Vol. 6, 1920 
STATISTICS: G. A. LINHART 
683 
In a paper now in press and another in preparation, it is shown that 
all the types of frequency curves thus far published, excepting those having 
a zero class, conform to one mathematical expression. It is pointed out 
also that it is futile to try to fit discontinuous irregular frequencies as 
well as those having a zero class to mathematical formulas, since the 
statistical constants obtained from such formulas are too often misleading. 
For these types of natural or perhaps unnatural frequencies a graphical 
method is developed capable of yielding any desired degree of accuracy; 
that is, within experimental error. The general equation referred to 
above is, 
— = 0- ^ (1) 
yo 
where m denotes the numerical value of any measurement, mo the value 
of the mean, e the base of natural logarithms, y any frequency, and yo 
a frequency of the probability of a deviation zero. 
Obviously, when in equation (1), ni is greater than zero and less than 
twice mo, we may express the exponent of e in series, thus, 
y -1)^ + ,/3(- -,)^...]^ , . 
yo ^ . ' 
Neglecting all terms but the first, we obtain, 
= ^- V mo ^ = ^- = e- , (3) 
yo 
which is identical with the usual form of the equation for the probability 
of errors, and which is but an approximation of equation (1). 
In order to minimize the labor and time required to perform a large 
set of calculations, often running into the thousands, equation (3) was 
transformed into a rectilinear one, thus. 
Log (Log yo/y) = 2Logx + 2 Log h — 0.3623 = 2 Log x + K (4) 
With a few determinative values for x read off on the frequency polygon 
constructed from the experimental data, we may obtain the value for K, 
which is the distance on the Log (Log yo/y) axis from the origin to the 
point of intersection. We can then calculate the several statistical 
constants from the value of K ; for example, the index of precision, h; 
the probable error, x; the probable error of the mean, Xo ; the standard 
deviation; the coefficient of variability, etc.; or, 
k + 0.3623 
h = (10) 2 , 
k + 0.3623 
X = 0.4769 (10)- 2 , 
X 0.4769 k_+jy^623 
Xo = -r = —r~ (10)- 2 , etc. 
