Submitted by Petr Vesely on

Founded 11-Jul-2004

Last update 12-Dec-2004

Ake-Ptolemais mint Damaskos mint Comparison References

## Ake-Ptolemais Mint

### 1. Examined type

Denomination: |
AR Tetradrachm |

Period: |
125 - 121 BC |

Obverse: |
Jugate heads of Kleopatra Thea, diademed and with stephane and veil, and Antiochos VIII, diademed, to right; dotted or fillet border |

Reverse: |
‘ΒΑΣΙΛΙΣΣΗΣ ΚΛΕΟΠΑΤΡΑΣ ΘΕΑΣ’ right, ‘ΚΑΙ ΒΑΣΙΛΕΩΣ ΑΝΤΙΟΧΟΥ’ left; Zeus Nikephoros seated on throne left holding Nike in right hand and scepter in left hand |

### 2. Acceptable weight range

Lower exclusion limit: |
15.75 grams |

Upper exclusion limit: |
17.25 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):

16.34, 16.40, 16.42, 16.44, 16.45, 16.51, 16.52, 16.52, 16.55, 16.57, 16.59, 16.60, 16.60, 16.63, 16.67, 16.69, 16.75, 16.77, 16.82

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

- American Numismatic Society: Accession No. 1948.19.2451
- Argenor Numismatique S.A.: Auction 4 (Apr 2001), Lot No. 53; Auction 5 (Apr 2002), Lot No. 110
- Baldwin's Auctions Ltd and M&M Numismatics Ltd: The New York Sale III (Dec 2000), Lot No. 165
- Classical Numismatic Group, Inc.: Auction 58 (Sep 2001), Lot No. 707; Triton V (Jan 2002), Lot No. 1506; Auction 60 (May 2002), Lots No. 922 and 924; Auction 61 (Sep 2002), Lots No. 848, 849 and 850; Auction 64 (Sep 2003), Lot No. 401
- Dr. Busso Peus Nachf.: Auction 372 (Oct 2002), Lot No. 563
- Fritz Rudolf Kuenker Muenzenhandlung: Auction 89 (Mar 2004), Lot No. 1477
- Gorny & Mosch Giessener Münzhandlung: Auction 114 (Mar 2002), Lot No. 140
- Leu Numismatik Ltd.: Auction 83 (May 2002), Lot No. 394
- Numismatik Lanz München: Auction 117 (Nov 2003), Lot No. 419
- Sylloge Nummorum Graecorum: Vol. I 44 Salting Collection (SNG_0101b0044); Vol. III 3175 Lockett Collection (SNG_0300_3175)

### 4. Descriptive statistics

No. of observations: |
19 | |

Mean: |
16.57 | (95% confidence interval: 16.51 ≤ mean ≤ 16.63) |

Standard deviation: |
0.13 | |

Interquartile range: |
0.20 | |

Skewness: |
0.18 | |

Kurtosis: |
2.28 | |

Minimum: |
16.34 | |

25th percentile: |
16.47 | (94.0% confidence interval: 16.40 ≤ 25th percentile ≤ 16.55) |

Median: |
16.57 | (93.6% confidence interval: 16.51 ≤ median ≤ 16.63) |

75th percentile: |
16.66 | (94.0% confidence interval: 16.59 ≤ 75th percentile ≤ 16.77) |

Maximum: |
16.82 |

**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 77.4% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 16.34 g and 16.82 g, and at least 64.1% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 16.40 g and 16.77 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 0.082 and Gaussian kernel with a bandwidth of 0.066). 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 0.095 is less than the cutoff value of 0.195 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.

## Damaskos Mint

### 1. Examined type

Denomination: |
AR Tetradrachm |

Period: |
125 - 121 BC |

Obverse: |
Jugate heads of Kleopatra Thea, diademed and with stephane and veil, and Antiochos VIII, diademed, to right; dotted or fillet border |

Reverse: |
‘ΒΑΣΙΛΙΣΣΗΣ ΚΛΕΟΠΑΤΡΑΣ ΘΕΑΣ’ right, ‘ΚΑΙ ΒΑΣΙΛΕΩΣ ΑΝΤΙΟΧΟΥ’ left; Zeus Nikephoros seated on throne left holding Nike in right hand and scepter in left hand |

### 2. Acceptable weight range

Lower exclusion limit: |
15.75 grams |

Upper exclusion limit: |
17.25 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):

15.99, 16.48, 16.70, 16.72, 16.77, 17.16

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

- Classical Numismatic Group, Inc.: Online e-Auction 24, Lot No. 61867; Auction 60 (May 2002), Lot No. 925; Auction 61 (Sep 2002), Lot No. 851
- Gorny & Mosch Giessener Münzhandlung: Auction 125 (Oct 2003), Lot No. 253
- Numismatik Lanz München: Auction 114 (May 2003), Lot No. 187
- Sylloge Nummorum Graecorum: Vol. I 437 Newnham Davis Coins (SNG_0102_0437)

### 4. Descriptive statistics

No. of observations: |
6 | |

Mean: |
16.64 | (95% confidence interval: 16.23 ≤ mean ≤ 17.04) |

Standard deviation: |
0.39 | |

Interquartile range: |
0.29 | |

Skewness: |
-0.49 | |

Kurtosis: |
2.71 | |

Minimum: |
15.99 | |

25th percentile: |
16.48 | (96.2% right-sided confidence interval: 25th percentile ≤ 16.72) |

Median: |
16.71 | (96.9% confidence interval: 15.99 ≤ median ≤ 17.16) |

75th percentile: |
16.77 | (96.2% left-sided confidence interval: 16.70 ≤ 75th percentile) |

Maximum: |
17.16 |

**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 41.8% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 15.99 g and 17.16 g, and at least 15.3% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 16.48 g and 16.77 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 0.169 and Gaussian kernel with a bandwidth of 0.135). 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 0.232 is less than the cutoff value of 0.319 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 Ake-Ptolemais and Damaskos Mints

### Basic characteristics

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 0.509 is less than the exact cutoff value of 0.614 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).