Submitted by Petr Vesely on

Founded 15-Apr-2007

Last update 15-Apr-2007

## 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; club surmounted by ‘ΤΥΡ’ monogram in left field; date above control mark in right 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):

13.71, 13.78, 13.88, 13.94, 13.98, 14.01, 14.12, 14.12, 14.17, 14.19, 14.19, 14.22, 14.41

**Note:** The following coins were included into the analysis (all these coins are dated either 167 SE or 169 SE):

- Argenor Numismatique S.A: Auction 5 (Apr 2002), Lot 103
- Classical Numismatic Group, Inc.: Triton V (Jan 2002), Lot 1498; Mail Bid Sale 64 (Sep 2003), Lot No. 397
- Dr. Busso Peus Nachf.: Auction 369 (Oct 2001), Lot 241
- Fritz Rudolf Künker Müzenhandlung: Auction 89 (Mar 2004), Lot No. 1470; Auction 97 (Mar 2005), Lot No. 922; Auction 111 (Mar 2006), Lot No. 6313
- Gemini, LLC (Harlan J. Berk, Ltd. & Freeman & Sear): Auction I (Jan 2005), Lot No. 210
- Gorny & Mosch Giessener Münzhandlung: Auction 125 (Oct 2003), Lot No. 252
- Hess-Divo AG: Auction 299 (Oct 2004), Lot 103
- Münzen & Medaillen AG Basel: Auction 95 (Oct 2004), Lot No. 68
- Numismatik Lanz München: Auction 117 (Nov 2003), Lot 411
- UBS Gold & Numismatics: Auction 64 (Jan 2006), Lot No. 85

### 4. Descriptive statistics

No. of observations: |
13 | |

Mean: |
14.06 | (95% confidence interval: 13.94 ≤ mean ≤ 14.17) |

Standard deviation: |
0.20 | |

Interquartile range: |
0.26 | |

Skewness: |
-0.16 | |

Kurtosis: |
2.39 | |

Minimum: |
13.71 | |

25th percentile: |
13.93 | (95.2% confidence interval: 13.71 ≤ 25th percentile ≤ 14.12) |

Median: |
14.12 | (97.8% confidence interval: 13.88 ≤ median ≤ 14.19) |

75th percentile: |
14.19 | (95.2% confidence interval: 14.12 ≤ 75th percentile ≤ 14.41) |

Maximum: |
14.41 |

**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 68.4% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 13.71 g and 14.41 g, and at least 50.5% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 13.78 g and 14.22 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 0.129 and Gaussian kernel with a bandwidth of 0.105). 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 0.168 is less than the cutoff value of 0.233 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.

### 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).