IO SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 149 



Statistical Significance of Numbers of Individuals 



I have suggested, however, that it is desirable not only to estab- 

 lish the relative abundance (species proportions) of the foraminiferal 

 population in a given area, but also to estimate the actual number of 

 individuals living and/or dead per unit area. To do so, it must be 

 assumed that the number of individuals in a given sample is a repre- 

 sentative portion of an unknown population which is homogeneously 

 distributed throughout the area the sample represents. 



If each member of a sample pair is a reliable estimate of the number 

 of individuals at a station, then a sample pair should be from the 

 same statistical population. Let the total number of individuals in a 

 sample pair be n. The probability of any individual belonging to one 

 or the other sample is p and q — \-p respectively. Therefore, we have 

 a binomially distributed variate with a mean of np and a variance 

 of npq. When n is large and p is close to \, the binomial distri- 

 bution closely approximates the normal distribution. The transfor- 

 mation is achieved by the formula: 



(r-np) 



x — 2 , where x is the standardized normal random variable, 



Vnpq 



r is the number of individuals in a sample, and ^ is a correction for 

 continuity. (Bradley, 1960, gives a discussion of tests based on the 

 binomial distribution.) 



The value of x was calculated for the total and living populations 

 in all the sample pairs. The results are shown in table 2 (page 71). 

 If a sample pair has a significant value of x, then we are confident 

 that each member of the pair is from the same population. In the 

 living population, 7 of the 12 sample pairs have a significant x 

 value. In the offshore areas (pairs 104-104' through 133-133'), 5 

 of the 7 pairs give a significant x value. In the total popu- 

 lation, 5 of the 12 pairs have a significant x value, while in the offshore 

 areas 3 of the 7 pairs are significant. In general, the number of 

 living individuals in the sample pairs give better results than the total 

 number, and the offshore areas give a more reliable estimate of the 

 number of individuals at a station than the near-shore areas. 



The possibility that the Foraminifera in L.I.S. are not homo- 

 geneously distributed throughout the area that a sample represents has 

 not been thoroughly investigated. As will be seen later, however, in 

 the offshore areas, the number of living individuals in samples from 

 the same traverse does not differ significantly. 



