Submitted by Petr Vesely on

Founded 27-Jul-2003

Last update 12-Dec-2004

Antioch Mint Seleukeia ad Kalykadnon Mint Comparison References

## Antioch Mint

### 1. Examined type

Denomination: |
AR Tetradrachm |

Period: |
c. 95 - c. 94 BC |

Obverse: |
Diademed head of Seleukos VI right; fillet border |

Reverse: |
‘ΒΑΣΙΛΕΩΣ ΣΕΛΕΥΚΟΥ’ right, ‘ΕΠΙΦΑΝΟΥΣ ΝΙΚΑΤΟΡΟΣ’ left; Zeus Nikephoros seated on throne left holding Nike in right hand and scepter in left hand; ‘ΔΙ’ monogram under throne; all within laurel wreath |

### 2. Acceptable weight range

Lower exclusion limit: |
14.75 grams |

Upper exclusion limit: |
16.75 grams |

Each coin is a priori excluded from the data sample if its weight is lesser than the lower exclusion limit or greater than the upper exclusion limit.

### 3. Data

Sorted data (weights in grams):

15.64, 15.77, 15.79, 15.79, 15.83, 15.84, 15.89, 15.91, 15.99, 16.04, 16.06, 16.09, 16.11, 16.14, 16.16, 16.21

**Note:** The following coins were included into the analysis:

- Classical Numismatic Group, Inc.: Auction 61 (Sep 2002), Lot No. 855; Triton VI (Jan 2003), Lot No. 463
- Fritz Rudolf Künker Müzenhandlung: Auction 67 (Oct 2001), Lot No. 456; Auction 71 (Mar 2002), Lot No. 443
- Gorny & Mosch Giessener Münzhandlung: Auction 108 (Apr 2001), Lot No. 1365; Auction 118 (Oct 2002), Lot No. 1522
- Jean Elsen s.a.: Auction 72 (Dec 2002), Lot No. 267; Auction 72 (Dec 2002), Lot No. 268; Auction 73 (Mar 2003), Lot No. 128; Auction 76 (Sep 2003), Lot No. 177
- Leu Numismatik Ltd.: Auction 81 (May 2001), Lot No. 346
- Münzen & Medaillen Deutschland GmbH: Auction 11 (Nov 2002), Lot No. 779; Auction 12 (Apr 2003), Lot No. 119
- Numismatik Lanz München: Auction 117 (Nov 2003), Lot No. 422
- Sylloge Nummorum Graecorum: Vol. III 3192 Lockett Collection (SNG_0300_3192); Vol. III 3193 Lockett Collection (SNG_0300_3193)

### 4. Descriptive statistics

No. of observations: |
16 | |

Mean: |
15.95 | (95% confidence interval: 15.86 ≤ mean ≤ 16.04) |

Standard deviation: |
0.17 | |

Interquartile range: |
0.29 | |

Skewness: |
-0.13 | |

Kurtosis: |
1.86 | |

Minimum: |
15.64 | |

25th percentile: |
15.81 | (96.3% confidence interval: 15.64 ≤ 25th percentile ≤ 15.91) |

Median: |
15.95 | (92.3% confidence interval: 15.83 ≤ median ≤ 16.09) |

75th percentile: |
16.10 | (96.3% confidence interval: 15.99 ≤ 75th percentile ≤ 16.21) |

Maximum: |
16.21 |

**Notes:** The unbiased estimation of the variance was used for the computation of the standard deviation (i.e. the number of observations minus one was used as a divisor). The sample skewness was computed without sample corrections (i.e. the skewness was computed as the square root of the number of observations times the sum of the third powers of deviations from the mean divided by the 3/2 power of the sum of the squares of deviations from the mean). Similarly, the sample kurtosis was computed as the number of observations times the sum of the fourth powers of deviations from the mean divided by the second power of the sum of the squares of deviations from the mean.

The confidence interval for mean was computed by using the Student t-distribution. The confidence intervals for median and percentiles were computed nonparametrically by using the binomial distribution (see, e.g., Conover, *Practical Nonparametric Statistics*, pp. 143 - 148).

### 5. Estimation of proportion of coins with weights within the observed range

At the 95% level of confidence, at least 73.6% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 15.64 g and 16.21 g, and at least 58.3% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 15.77 g and 16.16 g.

**Note:** These estimations are computed as tolerance limits based on the binomial distribution. See, e.g., Conover, *Practical Nonparametric Statistics*, pp. 150 - 155.

### 6. Histogram and probability density function

Histogram of the sample is presented in Figure 1. Kernel estimations of the probability density function are shown in Figure 2 (two kernel estimations were used: Epanechnikov kernel with a bandwidth of 0.109 and Gaussian kernel with a bandwidth of 0.087). The dotted curve in Figure 2 is a probability density function of a normal distribution estimated by the maximum likelihood method.

**Note:** The bandwidth of the Gaussian kernel was computed as h_{Gauss} = 0.9 × min(σ, SIQR) × n^{-1/5}, where σ is the standard deviation, SIQR is the standardised interquartile range (i.e. the interquartile range of the data sample divided by the interquartile range of the standard normal density) and n is the number of observations; see Silverman, *Density Estimation for Statistics and Data Analysis*, p. 48, formula (3.31). The bandwidth of the Epanechnikov kernel was chosen subjectively in the range from 0.75×h_{Gauss} to 1.25×h_{Gauss}.

### 7. Test of normality

The Lilliefors test of normality was used. The test statistic of 0.133 is less than the cutoff value of 0.213 for a 95% level test. Thus we cannot reject the hypothesis of normality at the 95% level of significance. Normal probability plot of the sample is presented in Figure 3.

