Coin catalogue section: |
Kelenderis |

Coin corpus datasets: |
Kelenderis, Third staters |

### Summary

Third staters can be divided into two groups. The first consists of Types 4.1–2 (Group 4A) and the second of Types 4.3–10 (Group 4B), with the second group characterized by a slightly lower weight standard than the first group and at the same time a higher dispersion of coin weights. The difference in the average observed weights of these two groups is 0.09 g, and the difference in medians is 0.07 g. However, due to the small number of observations, this segmentation of third staters cannot be considered completely certain and the only possible one, and a larger sample of data is needed to confirm or refine it.

### 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. Type 4.5 is represented by two specimens, one of which has an exceptionally low weight of 2.81 g (coin no. 17), which is quite different from the weights of the other coins. This coin, however, appears to be well preserved and there is no reason to doubt its authenticity and has therefore not been excluded from this weight analysis.

**Figure 1: **Box plots of individual coin types

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

4.1 |
1 | 3.59 | 3.59 | 0.00 | 0.00 |

4.2 |
5 | 3.55 | 3.56 | 0.04 | 0.07 |

4.3 |
3 | 3.52 | 3.50 | 0.04 | 0.05 |

4.4 |
7 | 3.48 | 3.50 | 0.09 | 0.14 |

4.5 |
2 | 3.19 | 3.19 | 0.53 | 0.75 |

4.6 |
3 | 3.47 | 3.46 | 0.08 | 0.12 |

4.7 |
1 | 3.45 | 3.45 | ||

4.8 |
10 | 3.49 | 3.49 | 0.09 | 0.10 |

4.9 |
1 | 3.48 | 3.48 | ||

4.10 |
1 | 3.49 | 3.49 |

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

As Figure 1 and Table 1 show, the weight standard of Types 4.1–2 aappears to be higher than that of the other types. Third staters can therefore be divided into the following two groups:

Group 4A: |
Types 4.1–2; |

Group 4B: |
Types 4.3–10. |

However, due to the small size of the data sample, this segmentation is still preliminary. Note, for example, that weights of two coins of Type 4.8 exceed the highest weight observed in Group 4A. These are coins nos. 28 and 31 weighing 3.64 g and 3.62 g, respectively, while the heaviest coin of Group 4A weighs 3.61 g (coin no. 6). Perhaps this can be explained by the fact that less attention was paid to the weight of the flans of this type.

Table 2 shows the descriptive statistics of all third staters and of Groups 4A and 4B.

Statistics | All coins | Group 4A | Group 4B |
---|---|---|---|

Number of coins: |
34 | 6 | 28 |

Mean: |
3.48 | 3.56 | 3.47 |

Standard deviation: |
0.14 | 0.04 | 0.15 |

Interquartile range: |
0.10 | 0.08 | 0.10 |

Skewness: |
-3.37 | -0.17 | -3.18 |

Kurtosis: |
17.07 | 1.59 | 15.05 |

Minimum: |
2.81 | 3.51 | 2.81 |

25th percentile: |
3.46 | 3.51 | 3.44 |

Median: |
3.50 | 3.56 | 3.49 |

75th percentile: |
3.56 | 3.59 | 3.54 |

Maximum: |
3.64 | 3.61 | 3.64 |

**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 2.80 to 3.65 g in increments of 0.05 g). The continuous curves represent approximations of the data by the Weibull distribution2 based on maximum likelihood estimates. Cumulative distributions are shown in Figure 4.

**Figure 2: **Box plots of Groups 4A and 4B

**Figure 3: **Relative frequency histograms of Groups 4A and 4B

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

The two-sample Kolmogorov-Smirnov test rejects the hypothesis of the equality of the weight distributions of groups 4A and 4B (p-value of 0.018). Similarly, the one-sided Welch’s t-test3 rejects the null hypothesis that the mean weights of both groups are equal in favour of the alternative that the mean weight of Group 4A is higher than that of Group 4B (p-value of 0.004). The percentile bootstrap method was also used to supplement these results. Table 3 shows the observed sample medians and bootstrap 95% confidence intervals.4 Table 4 shows the difference in sample medians, its bootstrap 95% confidence interval and p-value.5 Overall, we can conclude that the weight distributions of groups 4A and 4B are significantly different at the 5% significance level, but due to the small number of observations, this segmentation of third staters cannot be considered completely certain and the only possible one.

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

Group 4A |
3.56 | 3.51 | 3.60 |

Group 4B |
3.49 | 3.46 | 3.51 |

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

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

Group 4A vs Group 4B |
0.07 | 0.01 | 0.12 | 0.009 |

**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 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 4A and 4B, respectively:

*a*: 3.577, 3.516;

*b*: 111.159, 41.356.

3The two-sample t-tests with the effective degrees of freedom approximated by the Welch–Satterthwaite equation. The variances of these two groups are significantly different at the 5% significance level (the p-value of F-test is 0.010).

4Wilcox 2022, pp. 122–3. The number of bootstrap samples was 10^{6} (one million) for each group.

5Wilcox 2022, pp. 196–7. The number of bootstrap samples was 10^{6} (one million) for each comparison.

8 July 2023 – 21 October 2024