Coin catalogue section: |
Kelenderis |

Coin corpus datasets: |
Kelenderis, hemiobols |

### Summary

Hemiobols can be divided into two groups. The first consists of Types 6.1–2 (Group 6A) and the second of Types 6.3–6 (Group 6B), with the second group characterized by a lower weight standard than the first group. The differences in the mean and median observed weights of the two groups are 0.06 g and 0.07 g, respectively. However, this segmentation is problematic for several reasons, and to reliably determine the relationships between the weight standards of individual coin types, it is necessary to expand the data corpus.

### 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

Type | Count | Mean | Median | Std. Dev. | IQR |
---|---|---|---|---|---|

6.1 |
20 | 0.36 | 0.38 | 0.06 | 0.09 |

6.2 |
15 | 0.40 | 0.41 | 0.06 | 0.06 |

6.3 |
4 | 0.32 | 0.32 | 0.01 | 0.02 |

6.4 |
3 | 0.34 | 0.36 | 0.03 | 0.05 |

6.5 |
1 | 0.33 | 0.33 | ||

6.6 |
2 | 0.29 | 0.29 | 0.01 | 0.02 |

**Table 1:** Basic descriptive statistics of coin types

As Figure 1 and Table 1 show, the weight standard of Types 6.1–2 aappears to be higher than that of the other types. Third staters can therefore be divided into the following two groups:

Group 6A: |
Types 6.1–2; |

Group 6B: |
Types 6.3–6. |

However, this segmentation is not without problems. First, the weights of the coins of Types 6.1–6.2 have a large variability, and some of these coins correspond in their light weight to Types 6.3–6. This may be the result of corrosion of some specimens, or less attention was paid to the production of flans, or the weight standard was lowered during the minting of these types.. Second, the hemiobols of Types 6.1–5 correspond in design with the obols of Types 5.2, 5.4, and 5.6–8. However, all these types of obols belong to Group 5A characterized by approximately the same weight standard (see the weight analysis of Kelenderis obols), while here the weight distribution of Types 6.3–5 appears to be different from Types 6.1–2. Both of these mentioned facts require further analysis, the prerequisite of which, however, is the expansion of the data corpus.

Table 2 shows the descriptive statistics of all third staters and of Groups 6A and 6B.

Statistics | All coins | Group 6A | Group 6B |
---|---|---|---|

Number of coins: |
45 | 35 | 10 |

Mean: |
0.36 | 0.38 | 0.32 |

Standard deviation: |
0.06 | 0.06 | 0.03 |

Interquartile range: |
0.10 | 0.09 | 0.03 |

Skewness: |
0.13 | -0.26 | 0.38 |

Kurtosis: |
2.16 | 2.36 | 2.12 |

Minimum: |
0.26 | 0.26 | 0.28 |

25th percentile: |
0.31 | 0.33 | 0.30 |

Median: |
0.37 | 0.39 | 0.32 |

75th percentile: |
0.41 | 0.42 | 0.33 |

Maximum: |
0.49 | 0.49 | 0.36 |

**Table 2:** Descriptive statistics of coin groups

The following charts visualize the weight distributions of these groups. Figure 2 shows box plots and Figure 3 present relative frequency histograms (the bars represent the relative frequencies of observations ranging from 0.25 to 0.50 g in increments of 0.05 g). Cumulative distributions are shown in Figure 4.

**Figure 2: **Box plots of Groups 6A and 6B

**Figure 3: **Relative frequency histograms of Groups 6A and 6B

**Figure 4: **Cumulative distributions of Groups 6A and 6B

The Kolmogorov-Smirnov test rejects the equality the weight distributions of groups 6A and 6B (p-value of 0.001). The one-sided two-sample t-test with Satterthwaite’s approximation rejects the null hypothesis that the mean weights of both groups are equal in favour of the alternative that the mean weight of Group 6A is higher than that of Group 6B (p-value less than 0.001).2 The percentile bootstrap method was also used to supplement these results. Table 3 shows the observed sample medians and bootstrap 95% confidence intervals.3 Table 4 shows the difference in sample medians, its bootstrap 95% confidence interval and p-value.4 These values unsurprisingly confirm the higher weight standard in group 6A.

median | 95% confidence interval | ||
---|---|---|---|

Group 6A |
0.39 | 0.35 | 0.41 |

Group 6B |
0.32 | 0.30 | 0.34 |

**Table 3:** Medians and their confidence intervals

medians difference | 95% confidence interval | p-value | ||
---|---|---|---|---|

Group 6A vs Group 6B |
0.08 | 0.04 | 0.10 | 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.

2The weight variances in these two groups are clearly different, and therefore Satterthwaite’s approximation for the effective degrees of freedom was applied. A two-sample F-test rejects the hypothesis of the same variance in both groups at the 5% significance level (p-value of 0.014).

3Wilcox 2022, pp. 122–3. The number of bootstrap samples was 10^{6} (one million) for each group.

4Wilcox 2022, pp. 196–7. The number of bootstrap samples was 10^{6} (one million) for each comparison.

8 July 2023 – 11 December 2023