Submitted by Petr Vesely on

Founded 8-Aug-2003

Last update 11-Dec-2005

Antioch Mint Tyre Mint References

## Antioch Mint

### 1. Examined type

Denomination: |
AR Tetradrachm |

Period: |
138 - 129 BC |

Obverse: |
Diademed head of Antiochos VII right; fillet border |

Reverse: |
‘ΒΑΣΙΛΕΩΣ ΑΝΤΙΟΧΟΥ’ right, ‘ΕΥΕΡΓΕΤΟΥ’ left; Athena Nikephoros standing and facing left, holding Nike in right hand , and resting left hand on shield with a human face, spear propped against her left arm; ‘ΔΙ’ monogram above control mark in outer left field; all within laurel wreathwho faces left |

### 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, 16.05, 16.06, 16.14, 16.15, 16.18, 16.28, 16.28, 16.33, 16.39, 16.40, 16.46, 16.46, 16.46, 16.47, 16.50, 16.51, 16.52, 16.52, 16.54, 16.54, 16.54, 16.58, 16.59, 16.60, 16.60, 16.62, 16.64, 16.64, 16.66, 16.66, 16.71, 16.71, 16.73, 16.73, 16.74, 16.75, 16.75, 16.75, 16.76, 16.77, 16.77, 16.87, 16.89, 16.91, 17.02

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

- Argenor Numismatique S.A: Auction 5 (Apr 2002), Lot 105
- Classical Numismatic Group, Inc.: Online e-Auction 39 (Oct 2001), Lot 64436; Auction 58 (Sep 2001), Lot 704; Auction 60 (May 2002), Lot 916; Auction 61 (Sep 2002), Lots 842 and 843; Electronic Auction 59 (Feb 2003), Lot 51; Electronic Auction 60 (Mar 2003), Lot 37; Auction 63 (May 2003), Lots 651 and 652; Electronic Auction 71 (Aug 2003), Lot 18; Mail Bid Sale 66 (May 2004), Lots 689 and 690; Mail Bid Sale 67 (Sep 2004), Lots 882 and 883
- Dr. Busso Peus Nachf.: Auction 366 (Oct 2000), Lots 233 and 234; Auction 368 (Apr 2001), Lot 288; Auction 372 (Oct 2002), Lot 559; Auction 380 (Nov 2004), Lot 598; Auction 382 (Apr 2005), Lot 234
- Fritz Rudolf Kuenker Muenzenhandlung: Auction 77 (Sep 2002), Lot 239; Auction 89 (Mar 2004), Lots 1471 and 1472; Auction 94 (Sep 2004), Lot 1424; Auction 97 (Mar 2005), Lots 926 and 927; Auction 101 (Jun 2005), Lot 1041
- Gorny & Mosch Giessener Münzhandlung: Auction 108 (Apr 2001), Lots 1352 and 1353; Auction 114 (Mar 2002), Lot 139; Auction 118 (Oct 2002), Lot 1517; Auction 126 (Oct 2003), Lot 1442; Auction 130 (Mar 2004), Lots 1288 and 1289; Auction 134 (Oct 2004), Lot 1554
- Hess-Divo AG: Auction 299 (Oct 2004), Lot 104
- Jean Elsen s.a.: Auction 76 (Sep 2003), Lot 173
- Malter Galleries, Inc.: Fixed Price List (2002-2003), 1 item
- Münzen & Medaillen Deutschland GmbH: Auction 8 (May 2001), Lot 202; Auction 12 (Apr 2003), Lot 117; Auction 16 (May 2005), Lot 961
- Numismatik Lanz München: Auction 117 (Nov 2003), Lot 414
- Sylloge Nummorum Graecorum: Vol. I 425 Newnham Davis Coins (SNG_0102_0425); Vol. VI 1087 Fitzwilliam Museum (SNG_0601_1087); SNG Vol. VIII 1063 Blackburn Museum (SNG_0800_1063)

### 4. Descriptive statistics

No. of observations: |
46 | |

Mean: |
16.55 | (95% confidence interval: 16.48 ≤ mean ≤ 16.62) |

Standard deviation: |
0.24 | |

Interquartile range: |
0.27 | |

Skewness: |
-0.56 | |

Kurtosis: |
2.82 | |

Minimum: |
15.99 | |

25th percentile: |
16.46 | (96.1% confidence interval: 16.18 ≤ 25th percentile ≤ 16.52) |

Median: |
16.59 | (94.6% confidence interval: 16.51 ≤ median ≤ 16.66) |

75th percentile: |
16.73 | (96.1% confidence interval: 16.64 ≤ 75th percentile ≤ 16.77) |

Maximum: |
17.02 |

**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 90.1% 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 17.02 g, and at least 84.0% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 16.05 g and 16.91 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 0.105 and Gaussian kernel with a bandwidth of 0.084). 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 0.117 is less than the cutoff value of 0.131 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.

