At a glance — the formulae and what they mean

Part one · the date

For one findspot, take every stamp found there and the production range \([a_i,\,b_i]\) of the potter who made it.

1 Where the dates sit

Average the earliest and latest years, then take the centre.

\[ \overline{a} = \frac{1}{n}\sum a_i \qquad \overline{b} = \frac{1}{n}\sum b_i \qquad m = \frac{\overline{a} + \overline{b}}{2} \]

→ avg_datemin, avg_datemax, midpoint_year

2 How scattered they are

Two things blur the date: each potter's range is itself wide, and different potters point to different years. Both are added together.

\[ \sigma = \sqrt{\;\underbrace{\overline{\left(\frac{(b_i-a_i)^2}{12}\right)}}_{\text{each range is wide}} \;+\; \underbrace{\operatorname{Var}\!\left(\frac{a_i+b_i}{2}\right)}_{\text{the potters disagree}}\;} \]

→ \(\sigma\) in years. Small \(\sigma\) = the stamps point to a narrow window.

3 How far to trust it

The more stamps a findspot yields, the tighter the interval that may reasonably be claimed. \(k\) falls from 1.5 to 0.5 as the material accumulates.

\[ k = 1.5 - 1.0\left(1 - e^{-n/\tau}\right) \qquad \tau = 6 \]

→ \(k \approx 1.2\) for two stamps · \(k \approx 0.54\) for twenty · \(k \to 0.5\) beyond about forty

4 The date

The centre, widened by the scatter, scaled by the trust.

\[ \text{eff\_start} = m - k\sigma \qquad\qquad \text{eff\_end} = m + k\sigma \]

→ eff_start, eff_end — the box in the plot

Two constants, easily confused

The model carries two numbers whose symbols look alike and whose values are not alike. They are unrelated, and mixing them up has already cost this project one full set of figures.

	\(\tau\) = 6	\(t_0\) = 20
counted in	stamps	years
controls	how wide the box is drawn	what colour the whiskers are
comes from	calibration against findspots dated without ceramics — Dangstetten, Oberaden, Pompeii, Inchtuthil, and Velsen I shown but excluded as contested. Only Pompeii binds the value; the others leave it headroom	the expert thresholds: about 5 years' scatter is a sharp date, about 25 is unusable

Neither of them is Student's \(t\). Nothing here is a confidence interval, and nothing is being tested for significance. Both values travel with every row of output, as p_tau and p_t0, so any figure can be checked against the parameters that produced it. The long version is on the method page.

What the date means

The interval is a virtual fuzzy year: the central region of the date distribution the stamps imply. Read it as "the material at this findspot clusters here" — not as the foundation and abandonment of the site, and not as a replacement for a date known from a destruction horizon or a written source.

Interval	Read as
narrow, ≲ 20 years	Stamps agree closely and there is enough of them; a usable working date.
moderate, 20–50 years	A generation-scale window. Normal for most settlement assemblages.
wide, ≳ 50 years	Either the potters disagree, or too little material. Check which, using the quality measures below.

Part two · the quality

Two measures, deliberately kept apart. They answer different questions and a findspot may score high on one and low on the other.

Q1 Dating sharpness

Do the potters agree, relative to the length of the span they describe?

\[ q_{\text{interval}} = \exp\!\left(-\frac{\sqrt{s^2_a + s^2_b}}{\left|\overline{b}-\overline{a}\right|}\right) \]

→ q_interval, between 0 and 1 · drives the box colour

Value	Meaning
0.8 – 1.0	The potters point to nearly the same years. Sharp date.
0.5 – 0.8	Moderate agreement. The date is meaningful but broad.
below 0.5	The potters scatter more widely than the interval they define. Treat the date as indicative only.

Q2 Die repetition

Does the same stamp die recur? \(r\) is the number of attestations per distinct potter–die combination.

\[ r = \frac{\text{stamps with a die}}{\text{distinct potter–die pairs}} \qquad\qquad q_{\text{repetition}} = 1 - \frac{1}{\max(r,\,1)} \]

→ die_repetition, q_repetition

Value	Meaning
0.0	Every die occurs once. An ordinary accumulated assemblage.
0.3 – 0.6	Noticeable repetition. Possibly a partly closed group.
above 0.6	Strong repetition — the signature of a hoard, consignment or single delivery.
— (empty)	No dies recorded. Not measured, rather than zero.

Reading the two together

	Low repetition	High repetition
High sharpness	Well-dated settlement context.	Well-dated closed group — the strongest evidence available.
Low sharpness	Diffuse background material.	A closed group of chronologically mixed pottery.

A low repetition score is not a poor result. It says the deposit is not a hoard, which is true of most findspots and says nothing about how well they are dated.

A worked example

Three stamps at one findspot

Stamp	Potter dated	Width \(w\)	Midpoint \(c\)
1	AD 60 – 80	20	70
2	AD 65 – 85	20	75
3	AD 50 – 90	40	70

1 ā = 58.3 · b̄ = 85.0 · m = 71.7
2 within = (400+400+1600)/12/3 = 66.7 · between = 8.3 → σ = √75.0 = 8.7 years
3 n = 3 stamps → k = 1.5 − 1.0(1 − e^−3/6) = 1.11
4 71.7 ± 1.11 × 8.7 = 71.7 ± 9.6

Q1 √(58.3 + 25.0) = 9.1 · span = 26.7 → q_interval = e^−0.34 = 0.71
Q2 3 stamps ÷ 3 dies → r = 1 → q_repetition = 0.00

Date AD 62 – 81 | sharpness 0.71 | repetition 0.00

Reasonably sharp for the material available, and no hoard signature: a normal settlement deposit of the Neronian–Flavian period. The interval is wide chiefly because three stamps is very little — with thirty stamps of the same dates, \(k\) would fall to about 0.51 and the interval would tighten to roughly AD 67 – 76.