Weight Analysis

Seleukos VI

Seleukeia ad Kalykadnon 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):
15.76, 15.88, 15.91, 16.46, 16.72

Note: The following coins were included into the analysis:

• Jean Elsen s.a.: Auction 71 (Sep 2002), Lot No. 302
• Classical Numismatic Group, Inc.: Triton VI (Jan 2003), Lots No. 464 and 465; Auction 64 (Sep 2003), Lot No. 403
• Freeman & Sear: Fixed Price List (2003), 1 item

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 34.3% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 15.76 g and 16.72 g, and at least 7.6% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 15.88 g and 16.46 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 0.342 and Gaussian kernel with a bandwidth of 0.274). 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 0.313 is less than the cutoff value of 0.337 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).