Weight Analysis

Demetrios II, 2nd Reign

Tyre Mint

Contents

1. Examined type
2. Acceptable weight range
3. Data
4. Descriptive statistics
5. Estimation of proportion of coins with weights within the observed range
6. Histogram and probability density function
7. Test of normality

1. Examined type

2. Acceptable weight range

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):
13.58, 13.65, 13.91, 13.94, 13.95, 14.01, 14.02, 14.02, 14.05, 14.07, 14.12, 14.13, 14.16, 14.18, 14.21, 14.24, 14.29

Note: The following coins were included into the analysis:

• Classical Numismatic Group, Inc.: Auction 26, Lot No. 62161; Auction 58 (Sep 2001), Lot No. 706; Electronic Auction 53 (Nov 2002), Lot No. 24; Electronic Auction 59 (Feb 2003), Lots No. 52 and 53; Electronic Auction 72 (Sep 2003), Lot No. 41
• Fritz Rudolf Künker Münzenhandlung: Auction 83 (Jun 2003), Lots No. 413, 414 and 415
• Gorny & Mosch Giessener Münzhandlung: Auction 108 (Apr 2001), Lot No. 1356; Auction 118 (Oct 2002), Lot No. 1518
• Münzen & Medaillen Deutschland GmbH: Auction 11 (Nov 2002), Lot No. 772; Auction 14 (Apr 2004), Lot No. 594
• Sylloge Nummorum Graecorum: Vol. I 434 Newnham Davis Coins (SNG_0102_0434); Vol. VII 1348 Manchester University Museum (SNG_0700_1348); Vol. VII 1349 Manchester University Museum (SNG_0700_1349); Vol. VIII 1067 Blackburn Museum (SNG_0800_1067)

4. Descriptive statistics

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 75.0% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 13.58 g and 14.29 g, and at least 60.4% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 13.65 g and 14.24 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 0.089 and Gaussian kernel with a bandwidth of 0.082). 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.

Fig. 1: Histogram

Fig. 2: Probability density estimations

7. Test of normality

The Lilliefors test of normality was used. The test statistic of 0.162 is less than the cutoff value of 0.206 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

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).