's tedium, it will discover x_m, y_m, S_m, P, and all σs & ρs.)]

Application:

An example always helps. So let us examine the above-cited infamous case of the Hipparchos-Strabo klimata, data which are discussed and explanatorially tabulated in DIO 4.2 [1994] [pp.55-57] (Table 1); DIO 5 [2009] (Table 0 [p.7], seven Diller-predictive-success confirmatory items 1-7 [pp.8-9], n.25 [p.9]); and DIO 16 [2009] ‡3 (Tables 1&2 [pp.20-21]). These data Aubrey Diller discovered in 1934 (Klio 27:258-269 [1934] p.267) were anciently calculated via sph trig for obliquity ε = 23°2/3.
No modern scholar or extant ancient ms had previously recognized either [1] this obliquity or [2] the method by which Strabo's data had been calculated. (Diller's late-Hipparchan obliquity 23°2/3 is the most accurate value known to have been used by any ancient astronomer.)

Predictivity details: Since Diller 1934, two new Hipparchos-Strabo klimata have surfaced. Both are in EXACT accord with Diller's sph trig formula. During the same 75y period, several independent evidences for obliquity 23°2/3 have appeared, adding yet further predictivity-successes to the Diller scorecard. No other theory has produced any.

Yet, in one of the three statistically inelegant papers cited earlier here, A.Jones — to whose knowledge & creativity DR has substantial debts (e.g., DIO 11.1 [2002] ‡3 §§D1-2 [pp.22-23], and DIO 11.2 [2003] cover etc; DIO 14 [2008] ‡3 §I2 [p.44]) — proposes (JHA 33:15-19 [2002]) the following typically original theories:
[a] that Diller's obliquity ε should have been not 23°2/3 but 23°51'20", and
[b] that a hitherto-undetected constant A might have been added in antiquity to all Hipparchan klimata.
In the following response to this strange proposal, we will cooperatively treat ε as an unknown and (to test Alexander Jones' shift-speculation) add another unknown A to the anciently standard sph trig equation for latitude L (in degrees) as a function of longest-day M (in hours), thus:

Comparing this function to the Hipparchos-Strabo klimata data: we seek the best-fit choice of ε&A, as well as a way of gauging the probability of all other ε&A pairs. Here, L, M, ε, & A take the respective rôles of j, t, x, & y in our earlier explanatory paragraph. The data L are the latitude values given by Strabo, for 13 distinct klimata M, ranging from 12h3/4 to 19h.

A table is provided (below) of the Hipparchos-Strabo klimata latitude L data (in stades, consistently reported by Strabo in hundreds of stades), adjacent to competing theories' computed L (rounded here to 1 stade) and their residuals Δ, with sign-convention Δ = Hipparchos-minus-calculation.
The last rows of the table display residual-sum S, normalized r (see below), and probability P:

Klima	M	Hipp L	best-fit L	best-fit Δ	Diller L	Diller Δ	DIO L	DIO Δ	JHAL	JHA Δ	Princttt L	Princttt Δ
Cinnamon	12h3/4	8800	8812	−12	8824	−24	8808	−8	8848	−48	10202	−1402
Meroë	13h	11600	11602	−2	11609	−9	11608	−8	11611	−11	12800	−1200
Syene	13h1/2	16800	16797	3	16797	3	16800	0	16764	36	17569	−769
LowerEgypt	14h	21400	21399	1	21394	6	21408	−8	21338	62	21800	−400
Phoenicia	14h1/4	23400	23469	−69	23462	−62	23450	−50	23398	2	23726	−326
Rhodos	141/2	25400	25388	12	25380	20	25375	25	25309	91	25531	−131
Hellespont	15h	28800	28798	2	28788	12	28817	−17	28711	89	28800	0
Massalia	15h1/4	30300	30305	−5	30295	5	30275	25	30216	84	30273	27
Pontus	15h1/2	31700	31692	8	31683	17	31675	25	31604	96	31644	56
Borysthenes	16h	34100	34144	−44	34135	−35	34125	−25	34057	43	34100	0
Tanais	17h	38000	37981	19	37974	26	37975	25	37903	97	38000	0
S.LittleBritain	18h	40800	40752	48	40746	54	40775	25	40685	115	40800	0
N.LittleBritain	19h	42800	42761	39	42758	42	42758	42	42705	95	42800	0
Residual-Square Sums S (stades squared):				11279		12084		8495		73470		4284816
Residual-Square Sums S (arcmin squared):				82.86		88.78		62.24		539.8		31480
Isotropic Normalized Deviation r:				0		0.89				7.8
Probability P:				1		0.674				10−13

