Importance of Determining Limits of Error . 267 
school altogether is the largest element of uncertainty. In 
the calculation, of course, the total number who dropped out at 
any grade is made up both from those who were promoted and 
those who failed of promotion. No exact determination of the 
number in either case can be made, but the following additional 
considerations may render our inferences more accurate. t 
We must begin with the enrollment of the grades as corrected 
for increase in population (Table XIV, and p. 268). If we sub¬ 
tract, for example, from the third grade enrolled in 1887 the 
number promoted from the second grade in 1886, we have left 
approximately the number that were enrolled a second time in 
1887 in the third grade. This is about twenty-five per cent, of 
of the enrollment of 1886, and is about the average for the third 
grade, as will be found by trying different years. In the same 
way it will be found that a deduction of about 10 per cent, should 
be made from the fourth grade, 33 per cent, from the second grade, 
and 45 per cent, from the first grade to allow for double enroll¬ 
ments. Adding together our revised enrollments, we get a 
grand total of 1,270,000 (See Table IX. True Enrollment , p. 
290). Since now we have made reductions of 10, 25, 33, and 45 
per cent, in our percentages and a reduction of only 38 per cent, 
in our base, which was already three times too large (See p. 
266), it must be admitted that the resulting per cents are the 
smallest possible. 
Let us recapitulate the argument as regards the second grade. 
(1) We have as a base, 1,500,000, the total enrollment of all 
grades; as percentage, 282,000, the enrollment of the second 
grade; which is 18 per cent, of the total enrollment. (2) We 
have reduced the base to 1,270,000, while the true base, the num¬ 
ber that entered the first grade, is nearer 500,000 (p. 264); 
we have reduced the percentage to 212,000 or 33 per cent., 
which is about the true percentage; giving 16 as our per cent, 
instead of 18. (3) But since we have reduced the percentage as 
much as possible and left the base far too large, the resulting 
per cent, must be too small; that is, the “16 per cent.” of all 
that entered the first grade said to drop out in the second, is 
the smallest imaginable per cent., which was to be proven. The 
same is true of each of the other grades. 
