Staters

Summary Analysis

Coin catalogue section: Kelenderis
Coin corpus datasets: Kelenderis, staters, Group 1, Kelenderis, staters, Group 2,
Kelenderis, staters, Group 3

Summary

Although the data samples cannot yet be considered fully representative, the basic analysis below so far indicates the following:

  1. Coins in Groups 1 and 2 were probably minted in the same (or very similar) weight standard.
  2. The weight standards in Groups 1 and 2 and for some types of Group 3 are similar. However, within Group 3, two significant decreases in the weight standard can be observed. The first decrease can be observed in Types 3.13–14, the second in Types 3.4–6 and 3.15–17. These two declines divide Group 3 into three subgroups: 3A (Types 3.1–3 and 3.7–12), 3B (Types 3.13–14) and 3C (Types 3.4–6 and 3.15–17). The groups 3B and 3C probably correspond to the penultimate and final phase of the Kelenderis coinage, respectively.
  3. The decrease in the weight standard from the early issues (Group 1) to the last issues (Group 3C) is 7.7% in terms of average weight (from 10.73 g to 9.90 g) and 7.5% in terms of median weight (from 10.79 g to 9.98 g).
  4. Unlike the orientation of the rider on the obverse, the orientation of the kneeling goat on the reverse is not in itself an indicator of phases of the Kelenderis coinage, as both orientations of the goat are found on coins belonging to the final phase of the pre-Hellenistic Kelenderis coinage.

Analysis

Box plots1 of individual coin types and basic descriptive statistics are presented in Figure 1 and Table 1 (Std. Dev. denotes the standard deviation and IQR the interquartile range), respectively.

Figure 1: Box plots of individual coin types

Figure 1: Box plots of individual coin types

Type Count Mean Median Std. Dev. IQR
1.1 2 10.57 10.57 0.02 0.03
1.2 2 10.77 10.77 0.11 0.15
1.3 9 10.76 10.82 0.26 0.17
2.1 1 10.68 10.68 0.00 0.00
2.2 8 10.71 10.71 0.06 0.08
2.3 10 10.76 10.75 0.06 0.11
2.4 10 10.77 10.79 0.07 0.09
2.5 2 10.73 10.73 0.06 0.08
2.6 36 10.72 10.74 0.10 0.17
2.7 11 10.71 10.75 0.13 0.23
2.8 36 10.70 10.71 0.10 0.12
2.9 14 10.78 10.80 0.07 0.06
2.10 11 10.76 10.75 0.14 0.25
2.11 40 10.74 10.74 0.09 0.10
2.12 50 10.72 10.73 0.11 0.16
3.1 29 10.53 10.65 0.44 0.28
3.2 17 10.67 10.68 0.10 0.06
3.3 1 10.40 10.40 0.00 0.00
3.4 3 9.58 9.92 0.63 0.83
3.5 4 10.07 10.05 0.08 0.11
3.6 2 10.04 10.04 0.08 0.12
3.7 52 10.64 10.62 0.16 0.19
3.8 39 10.67 10.71 0.18 0.17
3.9 25 10.68 10.69 0.13 0.13
3.10 5 10.70 10.70 0.07 0.14
3.11 3 10.62 10.69 0.16 0.22
3.12 9 10.62 10.70 0.21 0.06
3.13 9 10.50 10.46 0.20 0.37
3.14 5 10.38 10.29 0.18 0.32
3.15 9 9.95 10.03 0.44 0.33
3.16 6 9.92 9.91 0.09 0.16
3.17 5 9.80 9.85 0.23 0.21

Table 1: Basic descriptive statistics of coin types

As Figure 1 and Table 1 show, Types 3.4–6 and 3.15–17 stand out in Group 3 with their low weights. The common features of Types 3.5–6 and 3.16–17 are dotted borders on the reverse and broader flans. Although Types 3.4 and 3.15 do not have the dotted border on the reverse, they share a slender goat with Types 3.5–6. These six types can thus probably be included in the last phase of the pre-Hellenistic Kelenderis coinage. It is also worth mentioning that in Types 3.4–6 the kneeling goat is oriented to the left, while in Types 3.15–17 to the right. This shows that the orientation of the kneeling goat is not in itself an indicator of phases of the Kelenderis coinage.

Lower median weights can also be observed for Types 3.3 and 3.13–14, which may thus represent the penultimate phase of the pre-Hellenistic Kelenderis coinage. However, since the Types 3.3 is so far represented by a single specimen, it must be left aside for the time being.

Based on these observations, Group 3 of Kelenderis staters can be divided into the following three subgroups:

Group 3A: Types 3.1–3 and 3.7–12, i.e. coins of Group 3 except Types 3.4–6 and 3.13–17;
Group 3B: Types 3.13–14;
Group 3C: Types 3.4–6 and 3.15–17.

Table 2 shows the descriptive statistics of Groups 1, 2, 3A, 3B and 3C. In Group 1, the coin no. 10 (Type 1.3a, ANS Collection 1944.100.53201) stands out from the other coins, weighing only 10.11 g. Although this coin is struck from worn dies, it appears to be well preserved and without signs of serious metal degradation. It was therefore probably indeed struck on a lighter flan.2 In Group 3, especially coins nos. 14 (Type 3.1f), 50 (Type 3.4b) and 204 (Type 3.15b) differ substantially from other coins of the same type, weighing only 8.70 g, 8.86 g and 8.91 g, respectively. Again, these coins seem to show no signs of serious metal degradation.

