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Founded 1-Nov-2003

Last update 12-Sep-2004

Ake-Ptolemais Mint Antioch Mint Damaskos Mint Tyre Mint References

## Ake-Ptolemais Mint

### 1. Examined type

Denomination: |
AR Tetradrachm |

Period: |
129 - 125 BC |

Obverse: |
Diademed and bearded head of Demetrios II right; fillet border |

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

### 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):

15.78, 16.38, 16.44, 16.47

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

- American Numismatic Society: Accession Nos. 1948.19.2450 and 1957.172.2066
- Classical Numismatic Group, Inc.: Triton VI (Jan 2003), Lot No. 457
- Sylloge Nummorum Graecorum: Vol. VI 1093 Fitzwilliam Museum (SNG_0601_1093)

### 4. Descriptive statistics

No. of observations: |
4 | |

Mean: |
16.27 | (95% confidence interval: 15.75 ≤ mean ≤ 16.79) |

Standard deviation: |
0.33 | |

Interquartile range: |
0.38 | |

Skewness: |
-1.11 | |

Kurtosis: |
2.30 | |

Minimum: |
15.78 | |

25th percentile: |
16.08 | (99.6% right-sided confidence interval: 25th percentile ≤ 16.47) |

Median: |
16.41 | (100.0% confidence interval: -Inf ≤ median ≤ Inf) |

75th percentile: |
16.45 | (99.6% left-sided confidence interval: 15.78 ≤ 75th percentile) |

Maximum: |
16.47 |

**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 24.9% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 15.78 g and 16.47 g, and at least 1.3% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 16.38 g and 16.44 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 0.237 and Gaussian kernel with a bandwidth of 0.190). 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 0.385 is greater than the cutoff value of 0.381 for a 95% level test. Thus we reject the hypothesis of normality at the 95% level of significance. Normal probability plot of the sample is presented in Figure 3.

## Antioch Mint

### 1. Examined type

Denomination: |
AR Tetradrachm |

Period: |
129/8 BC |

Obverse: |
Diademed and bearded head of Demetrios II right; fillet border |

Reverse: |
‘ΒΑΣΙΛΕΩΣ ΔΗΜΗΤΡΙΟΥ’ right, ‘ΘΕΟΥ ΝΙΚΑΤΟΡΟΣ’ left; Zeus Nikephoros seated on throne left holding Nike in right hand and scepter in left hand; ‘Ξ’ in left field or in exergue; all within laurel wreath |

### 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):

16.29, 16.43, 16.43, 16.44, 16.45, 16.53, 16.53, 16.54, 16.60, 16.61, 16.62, 16.65, 16.65, 16.66, 16.71, 16.77, 16.84, 16.95, 17.17

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

- Classical Numismatic Group, Inc.: eBay, Item No. 1210076004 (Jan 2001); Triton V (Jan 2002), Lot No. 1502; Auction 60 (May 2002), Lot No. 921; Auction 61 (Sep 2002), Lots No. 845 and 846; Auction 63 (May 2003), Lots No. 644 and 645
- Dr. Busso Peus Nachf.: Auction 372 (Oct 2002), Lot No. 561
- Edward J. Waddell, Ltd.: On-line Auction (Oct 2003), Inventory No. 40805
- Gorny & Mosch Giessener Münzhandlung: Auction 117 (Oct 2002), Lot No. 326; Auction 130 (Mar 2004), Lot No. 1291
- Leu Numismatik Ltd.: Auction 83 (May 2002), Lot No. 390; Auction 86 (May 2003), Lot No. 436
- Münzen & Medaillen Deutschland GmbH: Auction 12 (Apr 2003), Lot No. 118
- Numismatik Lanz München: Auktion 109 (May 2002), Lot No. 211; Auction 117 (Nov 2003), Lot No. 416
- Tkalec AG: Auction 2002, Lot No. 94
- Sylloge Nummorum Graecorum: Vol. I 431 Newnham Davis Coins (SNG_0102_0431); Vol. VII 1347 Manchester University Museum (SNG_0700_1347)

### 4. Descriptive statistics

No. of observations: |
19 | |

Mean: |
16.62 | (95% confidence interval: 16.53 ≤ mean ≤ 16.72) |

Standard deviation: |
0.21 | |

Interquartile range: |
0.23 | |

Skewness: |
0.95 | |

Kurtosis: |
3.96 | |

Minimum: |
16.29 | |

25th percentile: |
16.47 | (94.0% confidence interval: 16.43 ≤ 25th percentile ≤ 16.60) |

Median: |
16.61 | (93.6% confidence interval: 16.53 ≤ median ≤ 16.66) |

75th percentile: |
16.70 | (94.0% confidence interval: 16.62 ≤ 75th percentile ≤ 16.95) |

Maximum: |
17.17 |

**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 16.29 g and 17.17 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 16.43 g and 16.95 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 0.101 and Gaussian kernel with a bandwidth of 0.084). 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 0.169 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 6.

