CIRCULAR 816, U. S. DEPARTMENT OF AGRICULTURE 13 
per county has been estimated from survey practice at 12 hours. The time 
required per -square-foot sample has been estimated at 2 minutes; or 
4 minutes for taking a sample from the field and a corresponding sample 
from the margin. It is probably true that overhead costs per field would 
be reduced if the number of fields were increased; however, for this 
approximate approach we must assume fixed costs. 
This application of biometric theory represents a departure from the 
practice of estimating the work required for a desirable standard error, 
such as 0.125 pod per square foot. It substitutes for this practice the find¬ 
ing of a work arrangement that will give as low a standard error as pos¬ 
sible under given limitations. This standard error may not be as low as 
would be desired, but it will be the lowest possible under the stated 
conditions. 
The variance for fields (V f ) over and above that expected from within- 
field variation is estimated by subtracting the mean square within fields 
from the mean square between fields and dividing by the number of units 
per field. It seems best for this calculation to pool within-field variances 
(for locations, interaction, and within locations), since they do not differ 
much and since the variance thus estimated will be that to be expected 
in general sampling. This procedure is illustrated by using data from 
table 3 for Cascade County. All within-field variance (weighted average) 
is estimated as 0.42; between-field variance is 0.93; V f — (0.93—0.42)/10 = 
0.05. V w is simply within-field variance. By us e of calculus, the best com¬ 
bination may be derived as follows: k = V (V W 'CD)/(V/C) and n = 
T /(C D-^-kC) , 8 where k = number of half-square-foot units per field, 
n = number of fields, CD = overhead cost per field, C = cost per unit, and 
T = total cost. For Cascade County, if T = 720 minutes, CD = 60, C = 4, 
we may calculate k= V (0.42X60)/ (0.05X4), or about 11. Then n = 
720/(60-j-(ll X4)], or about 7. The best combination is thus indicated as 
7 fields and 11 units per field. The variance of the county mean for this 
combination is estimated as 0.05/7-|-0.42/77, or 0.0126. The standard er¬ 
ror is V 0.0126, or a little over 0.11; doubled to apply to a square foot basis it is 
0.23. A practical demonstration by varying k may be made. If k is taken 
as 7 instead of 11, n will be 8; the variance of the mean will be 0.05/8— 
0.42/56, or 0.0137. If k is 15, n will be 6, and variance will be 0.05/6— 
0.42/90, or 0.0130. In either case the variance is increased, although a 
wide latitude in number of fields sampled and number of samples per 
field gives little change with these low variances. 
Similar calculations including samples from the field margins give 
similar results. Their inclusion in this analysis evidently makes little 
difference unless the variance ratio departs sharply from that of field 
samples. Hence the influence of modifying marginal sampling is ex¬ 
pressed as a reduction of the time required per field unit. In previous 
studies, the 2-minute period allowed for each field unit was doubled be¬ 
cause an equal number of units was taken in the margin. The possibilities 
in taking fewer units in the margins and thereby reducing the time per 
within-field sample to 3 minutes per unit, as well as in allowing 16 hours 
»If it is desired to estimate k and n for constant standard error and minimum cost, 
k will be estimated as above and n as (K. V f + V w )/{K'V m ), where V m is the square 
of the standard error of the mean. 
