Coin catalogue section: |
Nagidos |

Coin corpus datasets: |
Nagidos, Staters – Group 1, Nagidos, Staters – Group 2, Nagidos, Staters – Group 3 |

### Summary

In the preliminary coin catalogue, the Nagidos staters are divided into three groups. The coins of Groups 1 and 2 were probably minted in the same weight standard, but a lower weight standard was used for the coins of Group 3. The two weight standards differ in terms of average and median weight by about 0.6 g. The exceptions, however, are Type 1.5e in Group 1 and Type 2.8 in Group 2, whose weights correspond to Group 3. These two types were thus probably struck later than the other types of Groups 1 and 2.

### 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 1.5 is divided into variants 1.5a–d and variant 1.5e, because specimens of variant 1.5e differ significantly in weight from other variants of this type and, in addition to the differently placed ethnic, they also differ in lower relief.

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

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

1.1 |
24 | 10.51 | 10.57 | 0.20 | 0.29 |

1.2 |
7 | 10.64 | 10.73 | 0.17 | 0.21 |

1.3 |
41 | 10.55 | 10.55 | 0.19 | 0.25 |

1.4 |
4 | 10.62 | 10.59 | 0.17 | 0.29 |

1.5a-d |
45 | 10.48 | 10.52 | 0.22 | 0.21 |

1.5e |
14 | 9.85 | 9.89 | 0.24 | 0.43 |

1.6 |
2 | 10.13 | 10.13 | 0.39 | 0.55 |

2.1 |
4 | 10.47 | 10.58 | 0.40 | 0.56 |

2.2 |
4 | 10.56 | 10.64 | 0.23 | 0.30 |

2.3 |
8 | 10.59 | 10.50 | 0.23 | 0.32 |

2.4 |
41 | 10.55 | 10.61 | 0.24 | 0.21 |

2.5 |
3 | 10.20 | 10.10 | 0.28 | 0.40 |

2.6 |
169 | 10.66 | 10.68 | 0.15 | 0.12 |

2.7 |
12 | 10.49 | 10.50 | 0.17 | 0.27 |

2.8 |
4 | 9.98 | 9.99 | 0.16 | 0.24 |

3.1 |
14 | 9.91 | 9.96 | 0.37 | 0.28 |

3.2 |
12 | 10.02 | 10.00 | 0.13 | 0.16 |

3.3 |
12 | 10.01 | 10.13 | 0.35 | 0.23 |

3.4 |
47 | 9.88 | 9.97 | 0.26 | 0.36 |

3.5 |
19 | 9.81 | 9.90 | 0.23 | 0.21 |

3.6 |
29 | 9.92 | 9.95 | 0.29 | 0.26 |

3.7 |
43 | 10.04 | 10.15 | 0.26 | 0.21 |

3.8 |
11 | 10.00 | 10.08 | 0.23 | 0.35 |

3.9 |
4 | 10.16 | 10.16 | 0.05 | 0.08 |

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

As shown in Figure 1 and Table 1, similar to Type 1.5e in Group 1, Type 2.8 in Group 2 is distinguished by its lower weight. The average and median weight of these two types correspond to Group 3. From the point of view of the weight standard, we can therefore adjust the catalogue coin groups as follows:

Group A: |
coins of Group 1 except Type 1.5e; |

Group B: |
coins of Group 2 except Type 2.8; |

Group C: |
coins of Group 3 with Types 1.5e and 2.8. |

Table 2 shows the descriptive statistics of Groups A, B and C. Figures 2–4 visualize the weight distributions of these groups: box plots, relative frequency histograms (the bars represent the relative frequencies of observations ranging from 8.70 to 11.10 g in increments of 0.10 g) with the continuous curves representing approximations of the data by the Weibull distribution2, and cumulative distributions.

Statistics | Group A | Group B | Group C |
---|---|---|---|

Number of coins: |
123 | 241 | 209 |

Mean: |
10.52 | 10.62 | 9.94 |

Standard deviation: |
0.22 | 0.19 | 0.27 |

Interquartile range: |
0.26 | 0.17 | 0.30 |

Skewness: |
-1.20 | -1.69 | -1.69 |

Kurtosis: |
5.15 | 6.32 | 6.99 |

Minimum: |
9.64 | 9.89 | 8.71 |

25th percentile: |
10.41 | 10.57 | 9.85 |

Median: |
10.55 | 10.66 | 9.99 |

75th percentile: |
10.67 | 10.74 | 10.15 |

Maximum: |
10.85 | 11.01 | 10.31 |

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

**Figure 2: **Box plots

**Figure 3: **Relative frequency histograms

**Figure 4: **Cumulative distributions

Coins of Groups A and B were apparently minted at a higher weight standard than those of Group C. This is confirmed by the one-sided Welch’s t-test3 which rejects the null hypothesis that the mean weights of Groups A and B are equal to the mean weight of Group C against the alternative that they are higher (p-value less than 0.001 for both tests). In addition, the two-sample Kolmogorov-Smirnov tests reject the hypothesis of the equality of the weight distributions of Groups A and C, and B and C (p-value less than 0.001 for both tests).

Table 2 and Figures 2-4 also show slightly higher weights in Group B versus Group A. This is due to the fact that the data of Types 2.6a and 2.6b contain few low-weight coins, see Figure 5. From this, however, it is probably not possible to conclude that a higher weight standard was used for the minting of these two types, although this cannot be ruled out. The most likely reason is that most of the coins of these two types in the analysed set come from hoards containing freshly struck specimens from good quality metal.

**Figure 5: **Relative frequency histograms of coin types in Group B

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 A, B and C, respectively:

*a*: 10.612, 10.700, 10.055;

*b*: 64.469, 80.807, 53.918.

3The two-sample t-tests with the effective degrees of freedom approximated by the Welch–Satterthwaite equation. The variances of these groups are significantly different at the 5% significance level (the p-value of F-test is 0.005 for Groups A and C, and less than 0.001 for Groups B and C).

14 October 2024