## Damaskos Mint

### 1. Examined type

Denomination: |
AR Tetradrachm |

Period: |
129 - 125 BC |

Obverse: |
Diademed and bearded head of Demetrios II right; fillet border |

Reverse: |
‘ΒΑΣΙΛΕΩΣ ΔΗΜΗΤΡΙΟΥ’ right, ‘ΘΕΟΥ ΝΙΚΑΤΟΡΟΣ’ left; Zeus Nikephoros seated on throne left holding Nike in right hand and scepter in left hand; control marks; all within laurel wreath |

### 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):

16.35, 16.53, 16.59, 16.59, 16.60, 16.61, 16.63, 16.67

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

- Classical Numismatic Group, Inc.: Auction 22, Lot No. 61555; Triton V (Jan 2002), Lot No. 1503; Auction 64 (Sep 2003), Lot No. 399; Auction 66 (May 2004), Lot No. 692
- Gorny & Mosch Giessener Münzhandlung: Auction 107 (Apr 2001), Lot No. 267
- Leu Numismatik Ltd.: Auction 81 (May 2001), Lot No. 345; Auction 83 (May 2002), Lot No. 391
- Spink and Son, Ltd.: Fixed Price List (2003), 1 item (Ref RWW03 - 1562)

### 4. Descriptive statistics

No. of observations: |
8 | |

Mean: |
16.57 | (95% confidence interval: 16.49 ≤ mean ≤ 16.65) |

Standard deviation: |
0.10 | |

Interquartile range: |
0.06 | |

Skewness: |
-1.58 | |

Kurtosis: |
4.49 | |

Minimum: |
16.35 | |

25th percentile: |
16.56 | (97.3% right-sided confidence interval: 25th percentile ≤ 16.60) |

Median: |
16.59 | (93.0% confidence interval: 16.53 ≤ median ≤ 16.63) |

75th percentile: |
16.62 | (97.3% left-sided confidence interval: 16.59 ≤ 75th percentile) |

Maximum: |
16.67 |

**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 52.9% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 16.35 g and 16.67 g, and at least 28.9% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 16.53 g and 16.63 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 7. Kernel estimations of the probability density function are shown in Figure 8 (two kernel estimations were used: Epanechnikov kernel with a bandwidth of 0.033 and Gaussian kernel with a bandwidth of 0.026). The dotted curve in Figure 8 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 0.326 is greater than the cutoff value of 0.285 for a 95% level test. Thus we reject the hypothesis of normality at the 95% level of significance. Normal probability plot of the sample is presented in Figure 9.

## Tyre Mint

### 1. Examined type

Denomination: |
AR Tetradrachm |

Period: |
129 - 125 BC |

Obverse: |
Diademed and draped head of Demetrios II right; dotted border |

Reverse: |
‘ΒΑΣΙΛΕΩΣ’ right, ‘ΔΗΜΗΤΡΙΟΥ’ left; eagle standing left on prow, palm over shoulder; ‘ΑΡΕ’ monogram above club surmounted by ‘ΤΥΡ’ monogram in left field; monogram above date in right field; control-mark between eagle’s legs; dotted border |

### 2. Acceptable weight range

Lower exclusion limit: |
13.50 grams |

Upper exclusion limit: |
14.50 grams |

### 3. Data

Sorted data (weights in grams):

13.58, 13.65, 13.91, 13.94, 13.95, 14.01, 14.02, 14.02, 14.05, 14.07, 14.12, 14.13, 14.16, 14.18, 14.21, 14.24, 14.29

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

- Classical Numismatic Group, Inc.: Auction 26, Lot No. 62161; Auction 58 (Sep 2001), Lot No. 706; Electronic Auction 53 (Nov 2002), Lot No. 24; Electronic Auction 59 (Feb 2003), Lots No. 52 and 53; Electronic Auction 72 (Sep 2003), Lot No. 41
- Fritz Rudolf Künker Münzenhandlung: Auction 83 (Jun 2003), Lots No. 413, 414 and 415
- Gorny & Mosch Giessener Münzhandlung: Auction 108 (Apr 2001), Lot No. 1356; Auction 118 (Oct 2002), Lot No. 1518
- Münzen & Medaillen Deutschland GmbH: Auction 11 (Nov 2002), Lot No. 772; Auction 14 (Apr 2004), Lot No. 594
- Sylloge Nummorum Graecorum: Vol. I 434 Newnham Davis Coins (SNG_0102_0434); Vol. VII 1348 Manchester University Museum (SNG_0700_1348); Vol. VII 1349 Manchester University Museum (SNG_0700_1349); Vol. VIII 1067 Blackburn Museum (SNG_0800_1067)

### 4. Descriptive statistics

No. of observations: |
17 | |

Mean: |
14.03 | (95% confidence interval: 13.93 ≤ mean ≤ 14.13) |

Standard deviation: |
0.19 | |

Interquartile range: |
0.22 | |

Skewness: |
-1.04 | |

Kurtosis: |
3.60 | |

Minimum: |
13.58 | |

25th percentile: |
13.95 | (98.0% confidence interval: 13.58 ≤ 25th percentile ≤ 14.05) |

Median: |
14.05 | (95.1% confidence interval: 13.95 ≤ median ≤ 14.16) |

75th percentile: |
14.16 | (98.0% confidence interval: 14.05 ≤ 75th percentile ≤ 14.29) |

Maximum: |
14.29 |

**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.

*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 75.0% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 13.58 g and 14.29 g, and at least 60.4% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 13.65 g and 14.24 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 10. Kernel estimations of the probability density function are shown in Figure 11 (two kernel estimations were used: Epanechnikov kernel with a bandwidth of 0.089 and Gaussian kernel with a bandwidth of 0.082). The dotted curve in Figure 11 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 0.162 is less than the cutoff value of 0.206 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 12.

### 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).