## Seleukeia ad Kalykadnon Mint

### 1. Examined type

Denomination: |
AR Tetradrachm |

Period: |
98 - 94 BC |

Obverse: |
Diademed head of Seleukos VI right; fillet border |

Reverse: |
‘ΒΑΣΙΛΕΩΣ ΣΕΛΕΥΚΟΥ’ right, ‘ΕΠΙΦΑΝΟΥΣ ΝΙΚΑΤΟΡΟΣ’ left; Athena Nikephoros standing and facing left, holding Nike in right hand who faces right, and resting left hand on shield, spear behind her left arm; five-lobed plant in outer left field; all within laurel wreath |

### 2. Acceptable weight range

Lower exclusion limit: |
14.75 grams |

Upper exclusion limit: |
16.75 grams |

Each coin is a priori excluded from the data sample if its weight is lesser than the lower exclusion limit or greater than the upper exclusion limit.

### 3. Data

Sorted data (weights in grams):

15.76, 15.88, 15.91, 16.46, 16.72

**Note:** The following coins were included into the analysis:

- Jean Elsen s.a.: Auction 71 (Sep 2002), Lot No. 302
- Classical Numismatic Group, Inc.: Triton VI (Jan 2003), Lots No. 464 and 465; Auction 64 (Sep 2003), Lot No. 403
- Freeman & Sear: Fixed Price List (2003), 1 item

### 4. Descriptive statistics

No. of observations: |
5 | |

Mean: |
16.15 | (95% confidence interval: 15.63 ≤ mean ≤ 16.67) |

Standard deviation: |
0.42 | |

Interquartile range: |
0.67 | |

Skewness: |
0.49 | |

Kurtosis: |
1.50 | |

Minimum: |
15.76 | |

25th percentile: |
15.85 | (98.4% right-sided confidence interval: 25th percentile ≤ 16.46) |

Median: |
15.91 | (93.8% confidence interval: 15.76 ≤ median ≤ 16.72) |

75th percentile: |
16.52 | (98.4% left-sided confidence interval: 15.88 ≤ 75th percentile) |

Maximum: |
16.72 |

**Notes:** The unbiased estimation of the variance was used for the computation of the standard deviation (i.e. the number of observations minus one was used as a divisor). The sample skewness was computed without sample corrections (i.e. the skewness was computed as the square root of the number of observations times the sum of the third powers of deviations from the mean divided by the 3/2 power of the sum of the squares of deviations from the mean). Similarly, the sample kurtosis was computed as the number of observations times the sum of the fourth powers of deviations from the mean divided by the second power of the sum of the squares of deviations from the mean.

The confidence interval for mean was computed by using the Student t-distribution. The confidence intervals for median and percentiles were computed nonparametrically by using the binomial distribution (see, e.g., Conover, *Practical Nonparametric Statistics*, pp. 143 - 148).

### 5. Estimation of proportion of coins with weights within the observed range

At the 95% level of confidence, at least 34.3% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 15.76 g and 16.72 g, and at least 7.6% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 15.88 g and 16.46 g.

**Note:** These estimations are computed as tolerance limits based on the binomial distribution. See, e.g., Conover, *Practical Nonparametric Statistics*, pp. 150 - 155.

### 6. Histogram and probability density function

Histogram of the sample is presented in Figure 4. Kernel estimations of the probability density function are shown in Figure 5 (two kernel estimations were used: Epanechnikov kernel with a bandwidth of 0.342 and Gaussian kernel with a bandwidth of 0.274). The dotted curve in Figure 5 is a probability density function of a normal distribution estimated by the maximum likelihood method.

**Note:** The bandwidth of the Gaussian kernel was computed as h_{Gauss} = 0.9 × min(σ, SIQR) × n^{-1/5}, where σ is the standard deviation, SIQR is the standardised interquartile range (i.e. the interquartile range of the data sample divided by the interquartile range of the standard normal density) and n is the number of observations; see Silverman, *Density Estimation for Statistics and Data Analysis*, p. 48, formula (3.31). The bandwidth of the Epanechnikov kernel was chosen subjectively in the range from 0.75×h_{Gauss} to 1.25×h_{Gauss}.

### 7. Test of normality

The Lilliefors test of normality was used. The test statistic of 0.313 is less than the cutoff value of 0.337 for a 95% level test. Thus we cannot reject the hypothesis of normality at the 95% level of significance. Normal probability plot of the sample is presented in Figure 6.

## Comparison of Antioch and Seleukeia ad Kalykadnon Mints

### Basic characteristics

The data sample of Seleukeia ad Kalykadnon mint is very small. Nevertheless, it seems that tetradrachms of Antioch mint have somewhat different distribution from the one of tetradrachms of Seleukeia ad Kalykadnon mint. Basic descriptive statistics of both samples are presented in Figure 7. Histograms are presented in Figure 8, kernel density estimations are presented in Figures 9 and 10, and empirical cumulative distribution functions are presented in Figure 11. Figure 12 shows box-percentile plots.1

### Test of differences

The Kolmogorov-Smirnov test was used to test the hypothesis that both distributions are the same. The test statistic of 0.400 is less than the exact cutoff value of 0.675 for a 95% level test. Thus, at the 95% level of significance, we cannot reject the hypothesis that the weight distributions of tetradrachms of both mints are the same. Note that the Wilcoxon rank sum test2 gives the same conclusion.

1 See Statistical Glossary, Box-percentile plot.

2 Also known as the Mann-Whitney test.

### References:

**Conover, W. J.:***Practical Nonparametric Statistics*, Third Edition. John Wiley & Sons, Inc., New York - Chichester - Weinheim - Brisbane - Singapore - Toronto, 1999.**Silverman, B.W.:***Density Estimation for Statistics and Data Analysis*. Chapman and Hall, London - New York - Tokyo - Melbourne - Madras, 1993 (reprint of the first edition published in 1986).