Seleukid Traces
Historical and Numismatic Information
Metrological Analyses of Seleukid Coins
Works of Other Authors
Excerpts about Greek and Hellenistic Civilization
Reflections of Antiquity in Modern Poetry
Petr Veselý © 2013
pvesely2002@yahoo.co.uk
Weight Studies of Well-Preserved Seleukid Coins
Demetrios II, 1st Reign
Antiochos VII
Demetrios II, 2nd Reign
Alexander II
Kleopatra Thea and Antiochos VIII
Antiochos VIII
Antiochos IX
Seleukos VI
Antiochos X
Demetrios III
Antiochos XIII
Antioch mint from Antiochos VII to Antiochos XIII
Tyre mint under Demetrios II and Antiochos VII
Other Weight Studies
Die Studies of Demetrios I and Alexander I
Methodological Notes
Metrological Analyses of Seleukid Coins /
Weight Studies of Well-Preserved Seleukid Coins /
Submitted by Petr Vesely on Tue, 06/18/2013 - 15:10
Founded 27-Jul-2003
Last update 25-Mar-2004
Antioch Mint References
Antioch Mint
1. Examined type
Denomination: AR Tetradrachm
Period: c. 94 - c. 93 BC
Obverse: Diademed head of Antiochos X right; fillet border
Reverse: 'ΒΑΣΙΛΕΩΣ ΑΝΤΙΟΧΟΥ' right, 'ΕΥΣΕΒΟΥΣ ΦΙΛΟΠΑΤΟΡΟΣ' 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):
15.28, 15.55, 15.56, 15.61, 15.62, 15.64, 15.67, 15.72, 15.73, 15.75, 15.76, 15.77, 15.83, 15.90, 15.91, 15.96, 15.97, 15.98, 15.98, 15.99, 16.02, 16.23
Note: The following coins were included into the analysis:
Classical Numismatic Group, Inc.: eBay, Item No. 443884967 (Sep 2000); Electronic Auction 59 (Feb 2003), Lot No. 56; Electronic Auction 60 (Mar 2003), Lot No. 38
Dr. Busso Peus Nachf.: Auction 369 (Oct 2001), Lot No. 246; Auction 372 (Oct 2002), Lot No. 571
Fritz Rudolf Kuenker Muenzenhandlung: Auction 67 (Oct 2001), Lot No. 457; Auction 71 (Mar 2002), Lot No. 442; Auction 77 (Sep 2002), Lot No. 247; Auction 77 (Sep 2002), Lot No. 248
Gorny & Mosch Giessener Münzhandlung: Auction 107 (Apr 2001), Lot No. 272
Jean Elsen s.a.: Auction 72 (Dec 2002), Lot No. 269; Auction 72 (Dec 2002), Lot No. 270; Auction 73 (Mar 2003), Lot No. 129; Auction 74 (Jun 2003), Lot No. 240; Auction 74 (Jun 2003), Lot No. 241; Auction 76 (Sep 2003), Lot No. 179; Auction 77 (Dec 2003), Lot No. 149
Münzen & Medaillen Deutschland GmbH: Auction 11 (Nov 2002), Lot No. 780
Numismatik Lanz München: Auction 117 (Nov 2003), Lot No. 423
Sylloge Nummorum Graecorum: Vol. I 446 Newnham Davis Coins (SNG_0102_0446); Vol. VIII 1071 Blackburn Museum (SNG_0800_1071)
Tkalec AG: Auction 2001, Lot No. 162
4. Descriptive statistics
No. of observations: 22
Mean: 15.79 (95% confidence interval: 15.70 ≤ mean ≤ 15.89)
Standard deviation: 0.21
Interquartile range: 0.33
Skewness: -0.25
Kurtosis: 3.15
Minimum: 15.28
25th percentile: 15.64 (95.6% confidence interval: 15.55 ≤ 25th percentile ≤ 15.75)
Median: 15.77 (94.8% confidence interval: 15.67 ≤ median ≤ 15.96)
Maximum: 16.23
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 80.2% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 15.28 g and 16.23 g, and at least 68.4% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 15.55 g and 16.02 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 0.125 and Gaussian kernel with a bandwidth of 0.102). 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 hGauss = 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×hGauss to 1.25×hGauss.
Fig. 1: Histogram
Fig. 2: Probability density estimations
7. Test of normality
The Lilliefors test of normality was used. The test statistic of 0.106 is less than the cutoff value of 0.183 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.
Fig. 3: Normal probability plot
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).