Inconvenientes do uso do valor médio do diâmetro para determinação da área basal - Gomes, F. Pimentel (Anais da Escola Superior de Agricultura Luiz de Queiroz, 1965)
This paper discusses the use of the arithmetic mean of diameters in the computation of basal area in forestry. This use, proposed in the Serviço Florestal of the Secretary of Agriculture of the State of São Paulo, leads to several difficulties, to be shown presently. Let be the diameter of a tree, with i = 1 referring to trees to be cut, and i = 2 to trees to be left uncut. If Nd is the number of trees to be cut, Nr the nummber of trees to be left uncut, and Np = Nd + Nr is the total number of trees, then the basal area of the total population (Ap) is Ap = (π/4) Σ/i,j D²ij , and the basal area of trees to be cut and to be left uncut are, respectively, Ad = (π/4) Σ/j D²lj , Ar = (π/4) Σ/j D²2j , The similar estimates A'p; A'd, A'r, computed with the aritmetic means of diameters, are: A'p = (π/4) D²../Np , A'p = (π/4) D²1./Nd , A'p = (π/4) D²2/Nr , It is shown that A'p A'd - A'r = - (π/4) [ sum of square of contrast cut thees v. uncut trees ] so that, since the contrast in question may be taken as alwys non-zero, we have A'p < A'd + A'r , which is certainly absurd from the point of view of the theory of measure. But indeed, in some cases we may even have A'p < A'r , which is really an absurd. On the other hand it is proved that Ap = A'p + .(π/4) [ sum of square of deviations of diameters in population ] so that the difference A'p, Ap - increases with the heterogeneity of diameteres of the trees.