The second random variable (TVA^) is the estimate of stand characters for the 

 stratum: volume per acre, excess trees per acre, silvicultural prescription, or what- 

 ever is the character associated with a sampling point in the subcompartments selected 

 for ground examination. For purposes of assessing the variance of the stand estimates, 

 the mean square deviation from the stand average is pertinent. However, for 

 assessing the variance of the forest summaries over the compartments, it should 

 be recognized that TVA, is the ratio of two random variables. 



Compartment Variances 



The variance computations can best be illustrated by considering the compartment 

 estimates on a per-acre basis: 



To simplify notation and to facilitate cross-reference to standard sampling 

 texts such as Cochran (1963), let 



n^^ = number of sample points in stratum in subcompartment "i" 



I, . = TVA, .Z, . 



"^hi ~ ^hi \ ^ik^^i 

 k 



'^h \ '^^^hi^hi \ ^ik!^i 

 ^ k 



= y TYA, .Z, , 

 ^ hi hi. 



'^h \ ^^hil ^ik^^i \ '^hi 

 V k t 



That is, 1-^ and Z^ are numerator and denominator of 'I^^y^ respectively. Denote 

 the variance of their ratio by 



The variance of such a ratio can be approximated by the relation 



^hl \ h ^ h '^h ^h 



With clustered samples such as are used in this des^ign_, the variance depends on 

 the mean-square deviation of the subcompartment means (^/jjZ^) about the corresponding 

 mean for the forest. This mean-square deviation includes two components of variation: 

 (1) the variance between subcompartments, and (2) the contribution from the sampling 

 within the subcompartment that yields an estimate of the mean that differs from the 

 true and unknown mean for that subcompartment. 



15 



