190 TRANSURANIC ELEMENTS IN THE ENVIRONMENT 



which is obtained from Eq. 1 by dividing both sides by X. The graphic representation is 

 given in Fig. 1, where the concentration is the same regardless of aliquot size. 



The fundamental assumption that underlies the mathematical assumption of a 

 multiplicative relationship between activity and aliquot size is that the radionuclide 

 activity is homogeneously dispersed throughout the medium on which it is measured. 

 This is the justification for the aliquoting procedure in which, for example, a 1-g aliquot 

 is taken for analysis from a 100-g soil sample and the observed aliquot concentration is 

 ascribed to the entire sample. 



< 



o 



21 



o 

 (J 



X 



X (ALIQUOT SIZE) 

 Fig. 1 Theoretical relationship between concentration and aliquot size. 



The classical statistical approach to estimating a ratio from a supposedly homoge- 

 neous set of data is linear regression. (See Snedecor and Cochran, 1967, pp. 167-171 , for 

 a complete discussion.) The model is 



Y = 7X + e 



(2) 



where Y, X, and 7 are as in Eq. 1 and e is the deviation of the data from the fitted model. 

 The estimate of 7 is the ratio estimate. 



An assumption underlying this approach is that Y is a random variable and X is 

 known without error. For a radionuclide concentration, tliis is a reasonable assumption. 

 The denominator is usually a weight or volume and is considered to vary through 

 measurement error only. This error is usually negligible compared with the sampling and 

 measurement errors associated with radionucHde activity. For that reason all the 

 variability in the ratio is attributed to the numerator. 



Another statistical assumption underlying the usual unweighted least-squares fit of 

 Eq. 2 is that the variance of Y is the same at each value of X; i.e., 



- ^2 



a'(Y|X)=a 



Both Y and X are assumed to be positive, and, from Eq. 2, the fitted line is forced to pass 

 through the origin (Fig. 2). If both Xand Y are assumed to be positive, the consequence. 



