Antiochos IX

Founded 1-May-2006
Last update 1-May-2006

Antioch Mint, 3rd Reign References


Antioch Mint, 3rd Reign

1. Examined type

Denomination: AR Tetradrachm
Period: 96 - 95 BC
Obverse: Diademed, beardless head of Antiochos IX right; fillet border
Reverse: ‘ΒΑΣΙΛΕΩΣ ΑΝΤΙΟΧΟΥ’ right, ‘ΦΙΛΟΠΑΤΟΡΟΣ’ left; Zeus Nikephoros seated left on throne, holding Nike in outstretched right hand and scepter in left hand; laurel wreath border

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):
15.30, 15.48, 15.53, 15.63, 15.69, 15.71, 15.76, 15.92, 16.03, 16.12, 16.12, 16.28, 16.41

Note: The following coins were included into the analysis:

  • Classical Numismatic Group, Inc.: Electronic Auction 76 (Oct 2003), Lot 15; Electronic Auction 119 (Jul 2005), Lot 57
  • Gorny & Mosch Giessener Münzhandlung: Auction 134 (Oct 2004), Lots 1563 and 1564
  • Houghton, CSE, 352 - 354
  • SNG Spaer, 2709 - 2710
  • Spink and Son, Ltd.: Sale 4026 (Apr 2004), Lot 116
  • UBS Gold & Numismatics: Auction 57 (Sep 2003), Lot 314
  • The British Sylloge Nummorum Graecorum: Vol. III 3184 Lockett Collection (SNGuk_0300_3184); Vol. IV 5806 Fitzwilliam Museum (SNGuk_0408_5806)

4. Descriptive statistics

No. of observations: 13  
Mean: 15.84 (95% confidence interval: 15.64 ≤ mean ≤ 16.05)
Standard deviation: 0.33  
Interquartile range: 0.52  
Skewness: 0.13  
Kurtosis: 2.00  
Minimum: 15.30  
25th percentile: 15.61 (95.2% confidence interval: 15.30 ≤ 25th percentile ≤ 15.76)
Median: 15.76 (97.8% confidence interval: 15.53 ≤ median ≤ 16.12)
75th percentile: 16.12 (95.2% confidence interval: 15.76 ≤ 75th percentile ≤ 16.41)
Maximum: 16.41  

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 15.30 g and 16.41 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 15.48 g and 16.28 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 1. Kernel estimations of the probability density function are shown in Figure 2 (two kernel estimations were used: Epanechnikov kernel with a bandwidth of 0.219 and Gaussian kernel with a bandwidth of 0.179). 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. 1: Histogram

Fig. 2: Probability density estimations

Fig. 2: Probability density estimations

7. Test of normality

The Lilliefors test of normality was used. The test statistic of 0.139 is less than the cutoff value of 0.234 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

Fig. 3: Normal probability plot

 


References:

Conover, W. J.:Practical Nonparametric Statistics, Third Edition. John Wiley & Sons, Inc., New York - Chichester - Weinheim - Brisbane - Singapore - Toronto, 1999.
Houghton, Arthur:Coins of the Seleucid Empire from the Collection of Arthur Houghton. The American Numismatic Society, New York, 1983. (abbr. CSE)
Houghton, Arthur; Spaer, Arnold (with the assistance of Catharine Lorber):Sylloge Nummorum Graecorum. Israel I. The Arnold Spaer Collection of Seleucid Coins. Italo Vecchi Ltd., London, 1998. (abbr. SNG Spaer)
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).