Last update 25-Mar-2004
1. Examined type
|Period:||93 - 83 BC|
|Obverse:||Diademed head of Philip I right; fillet border|
|Reverse:||‘ΒΑΣΙΛΕΩΣ ΦΙΛΙΠΠΟΥ’ right, ‘ΕΠΙΦΑΝΟΥΣ ΦΙΛΑΔΕΛΦΟΥ’ 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.
Sorted data (weights in grams):
15.17, 15.32, 15.34, 15.40, 15.42, 15.49, 15.62, 15.63, 15.72, 15.72, 15.75, 15.76, 15.77, 15.77, 15.81, 15.82, 15.85, 15.86, 15.86, 15.87, 15.88, 15.88, 15.90, 15.94, 15.96, 15.97, 15.98, 16.01, 16.09, 16.16
Note: The following coins were included into the analysis:
- Classical Numismatic Group, Inc.: eBay, Items No. 318350884 (May 2000), 340717672 (May 2000), 346315630 (Jun 2000) and 357966295 (Jun 2000); Fixed Price List (2002), 1 item; Auction 60 (May 2002), Lot No. 930; Auction 61 (Sep 2002), Lot No. 859; Electronic Auction 55 (Dec 2002), Lot No. 59; Auction 63 (May 2003), Lot No. 664
- Fritz Rudolf Künker Müzenhandlung: Auction 83 (Jun 2003), Lot No. 419
- Harlan J. Berk, Ltd.: eBay, Items No. 1229725205 (Apr 2001) and 1259112992 (Aug 2001)
- Jean Elsen s.a.: Auction 74 (Jun 2003), Lots No. 242, 243, 244, 245 and 246; Auction 76 (Sep 2003), Lots No. 180, 181, 182, 183 and 185; Auction 77 (Dec 2003), Lots No. 150, 151 and 152; Auction 78 (Mar 2004), Lots No. 102, 103 and 104
- Münzen & Medaillen AG Basel: Auction 29 (Jun 2003), Lot No. 753
- Numismatik Lanz München: Auktion 102 (May 2001), Lot No. 296
4. Descriptive statistics
|No. of observations:||30|
|Mean:||15.76||(95% confidence interval: 15.67 ≤ mean ≤ 15.85)|
|25th percentile:||15.63||(94.1% confidence interval: 15.40 ≤ 25th percentile ≤ 15.77)|
|Median:||15.82||(95.7% confidence interval: 15.72 ≤ median ≤ 15.88)|
|75th percentile:||15.90||(94.1% confidence interval: 15.86 ≤ 75th percentile ≤ 15.98)|
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 85.1% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 15.17 g and 16.16 g, and at least 76.1% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 15.32 g and 16.09 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 0.068 and Gaussian kernel with a bandwidth of 0.091). 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.
7. Test of normality
The Lilliefors test of normality was used. The test statistic of 0.171 is greater than the cutoff value of 0.161 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.
- 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).