48 BOTANICAL GAZETTE [JANUARY 
HAYNES and Jupp (3) have studied the requirements under the 
first condition. They proposed the following formula for use in 
calculating the number of individuals to include in a sample in 
order that a certain difference between two averages may be 
: er SDV ows « 
considered significant: N =2(322) .' N is the ‘number of 
samples which must be taken in order that there may be a proba- 
bility of 0.957? that a 5 per cent difference is significant”; 3 is the 
coefficient in the ‘‘table of odds” (table II), and thus is equivalent 
to odds of 22 to 1; ‘“‘p”’ is the probable error of a single sample and 
must be determined experimentally (in this case by analyzing 
individual fruits). 
Other values than 3 and 5 may be assumed to meet the condi- 
tions of the experiment; therefore, in order to make comparisons 
with what is to follow, it is desired to express the preceding formula 
coefficient of odds X P.E. sme)" 
difference 
(formula 1). To illustrate the use of this formula, data may be 
taken from Haynes and Jupp’s paper. Working with apples, 
they found the mean titration value to be 10.20 with a P.E. sing. 
of o.78, and the latter is thus 7.7 per cent of the mean. To get an 
assurance of 30 to 1 that a 5 per cent difference is significant: 
2 
N=2 (S2sia) = 49 apples. 
The problem under the second condition may now be considered. 
We wish a general formula that will connect the number in the 
sample with the probable error of a single fruit and with the 
coefficients in the ‘table of odds’’ (table IIT). In table Lit was shown 
that the mean sugar content was 1089.+0.06. What are the 
chances that the ‘‘true”’ value is within the limits 0.17? The 
0.17 
0.06 - is 
and looking up the coefficient 2.8 in table II, we find the 
chances are about 16 to 1 that the error in 10.89 is not more 
than +0.17. 
in general terms as follows: N = 2( 
chances are found in the following way (MERRIMAN 5): 
2 The expression 0.957 may be thought of as indicating a probability of 957 out of 
1000, which represents a ratio of 957 to 43, or about 22 to 1. 
