Interpretaciones y reflexiones de las narrativas de Ana

Creighton firstly discussed what the variability around the mean (i.e., his ‘average’ hoard) might represent and concluded (pp. 71–3):

If there is a wide variation around the mean, i.e., high inter-hoard variability, this must either

imply a slow velocity of coin circulation, or new issues being produced.

If the hoards are uniform, there must either be a fast velocity, or no new issues being produced.

Chapter 9 examines this hypothesis in some detail; here it is sufficient to note that this interpretation is far too simplistic.

18

Cochran’s C test for equality of variances: 0.53, P

=

0.62. Figures fromSTATGRAPHICS.

19_{In his Fig. 23.02 there appears to be only 11 hoards used, and of those three have only four coins between them, and}

3.12. The work of J. D. Creighton 87

Creighton then attempted to quantify variability by the use of Cram´er’s contingency coefficient ‘’ (pp. 73–76). This statistic, usually known as Cram´er’sV

2

(Bishop et al. 1975, pp. 386–387),20 is a measure of association based on the

2

statistic and is basically a scaling of 2

which removes the effects of sample size. Creighton used this statistic as a measure of similarity: if V

2

is near to zero he interpreted this as there being little variation between hoards, ifV

2

is near to 1 there was a high degree of variability between hoards. Creighton divided the hoards into batches of ten years and computedV

2

for each group. The values ofV 2

were then plotted on a graph (Fig. 24.08–24.09).21 _{The details of this technique were given in Appendix 2.43 and the details of the}

contingency tables in Appendix 2.44. Creighton stated that the pattern revealed is (p. 76): 1. Invasion to mid-second century: no systematic picture.

2. Mid-second–early-third centuries: very similar suggesting high rate of circulation.

3. Early to mid-third century: antoninianus introduced, abandoned and then re-introduced. No systematic picture but indication that the circulation rate is slowing down.

4. Very little similarity as the denarius is driven out of circulation.

There are two criticisms of this analysis: one methodological and one numismatic. The method- ological problem lies in the manner Creighton calculatedV

2

. In his Appendix 2.44 the contingency tables used were presented. As would be expected, the number of rows and columns in these tables varied according which decade the table represents. Apart from the first table, Creighton restricted the number of columns to four. However, in combining phases to create the columns, he was not consistent. For example, for hoards closingAD110–119, the columns were Phase A (Republican), B–E (Mark Antony to the Civil War), F (Flavian I) and G–I (Flavian II to Hadrianic), whereas for hoards closingAD120–128, the columns were A–E, F, G and H–I (Trajan to Hadrian). Presumably, the column combinations were an attempt to ensure that the expected value of each cell in the tables was greater than 5, a conservative rule of thumb often used to avoid the

2

test giving erroneous results in cases where there are either small sample sizes or sparse contingency tables. The problem is that the value of

2

, and thus of V 2

, will change according to which columns are combined. For example, the two tables cited above had values ofV

2

of 0.1344 and 0.0860 respectively. If we combine the columns in the two tables so that we only have three, A–E, F and G–I, the value ofV

2

rises to 0.1870 for the first table and 0.1214 for the second. One should also note that the associations appear greater ifV is used, rather thanV

2

. In any case, (Bishop et al. 1975, p. 393) note that “the major difficulty in their [

2

based measures] use is lack of clear interpretation.” The second problem is numismatic in that we are not comparing like with like, e.g., hoards closing just after the conquest can contain coins from 211BC up to their closing date (? AD50);

20

Note, however, that Bishop et al. (1975) actually define

V

, not

V

2

whereas both Shennan (1988) and Iman & Conover (1983) define

V

2

(although in the later case it is called

). Both theSTATGRAPHICSand SPSSpackages, however, provide Cr´amer’s

V

. For the purposes of this section, I will use

V

2

for ease of comparison with Creighton’s results.

21

Creighton produces a similiar graph in his paper on this topic (Creighton 1992b), but in this case the

y

-axis has been plotted on a logarithmic scale to emphasise the difference between the values of

V

2

hoards closing during Trajan can have coins from 211BCtoAD117, and thus coins minted at about the time of the conquest would have had 70 years for their distribution to further homogenise. This is most clear in the latter 3rd century when the debased antoniniani drove out the finer denarii; hoards of that period are much less ‘similar’ according to Cr´amer’sV than previously.

Creighton’s approach has some merit, but is weak in application. Although it may have reduced the number of usable hoards, and would have possibly taken longer to collect the data, a better approach would have been to divide the coins into the same ten year groups into which the hoards were divided. Having done this, Cr´amer’sV could then be calculated for coins dating to the last

few (? four) decades of the hoard. Although this would not remove the problem of the variable number of hoards in each table, it would ensure that the tables were more comparable. At present, Creighton’s results must be treated with extreme caution, and his interpretations more so.

In document La mirada de la reflexión docente a través de narrativas Historiando nuestra comprensión de la conversión del lenguaje retórico al algebraico en el aula (página 93-97)