52 BOTANICAL GAZETTE [JANUARY 
by lot and the values of P.E. sing. and N calculated. Strictly 
speaking, when the number involved is small, say ten, the formula 
for P.E. only gives approximate results (BRUNT 1). The value of 
P.E. sing. for the ratio is thus shown to vary from 0.5 in group 10 
to 1.8 in group 11, causing a change in N from 5 to 59 (table IV). 
One trial with a small number of fruits would not be adequate for 
the determination of the value of P.E. sing. and of N, at least with 
such variable material as oranges. 
Probable error of a probable error 
The preceding discussion indicates that variable values were 
found for N, depending on the value found for P.E. sing. To obtain 
an idea of the variability of P.E. sing. and of N in the manner 
described (that is, by obtaining the results given by several different 
groups containing different numbers) is tedious and unsatisfactory. 
A more convenient method of judging the accuracy of P.E. sing. 
and N is desired. It is plain that the probable error calculated 
from the analysis of fifty fruits is more representative of the lot 
than that calculated from ten fruits. The relation of the error in 
the probable error to the number of fruits analyzed is given by the — 
expression (BRUNT 1, p. 57): Probable error of P.E. sing. =P.E. 
sing. x ALO (formula 3). Thus if 1.4 is the P.E. sing. for 
the soluble-solids-acid ratio (table I), then the probable error 
of 1.3=1.3X 782 = 0.09, or about o.1. In other words, the 
5I—1 
“true” value of P.E. sing. is probably between 1.2 and 1.4. We 
may obtain an estimate of the limits of N by substituting 1.2 and 
1.4 successively in the formulas; in this case N is found to be 26 
or 36 for formula 1, and 13 or 18 for formula 2. 
Ordinarily it will be sufficient to consider the probable limits 
of the value of N by approximations made by the use of formula 3 
in the manner indicated. If it is found desirable to do so, however, 
a formula may be used for the correction. If we rearrange formula 
1 to read: N=, ( cece 
described by Goopwin (2), we find that deviation produced in the 
2 
) (P.E. sing.)?, and apply the method 
