XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 311
unreliability of observed statistical results, and the term probable
error is given to this quantity. It should be noted that the word
“probable” is hardly used in its usual sense in this connection :
the probable error is merely a quantity such that we may expect
greater and less errors of simple sampling with about equal
frequency, provided always that the distribution of errors is
normal. On the whole, the use of the ‘probable error” has little
advantage compared with the standard, and consequently little
stress is laid on it in the present work ; but the term is in constant
use, and the student must be familiar with it.
It is true that the “ probable error ” has a simpler and more direct
significance than the standard error, but this advantage is lost as
soon as we come to deal with multiples of the probable error.
Further, the best modern tables of the ordinates and area of the
normal curve are given in terms of the standard-deviation or
standard error, not in terms of the probable error, and the mul-
tiplication of the former by 0:6745, to obtain the probable error,
is not justified unless the distribution is normal. For very large
samples the distribution is approximately normal, even though p
and ¢ are unequal ; but this is not so for small samples, such as
often occur in practice. In the case of small samples the use of
the “probable error” is consequently of doubtful value, while the
standard error retains its significance as a measure of dispersion.
The ¢ probable error,” it may be mentioned, is often stated after
an observed proportion with the + sign before it; a percentage
given as 205 + 2-3 signifying “20'5 per cent., with a probable
error of 2'3 per cent.”
If an error or deviation in, say, a certain proportion p only just
exceed the probable error, it is as likely as not to occur in simple
sampling : if it exceed twice the probable error (in either direction),
it is likely to occur as a deviation of simple sampling about 18
times in 100 trials—or the odds are about 4'6 to 1 against its
occurring at any one trial. For a range of three times the probable
error the odds are about 22 to 1, and for a range of four times the
probable error 142 to 1. Until a deviation exceeds, then, 4 times
the probable error, we cannot feel any great confidence that it is
likely to be “significant.” Itis simpler to work with the standard
error and take + 3 times the standard error as the critical range :
for this range the odds are about 370 to 1 against such a devia-
tion occurring in simple sampling at any one trial.
18. The following are a few miscellaneous examples of the use
of the normal curve and the table of areas.
Example i.—A hundred coins are thrown a number of times.
How often approximately in 10,000 throws may (1) exactly 65
heads, (2) 65 heads or more, be expected §