### 8. Control marks

Control bellow ΔΙ | Secondary Controls | Frequency | Weight | ||||||
---|---|---|---|---|---|---|---|---|---|

left | right | Mean | Std | Min | Max | ||||

1 | none |
none |
10 | 21.7% | 16.60 | 0.28 | 16.15 | 17.02 | |

2 | none |
5 | 10.9% | 16.53 | 0.21 | 16.18 | 16.71 | ||

3 | none |
3 | 6.5% | 16.60 | 0.08 | 16.51 | 16.66 | ||

4 | none |
3 | 6.5% | 16.47 | 0.02 | 16.46 | 16.50 | ||

5 | none |
2 | 4.3% | 16.67 | 0.28 | 16.47 | 16.87 | ||

6 | none |
2 | 4.3% | 16.27 | 0.18 | 16.14 | 16.39 | ||

7 | none |
unclear |
2 | 4.3% | 16.34 | 0.40 | 16.06 | 16.62 | |

8 | none |
1 | 2.2% | 16.28 | 0.00 | 16.28 | 16.28 | ||

9 | none |
1 | 2.2% | 15.99 | 0.00 | 15.99 | 15.99 | ||

SUB-TOTAL |
29 |
63.0% |
16.51 |
0.26 |
15.99 |
17.02 |
|||

10 | none |
none |
8 | 17.4% | 16.52 | 0.20 | 16.05 | 16.73 | |

11 | none |
none |
4 | 8.7% | 16.63 | 0.20 | 16.33 | 16.75 | |

12 | none |
none |
2 | 4.3% | 16.79 | 0.18 | 16.66 | 16.91 | |

13 | none |
none |
1 | 2.2% | 16.77 | 0.00 | 16.77 | 16.77 | |

14 | none |
none |
1 | 2.2% | 16.76 | 0.00 | 16.76 | 16.76 | |

15 | none |
none |
1 | 2.2% | 16.74 | 0.00 | 16.74 | 16.74 | |

TOTAL |
46 |
100% |
16.55 |
0.24 |
15.99 |
17.02 |

## Tyre Mint

### 1. Examined type

Denomination: |
AR Tetradrachm |

Period: |
138 - 129 BC |

Obverse: |
Diademed and draped bust of Antiochos VII right; dotted border |

Reverse |
‘ΒΑΣΙΛΕΩΣ’ right, ‘ΑΝΤΙΟΧΟΥ’ left; eagle standing left on prow, palm over shoulder; club surmounted by ‘ΤΥΡ’ monogram in left field; dotted border |

### 2. Acceptable weight range

Lower exclusion limit: |
13.50 grams |

Upper exclusion limit: |
14.50 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):

13.81, 13.83, 13.92, 13.95, 13.96, 14.00, 14.01, 14.02, 14.06, 14.06, 14.09, 14.18, 14.28, 14.33

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

- Baldwin's Auctions Ltd and M&M Numismatics Ltd: The New York Sale III (Dec 2000), Lot No. 164
- Classical Numismatic Group, Inc.: Auction 58 (Sep 2001), Lot No. 705; Auction 60 (May 2002), Lot No. 920; Auction 61 (Sep 2002), Lot No. 844; Electronic Auction 52 (Nov 2002), Lot No. 64
- Freeman & Sear, Inc.: Fixed Price List (2003), 1 item
- Fritz Rudolf Kuenker Muenzenhandlung: Auction 77 (Sep 2002), Lot. No. 238
- Gorny & Mosch Giessener Münzhandlung: Auction 107 (Apr 2001), Lot No. 265; Auction 108 (Apr 2001), Lot No. 1354
- Numismatik Lanz München: Auction 117 (Nov 2003), Lot No. 415
- Sylloge Nummorum Graecorum: Vol. I 430 Newnham Davis Coins (SNG_0102_0430); Vol. III 3165 Lockett Collection (SNG_0300_3165); Vol. III 3166 Lockett Collection (SNG_0300_3166); Vol. VI 1086 Fitzwilliam Museum (SNG_0601_1086)

### 4. Descriptive statistics

No. of observations: |
14 | |

Mean: |
14.04 | (95% confidence interval: 13.95 ≤ mean ≤ 14.12) |

Standard deviation: |
0.15 | |

Interquartile range: |
0.14 | |

Skewness: |
0.48 | |

Kurtosis: |
2.63 | |

Minimum: |
13.81 | |

25th percentile: |
13.95 | (94.4% confidence interval: 13.81 ≤ 25th percentile ≤ 14.01) |

Median: |
14.02 | (94.3% confidence interval: 13.95 ≤ median ≤ 14.09) |

75th percentile: |
14.09 | (94.4% confidence interval: 14.02 ≤ 75th percentile ≤ 14.33) |

Maximum: |
14.33 |

**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 70.3% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 13.81 g and 14.33 g, and at least 53.4% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 13.83 g and 14.28 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 0.051 and Gaussian kernel with a bandwidth of 0.055). 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 0.150 is less than the cutoff value of 0.227 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.

### 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).