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JOURNAL OF ECONOMIC ENTOMOLOGY 
[Vol. 17 
a gain of 55.14 sound apples per tree. If, however, the probable errors 
of the means be calculated, a measure of comparison is provided. 
This gives the following data: Plat 2, 158.00 ±16.99; Plat 3, 102.86 
± 9.76; difference, 55.14 ± 19.59. The probable error of a sum or 
difference of two quantities A ± a and B ± b is given by the formula 
E d = Va- + b 2 . 
Before the significance of these numbers can be appreciated, it is 
necessary that some criterion be established as to the relation of a 
probable error to the number to which it is attached. Without going 
into a lengthy discussion fit may be stated that biometricians demand 
odds of at least 22 to 1 and many require odds of at least 30 to 1 that, 
should the experiment be repeated, the mean shall fall within certain 
limits before the result can be called significant. Odds of not less than 
22 to 1 require that a number shall be at least 3 times its probable 
error, and odds of not less than 30 to 1 require that a number be not 
less than 3.17 times its probable error. The writer prefers the latter 
odds. A simple method of comparison is to take the difference of two 
means and compute the probable error. The means of the two plats 
show a difference of 55.14 ± 19.59 and the difference is only 2.81 times 
its probable error, so is not significant. Stated otherwise, the odds are 
only about 16 to 1 that, should the experiment be repeated, the mean of 
Plat 3 would be higher than that of Plat 2, therefore, the data are not 
conclusive. It should be noted, however, that if total infested fruit is 
considered, the difference between the two plats is 4.93 times its probable 
error, so is significant. 
The Probable Error of the Mean of a Small Number 
of Observations 
With large numbers of observations, the formula for the calculation 
a 
of the probable error of a mean is E m = ± 6745 ~~A where a is the stan- 
V n 
dard deviation and n the number of observations. This wfill be referred to 
as the Gaussian Formula. If the number of observations (trees in this 
instance) are less than 16 in any plat, the probable error of the mean 
computed by the Gaussian formula will be too low, thus giving a higher 
degree of confidence than the data justify. To overcome this difficulty, 
some workers, who have been compelled to use few observations, have 
used modified formulae for the calculation of the probable error of a 
