Coin catalogue sections: |
Kelenderis, Fourrées, Nagidos, Fourrées |

Coin corpus datasets: |
Kelenderis, Fourrées, Nagidos, Fourrées |

### Summary

The weights of fourrées that imitate the official staters of cities of Kelenderis and Nagidos are examined. The analysed specimens of fourrées imitating Nagidos staters are mostly heavier than the analysed specimens of fourrées imitating Kelenderis staters: the differences between the average and median weights of these two groups are 1.29 g and 1.50 g, respectively. This surprising fact is due to the different weight characteristics of the Kelenderis fourrées classified in Types 2 and 3. These two types (especially Type 2) are characterized by a square incuse on the reverse. Perhaps a different technology was used in their production or they come from a different unofficial mint than the other fourrées (the fact that they come from an unofficial mint is evidenced by the low quality of the dies).

### Analysis

Descriptive statistics of all fourrées and the Kelenderis and Nagidos fourrées are presented in Table 1, box plots1 of the Kelenderis and Nagidos fourrées are shown in Figure 1.

Statistics | All fourrées | Kelenderis | Nagidos |
---|---|---|---|

Number of coins: |
28 | 21 | 7 |

Mean: |
7.63 | 7.31 | 8.60 |

Standard deviation: |
1.46 | 1.53 | 0.60 |

Interquartile range: |
2.30 | 1.93 | 0.77 |

Skewness: |
-0.21 | 0.24 | -0.74 |

Kurtosis: |
2.04 | 2.10 | 2.42 |

Minimum: |
4.87 | 4.87 | 7.53 |

25th percentile: |
6.54 | 6.17 | 8.22 |

Median: |
7.67 | 7.24 | 8.74 |

75th percentile: |
8.84 | 8.10 | 8.99 |

Maximum: |
9.90 | 9.90 | 9.29 |

**Table 1:** Descriptive statistics of coin groups

**Figure 1: **Box plots of the Kelenderis and Nagidos fourrées

These results suggest that fourrées imitating Nagidos staters were usually heavier than fourrées imitating Kelenderis staters. Indeed, the two-sample Kolmogorov-Smirnov test rejects the hypothesis of the equality of the weight distributions of the Kelenderis and Nagidos fourrées (p-value of 0.028). Similarly, the one-sided Welch’s t-test2 rejects the null hypothesis that the mean weights of both groups are equal in favour of the alternative that the mean weight of the Kelenderis fourrées is less than that of the Nagidos fourrées (p-value of 0.002).

A closer look at the corpus of the Kelenderis fourrées, however, shows that most of their lightweight specimens belong to Types 2 and 3, which are characterized by a square incuse on the reverse (for Type 3 it is not certain, but quite likely). Table 2 presents basic descriptive statistics (Std. Dev. denotes the standard deviation and IQR the interquartile range) and Figure 2 shows box plots when the Kelenderis fourrées are divided into two groups. Types 2 and 3 appear to form a separate group within the Kelenderis fourrées. Perhaps a different technology was used in their production or they come from a different unofficial mint than the other fourrées (the fact that they come from an unofficial mint is evidenced by the low quality of the dies).

Note that Table 2 and Figure 2 indicate that the Kelenderis fourrées without Types 2 and 3 are slightly heavier than the Nagidos fourrées. However, the very low number of observations (6 Kelenderis fourrées without Types 2 and 3, and 7 Nagidos fourrées) does not allow us to draw any conclusions. As expected, the one-sided Welch’s t-test does not reject the null hypothesis that the mean weights of both groups are equal (p-value of 0.166).

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

Kelenderis, Types 2 and 3 |
15 | 6.59 | 6.78 | 1.02 | 1.64 |

Kelenderis, other types |
6 | 9.10 | 9.51 | 1.04 | 1.19 |

Nagidos |
7 | 8.60 | 8.74 | 0.60 | 0.77 |

**Table 2:** Basic descriptive statistics when divided the Kelenderis fourrées into two groups

**Figure 2: **Box plots when divided the Kelenderis fourrées into two groups

The following two charts visualize the weight distributions of fourrées of these two cities in greater detail. Figure 3 presents relative frequency histograms (the bars represent the relative frequencies of observations ranging from 4.50 to 10.00 g in increments of 0.50 g), with the continuous curves representing approximations of the data by the Weibull distribution3 based on maximum likelihood estimates (one common curve for all the Kelenderis fourrées). Cumulative distributions are shown in Figure 4. In both charts, Types 2 and 3 of the Kelenderis fourrées are distinguished by a lighter blue colour.

**Figure 3: **Relative frequency histograms of the Kelenderis and Nagidos fourrées

**Figure 4: **Cumulative distributions of the Kelenderis and Nagidos fourrées

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 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 0.028).

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 Kelenderis and Nagidos fourrées, respectively:

*a*: 7.924, 8.841;

*b*: 5.346, 20.436.

10 October 2024 – 16 October 2024