simulation study and concluded that the estimate 

 of 5-variance obtained by Eq. 4 provided quite 

 accurate coverage, in both 95% and 99% confi- 

 dence intervals, for samples with more than 15 

 observations. Therefore, an approximate 95% 

 confidence interval for large samples (n > 1 00) can 

 be constructed as: 



95%C/ = 5-mean ± 1 .96^/ 8-variance . (6) 



For smaller sample sizes ( 15 < n < 100 ) the 

 corresponding two-tail t-value (a = 0.05) replaces 

 the value of 1.96 above. 



Comparison of the h-mean to Other 

 Statistics 



A numerical simulation was conducted to com- 

 pare the 5-mean to other statistics often used to 

 describe the abundance of marine organisms at 

 NUEL and elsewhere. The statistics chosen for 

 this study where the sample mean, the sample 

 median and the geometric mean. The properties 

 and common usage of these three statistics are 

 briefly discussed first. 



The sample mean: The arithmetic mean of a 

 sample or "sample mean" is the unbiased estimator 

 of the true population mean under normality, but 

 it is biased in the case of non-normal data (spe- 

 cially with small samples). The actual estimator 

 of the population mean has specific forms other 

 than a simple arithmetic mean for each known 

 statistical distribution (e.g., Eq. 1 is the form of 

 the unbiased estimator of the delta-distribution 

 mean). However, the sample mean is generally 

 an acceptable estimator of the population mean 

 for symmetric distributions not far from normal 

 when the sample is large. Tor lognormal distri- 

 butions with high variance (i.e., long tails), the 

 sample mean is a poor estimator of the population 

 mean and the standard error of the sample mean 

 underestimates the true variance of th^ lognormal 

 mean (Stuart and Ord 1987). The reason for this 

 is that both lognormal mean and variance increase 

 exponentially with the variance (a^) of the log- 

 transformed data. As a^ approaches zero the 



distribution becomes symmetric and the sample 

 mean, lognormal mean and median coincide 

 (Hastings and Peacock 1975). 



The sample median: The median of a sample 

 is the value corresponding to the mid-point of 

 the ranked observations in the sample. Unlike 

 the sample mean, the sample median always es- 

 timates the mid-point of a distribution. If the 

 distribution is symmetrical its mean and its median 

 coincide (e.g., the normal distribution). For nor- 

 mally distributed random samples, however, the 

 sample mean is a more accurate or "efficient" 

 estimator of the tme mean than the median be- 

 cause the standard error is larger for the median 

 than for the mean (Snedecor and Cochran 1980). 

 For this reason, and also because of the superior 

 statistical properties of the mean, the latter is pre- 

 ferred over the median with symmetrical distribu- 

 tions not far from normal. On the other hand, 

 with data whose distribution is highly skewed, the 

 median should be the preferred statistic because 

 it conforms with the concept of an "average" bet- 

 ter than the mean. An additional advantage of 

 the median is that it is not affected by extreme 

 values in the sample (i.e., outliers), whereas the 

 sample mean can be greatly affected, more so in 

 the case of small samples. A final consideration 

 regarding the median is that, for data far from 

 normal, nonparamctric confidence intervals and 

 tests to compare medians are readily available. 

 The performance of the median and its confidence 

 interval with lognormal data that contain many 

 zeros had not been reported prior to this study. 



Tlic geometric mean: The chief application of 

 the geometric mean lies in lognormally distributed 

 data for which the sample mean is a biased esti- 

 mator of the lognormal mean and the median has 

 only nonparamctric (very conservative) estimators 

 of its variance. Unlike the sample mean, the 

 geometric mean is also an estimator of the middle 

 point of a lognormal distribution because it co- 

 incides with the median. In practice, a simple 

 logarithmic transformation of the lognormal data 

 allows the estimation of the logarithmic sample 

 mean and variance which then can be used to 

 estimate the geometric mean and its asymmetric 



DELTA Means 



313 