One easily sees that the column which perfectly matches Strabo's figures (to the 100-stade precision of his reportage) is the DIO one, which realizes (e.g., DIO 4.2 [1994] p.55 n.6) that each L in degrees had previously (before conversion to stades at 700 /degree) been rounded to the nearest 1°/12 (5'), according to ancient geographical convention for klimata (Geographical Directory 1.23).
[The ancient calculator appears to have achieved extreme precision. (Quite a revelation, considering that we are discussing trig tables virtually at what has over-confidently been regarded as the time of trigonometry's invention.) When carrying out the scheme Diller discovered: an L error of merely 1" could have ended up misplacing the 18h klima by 100 stades, since application of the sph trig equation for M = 18h produces L = 58°12'31". (Or 40746 stades raw, vs 5'-rounding to 58°1/4 = 40775 stades or 40800 stades by DIO's theory. See both entries in the 18h row of the table.) The residuals of the DIO solution are not converted into probability since (as the table shows) their S is actually 25% smaller than the best-fit S_m for unrounded L data!
(Choosing 5' for rounding-precision, might be regarded as a 2nd unknown for the DIO solution; but this is an attested standard ancient value, so actually it's a known.)
At the other extreme, the grotesquely outsized S of the long-sancrosanct Princetitute scheme occurs despite Neugebauer's manipulation of a revolver-complex of four unknowns (vs Dillers' one — thus giving Diller's theory a clear Occamite preferability). Neugebauer's four unknowns are the coefficients of his proposed cubic-polynomial solution, L = 50(M³ − 62M² + 1307M − 8454). (Computations for the Princetitute theory should use F = 13 − 4 = 9, not 12, as for Diller's 1-unknown theory; nor 11, as for Jones' 2-unknown theorizing. But, when comparing, keep in mind that the higher-dimension normal hyper-surface for fitting the cubic is nothing like that for the Diller sph trig solution.)
If we solve by least squares [which was perhaps beyond Neugebauer's experience] to find the 4 best-fit cubic-polynomial coefficients, we find that (though far better than the Princetitute cubic) even with its 4 unknowns it leaves a σ (32 stades) about the same as that of the 2-unknown Diller-Jones function and slightly inferior to that of a 1-unknown fit to Diller's pure sph trig function (31 stades) Going quartic does not improve matters due to F 's shrinking from 9 to 8.) But of course resorting to ever-higher order polynomials can tighten σ more than the lower orders.
This is because of a crucial point that will make clear to any experienced mathematician that the Diller-DR final theory is valid: the reason that high-power polynomials (quintic or higher) are needed to get σ below 30 stades is that sufficient numbers of unknowns allow some threading through the local roughnesses caused by roundings, since no smooth function can overcome the kinks in the data-curve that correspond to the two anciently-conventional roundings (100 stades and 5') that inject ripples into the pure sph trig equation's curve. These kinks (deviations from smoothness) are about 0°.1 at 2 rough points: [a] From M = 14h1/4 to M = 14h1/2, and [b] from M = 16h to M = 18h. Each kink is primarily due to one of the two klimata which failed for the original Diller scheme (but was salvaged by the introduction of accounting for ordinary ancient 5' rounding, which Diller 1934 suspected was a potential source of non-fits): M = 14h1/4 (DIO 4.2 [1994]) n.10 [pp.56-57]) and 18h. From sixshooter 6-unknowns (quintic polynomial: σ = 11 stades) on up [none of which, as Keith Pickering notes, has anything to do with how ancients would solve the klimata problem!], σ is suddenly much tighter. I.e., it requires a six-unknown polynomial (to match the data's general curve and additionally snake through its rounding-kinks) in order to exceed Diller's plain 1-unknown solution. (A 6th order polynomial fits all klimata within 2 stades. By the time we get to a 12th-power polynomial with 13 unknowns, we of course get a perfect fit, but σ will be indeterminate since the equation for it will have null numerator & denominator, and — having thus eliminated over-determination — we have made the problem a no-longer-statistical one.)]
I.e., a philologist chose a better function (to fit to the klimata data) than did an eminent Ivy League & Princetitute mathematician, the Muffia's ultimo-gooroo: O.Neugebauer. (Who multiplied his initial error by not listening to the wiser chooser — and then abusing [and branding as a fool] the latter scholar — for the offense of disagreement.) One can see why admitting the folly and the offense has been harder for the Muffia's Ivy Leaguers & Princetitooters than submitting to torture. (Unless torture entails debating DR.)
NB: The Neugebauer-Princetitute cubic-polynomial theory was promoted for over 2/3 of a century as the highest wisdom on the matter entirely because it ascribed the klimata to Babylonianesque math (Neugebauer HAMA 1975 pp.305-306 & 334; earlier: Exact Sciences in Antiquity 2nd Ed. BrownU 1957 pp.157&176), though nothing in Babylon's voluminous extant records mentions klimata. (See, e.g., ibid p.158.) Indeed, Babylon's astrologer-priests didn't even have plane trig, much less the sph trig required to compute klimata.

From the numbers in the table, let's get an idea of just how outré the Princetitute solution is. If we artificially compare this 4-unknown solution's S (4284816 arcmin squared) to the 2-unknown minimum (11279 arcmin squared), we have D = 4284816/11279 − 1 ≈ 379. So the artificial normalized probability density is:

In other words, if we use the probablity density at the 2-unknown minimum-point as a standard (though it corresponds to a solution that is distinctly inferior to DIO's), the probability density for Neugebauer's Babylonianesque scheme is smaller by a factor of 1 followed by 740 zeroes. That's:

A more generous and less hybrid approach would not ask the lavish 4-unknown Princetitute cubic polynomial solution to match Diller-DR's sph trig sharp-shot Derringer — but instead just compare it to the best of its own type & no. of unknowns. As already noted, the Princetitute S = 4284816 arcmin squared.
However, the cubic polynomial that best fits the 13 klimata data reduces the residual-square sum (in arcmin squared) to S_m = 15995. So

And we know that F = 13 − 4, thus