Statistics Group 1 Group 2 Group 3A Group 3B Group 3C
Number of coins: 13 229 180 14 29
Mean: 10.73 10.73 10.64 10.45 9.90
Standard deviation: 0.23 0.10 0.23 0.20 0.34
Interquartile range: 0.21 0.13 0.18 0.36 0.19
Skewness: -1.54 -0.45 -3.85 0.28 -1.86
Kurtosis: 5.32 3.56 30.31 1.71 6.84
Minimum: 10.11 10.42 8.70 10.20 8.86
25th percentile: 10.65 10.67 10.57 10.25 9.86
Median: 10.78 10.74 10.68 10.43 9.98
75th percentile: 10.86 10.80 10.75 10.61 10.04
Maximum: 11.01 11.02 11.07 10.77 10.48

Table 2: Descriptive statistics of coin groups

The following charts visualize the weight distributions of all these groups. Figure 2 shows box plots. Figures 3 and 4 present relative frequency histograms (for clarity, the histograms are divided into two charts). The bars represent the relative frequencies of observations ranging from 8.70 to 11.10 g in increments of 0.10 g and the continuous curves represent approximations of the data by the Weibull distribution3 based on maximum likelihood estimates. Cumulative distributions are shown in Figure 5.

Figure 2: Box plots of coin groups

Figure 2: Box plots of coin groups

Figure 3: Relative frequency histograms of groups 1, 2 and 3

Figure 3: Relative frequency histograms of Groups 1, 2 and 3

Figure 4: Relative frequency histograms of groups 1, 2 and 3

Figure 4: Relative frequency histograms of Groups 3a, 3b and 3c

Figure 5: Cumulative distributions coin groups

Figure 5: Cumulative distributions of coin groups

These results suggest that the coins from Groups 1 and 2 were probably struck in the same (or very similar) weight standard. In Group 1, Type 1.1 stands out as having the lowest mean and median weight of all the types in Groups 1 and 2 (see Figure 1 and Table 1). However, this type is represented by only two coins, so nothing can be deduced from it. Group 3A shows a small decrease in the weight standard, although the heaviest coins in Group 3A match the weight of the heaviest coins in Groups 1 and 2. This decrease continues significantly in Groups 3B and 3C. This indiciates that Group 3C corresponds to the last phase of the pre-Hellenistic Kelenderis coinage and Group 3B to its penultimate phase. This is consistent with the late style of these coins.

The distributions of coin weights in individual groups have different shapes and, with the exception of Group 2, are asymmetric. Instead of comparing means, it is therefore statistically more appropriate to compare medians. Since the analyzed data have many tied values, the percentile bootstrap method was chosen. Table 3 shows the observed sample medians and bootstrap 95% confidence intervals.4 Table 4 shows the differences in sample medians, their bootstrap 95% confidence intervals and p-values.5 These results confirm the conclusions presented in the previous paragraph. However, it should be emphasized that this is not a statistically significant confirmation at the 5% confidence level, as this would require a multiple testing method for comparison of random samples from different asymmetric distributions.6

median 95% confidence interval
Group 1 10.78 10.67 10.85
Group 2 10.74 10.72 10.76
Group 3A 10.68 10.66 10.70
Group 3B 10.43 10.27 10.61
Group 3C 9.98 9.91 10.02

Table 3: Medians and their confidence intervals

medians difference 95% confidence interval p-value
Group 1 vs Group 2 0.04 -0.08 0.12 0.389
Group 2 vs Group 3A 0.06 0.03 0.09 <0.001
Group 3A vs Group 3B 0.25 0.08 0.42 0.009
Group 3B vs Group 3C 0.45 0.28 0.64 <0.001

Table 4: Differences in medians

 

1The bottom and top of each box are the 25th and 75th percentiles of the dataset, respectively (the lower and upper quartiles). Thus, the height of the box corresponds to the interquartile range (IQR). The red line inside the box indicates the median. Whiskers (the dashed lines extending above and below the box) indicate variability outside the upper and lower quartiles. From above the upper quartile, a distance of 1.5 times the IQR is measured out and a whisker is drawn up to the largest observed data point from the dataset that falls within this distance. Similarly, a distance of 1.5 times the IQR is measured out below the lower quartile and a whisker is drawn down to the lowest observed data point from the dataset that falls within this distance. Observations beyond the whisker length are marked as outliers and are represented by small red circles.

2John Thomassen (Collections Manager, American Numismatic Society), email to author, 13 July 2023.

3The probability density function of the Weibull distribution is f(x;a,b) = b/a×(x/a)b-1×exp(-(x/a)b) for x≥0, and f(x;a,b) = 0 for x<0, where a>0 is the shape parameter and b>0 is the scale parameter of the distribution. The estimated values of the parameters for Groups 1, 2, 3A, 3B and 3C, respectively:
 a: 10.825, 10.777, 10.723, 10.549, 10.034;
 b: 71.387, 116.204, 68.680, 58.051, 42.469.

4Wilcox 2022, pp. 122–3. The number of bootstrap samples was 106 (one million) for each group.

5Wilcox 2022, pp. 196–7. The number of bootstrap samples was 106 (one million) for each comparison.

6If we merged Groups 1 and 2 and tested the decrease in the weight standard in Groups 1+2, 3A, 3B and 3C, then a total of 6 comparisons between these 4 samples would be needed. The Bonferroni correction would be too conservative in this case and I am not yet aware of another suitable method whose assumptions are met for these data.

 

8 July 2023 – 25 February 2024