Two characteristics are common when describing functional relationships between 

 tree or stem size and volume. First, these relationships are nonlinear; and secon"d, 

 the variation within size classes is neither constant nor directly proportional to tree 

 size. When the data are plotted on logarithmic paper, the form may appear as a straight 

 line with a reduced heterogeneity of variance. The coefficients of the resulting log 

 linear model can be estimated by the method of least squares. 



To provide comparative statistics, the tables incorporating logarithmic models in- 

 clude expressions of mean error, average bias, and average deviations, all expressed in 

 percentage. These expressions indicate the relative reliability of the tables with 

 which they are associated. 



Percent average is obtained as follows: 



100(E Yi - E Yi)/E Yi = 100 (Y - Y) /Y 



where E Yi = sum of arithmetic estimates 



E Yi = sum of observations 



mean of arithmetic estimates 



Y = mean of observations 



Mean error, expressed in percentage, is an indication of the average variation of 

 the sample. Expressing the deviation in percentage tends to overcome the inherent 

 problem of heterogeneous variance. The procedure for calculating the mean error is: 



E ((Yi - Yi)100/Yi)2/n-k-l 

 i = l 



where Yi - observed value 



Yi = arithmetic estimated value 



n - number of observations in the data set 



k = number of independent variables in the model 



Inasmuch ^as bias is included in the calculation, the value obtained is always somewhat 

 larger than if an unbiased procedure had been used initially. From an interpretive 

 standpoint, the percent mean error is analogous to the usual standard error (deviation) 

 of regression. 



Although not appropriate from a purely statistical sense, average percent deviation 

 provides an estimate of the average deviation of the individual observations from their 

 respective means. In the past the value has often accompanied volume tables to provide 

 the user with a rough indication of variability. Average percent deviation is always 

 less than mean error, being approximately 75 percent that of mean error, depending upon 

 the magnitude of bias. Average deviation equals 



n 



E 

 i = l 



(Yi - Yi)100/Yi 



)/n 



5 



