the 95 %C/ increased noticeably with the data 

 variability, but it was almost unaffected by the 

 proportion of zeros in the data regardless of vari- 

 ability. This is a very desirable property of the 

 5-mean because, in actual applications, the pres- 

 ence of zeros in the data should only affect the 

 estimate of the mean without unduly inflating its 

 standard error. The 5-mean was not a good es- 

 timator of the middle point of the data. The only 

 correct estimator of central tendency, the median, 

 cannot be recommended for delta-type data be- 

 cause its 95%C/ was unreliable except for low 

 variability data with less than 30% of zeros (i.e., 

 less than 55 zeros in Fig. lb). Although for 

 highly variable data the geometric mean tracked 

 the median quite well, its 95%C/ was also unre- 

 liable. 



Application to NUEL's Monitoring 

 Data 



The 5-mean should be the preferred statistic to 

 estimate the population mean when the data con- 

 tains many zeros (e.g., data from NUEL's pro- 

 grams listed in Table 1). Its usage, however, 

 should be restricted to cases where the nonzero 

 observations are approximately lognormal and the 

 sample size is 15 or larger. A simple approach 

 to testing for lognormality is to log-transform the 

 data and then test for normality using the proce- 

 dure UNIVARIATE (SAS 1985). The most de- 

 sirable features of the S-mean mean are an accurate 

 standard error for delta-distributed data and 

 95%C/'s whose relative width (i.e., scaled by the 

 mean) is almost unaffected by the proportion of 

 zeros in the data. The weaknesses of the h-mean 

 are that it does not coincide with the middle point 

 of the data and that, unlike the median, it is not 

 resistant to outliers. The median, however, was 

 shown to have a 95%C/ which was unreliable 

 except for low variability data with less than 30% 

 of zeros. When the data contain only a few zeros 

 (less than 10%), either the median or the geometric 

 mean should perform reasonably well with low 

 to moderate variability (CV < 50% for log- 

 transformed data). 



In actual applications to monitoring data on 

 species that occur seasonally, the cut-off points 

 in the data series must be chosen consistently to 

 insure comparability among years. The following 

 procedure is suggested: 1) for each species, start 

 and end the data series on the dates of the first 

 and last occurrence of that species each year and 

 ignore any zero data collected before and after; 

 and 2) compute the cumulative distribution of 

 the data series (i.e., by adding observations se- 

 quentially) and trim the two tails of the distribu- 

 tion by 2.5%, so that only the central observations 

 adding up to 95% of the total sum are retained. 

 It should be noted that this procedure will result 

 in annual data series that will generally vary in 

 length and starting and ending dates from year to 

 year. This poses no problem for estimating cor- 

 rect annual h-means and insures that the data 

 series are a consistent and fair representation of 

 the annual occurrence of each species. 



Finally, the lengthy computation of the h-mean 

 and its standard error should be carried out using 

 a computer program. Such a program was written 

 for the simulation study described above and it 

 is available at NU's computer system. The ac- 

 curacy of this program was tested by reproducing 

 the results of Peimington (1983). 



References Cited 



Aitchison, J. 1955. On the distribution of a pos- 

 itive random variable having a discrete proba- 

 bility mass at the origin. J. Amer. Stat. Assoc. 

 50:901-908. 



Aitchison, J., and J.A.C. Brown. 1969. The 

 lognormal distribution. Cambridge University 

 Press, New York. 156 pp. 



Demetrius, L. 1971. Multiplicative processes. 

 Math. Biosci. 12:261-272. 



Hastings, N.A.J. , and J.B. Peacock. 1975. Sta- 

 tistical distributions: a handbook for students 

 and practitioners. John Wiley & Sons., New 

 York. 130 pp. 



DELTA Means 



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