Miscellanea 
113 
Now this value (ii) is the one ahnost universally used in calculating the probable error 
of the standard deviation, quite without regard to the type of the distribution under discussion. 
Indeed one suspects, from statements made in memoirs and handbooks of biometrical methods, 
that the impression prevails among biologists that (ii) is the true and unique expression for 
this probable error in every case. Obviously (i) will not equal (ii) except in the special case 
of the normal cur\e. In the common use of (ii) for (i) it is, of course, assumed, consciously 
or unconsciously, that the deviation of the curve from normality is not such as to cause, for 
practical purposes, any significant error as far as these probable errors are concerned. So far 
as the writer is aware no one has attempted to test in a concrete way the validity of this 
assumption in specific instances. Yet the matter is obviously a very important one. Probable 
errors lie at the vex'y basis of all biometrical reasoning. If in certain cases the probable errors 
are likely to be considerably in error when calculated in a certain way, any reasoning based 
upon them will have a most insecure foundation. 
These considerations suggest the following practical problem to the working biometrician : 
When a frequency curve is not a normal curve, is the error made in the probable error of the 
standard deviation by assuming normality (using (ii) for (i)) ever likely to be great enough 
to be of practical significance in drawing conclusions from data? This is our first problem. 
II. It has recently been pointed out by Sheppard* that in calculating the probable errors 
of the mean and standard deviation it is theoretically not justifiable to use the moments to 
which his corrections have been applied. Instead the "raw" moments should be used. The 
theory of this is stated as follows by Sheppard {loc. cit.), n being used to denote corrected and 
TT to denote raw moments : " Similarly if we take /nj to be equal to Tr-l — ^h}, where ir^ is the 
value of ■n<i as calculated from the actual measurements, the error in ^x^^ due to the limitation of 
number of observations, is equal to the error in tt^ ; and the mean square of this latter error 
is (774 — 7r2^)/«. Hence the probable error in the standard deviation will be, not 
Now while it is clear that theoreticcdh/ the uncorrected moments should be used, will it 
practically make any significant difiference in the results if the probable errors of the mean and 
standard deviation are calculated from the corrected moments ? This is our second problem. 
To get a ''working" answer to these two questions it was decided to calculate for a series 
of curves the probable error of the standard deviation by each of the different methods possible. 
These possibilities are, of course, first to calculate the probable erroz' by (i) with ijx) corrected 
and (6) "raw" moments, and then by (ii) using again the corrected and "raw" values of cr 
successively. With these results in hand for a number of curves we shall be able to form a 
practical judgment of the necessity or desirability of using the more refined methods in ordinary 
biometrical investigations. In choosing curves on which to make these tests it was decided 
to take a rather extreme example of each of Pearson's six types of curves. We may thus expect 
to get some idea of the maximum effect produced by calculating the probable error under 
discussion in the several different ways. 
The curves actually chosen for the work were the following : 
Type I. Curve for variation in leaf number of whorls on " all branches " of certain Ccrato- 
phyllum plants t. This is an extreme example of a skew curve of Type I. It is figured on 
p. 34 of the memoir cited. 
* Biometrika, Vol. v. p. 455, 1907. 
t Pearl, R., Pepper, 0. M., and Hagle, F. J. "Variation and Differentiation in Ceratopliylhim." 
Carnegie Institution of Washington, Publication No. 58, p. 32, 
Biometrika vi 15 
but 
