DECEMBER 1, 1899.] 
each person for each occasion, and that each 
person has his peculiar range within which 
he varies. Too short or too long a period 
between movements is more tiring than the 
natural one in walking, running, rowing, 
bicycling, and so on. 
It is highly desirable to get some definite 
measurement of the difficulty of a free 
rhythmical action. This cannot well be 
done by any of the methods applicable to 
the force or quickness of act, but it may be 
accomplished in the following manner : 
As a measure of the irregularity in a 
voluntary act we may use the probable error. 
When a series of measureable acts are per- 
formed they will differ from one another, if 
the unit of measurement is fine enough. 
Thus, let 2,, x,,---, «, be successive intervals 
of time marked off by a subject beating 
time, or walking, or running, at the rate he 
instinctively takes. The average of the 
measurements, 
Dbz ats auch ate 
a n 
a 
can be considered to give the period of nat- 
ural rhythm under the circumstances. The 
amount of irregularity in the measurements 
is to be computed according to the well- 
known formula: 
Yy ate a, quneanos 
n—1 
p= 
where v,= %, — 4, v,= %,—4, ---, U,=2,— a. 
The quantity p is known as the ‘ probable 
error,’ or the ‘probable deviation.’ The 
quantity 
the ‘relative probable error,’ expresses the 
probable error as a fraction of the average. 
If all errors in the apparatus and the ex- 
ternal surroundings have been made neg- 
ligible, this ‘ probable error’ is a personal 
quantity, a characteristic of the irregularity 
of the subject in action. If,as may be 
SCIENCE. 
809 
readily done, the fluctuations in the action 
of the limbs of the subject be reduced to a 
negligible amount, this probable error be- 
comes a central, or subjective, or psycho- 
logical, quantity. Strange as it may appear, 
psychologists have never understood the 
nature and the possibilities of the probable 
error (or of the related quantities, ‘average 
deviation,’ ‘mean error,’ etc.). In psycho- 
logical measurements it is—when external 
sources of fluctuation are rendered negli- 
gible—an expression for the irregularity of 
the subject’s mental processes. Nervous or 
excitable people invariably have large rela- 
tive probable errors; phlegmatic people 
have small ones. 
Thus a person with a probable error of 
25% in simple reaction time will invariably 
have a large error in tapping on a telegraph 
key, in squeezing a dynamometer, and so 
on. JI have repeatedly verified this in 
groups of students passing through a series 
of exercises in psychological measurements. 
I do not believe it going too far to use the 
probable error as a measure of a person’s ir- 
regularity. This is equivalent to asserting 
that a person with a probable error twice 
as large as another’s is twice as irregular, or 
that if a person’s probable error in beating 
time at one interval is r, and at another in- 
terval r,, his irregularity is 7, times as great 
in the second case as inthe first. This con- 
cept is analogous to that of precision in 
measurements. We might use the recip- 
rocal of the probable error as a measure 
of regularity. The positive concept, how- 
ever, is in most minds the deviation, vari- 
ation or irregularity, and not the lack of 
deviation, the non-variability, or the regu- 
larity. In the case of the word ‘irregu- 
larity’ the negative word is applied to a 
concept that is naturally positive in the 
average mind. 
The irregularity in an act isa good ex- 
pression of its difficulty. Thus, if a person 
beating time at the interval 7 has an ir- 
