Miscellanea 
117 
From these results the answer to our second question is, I think, clear, and may be stated in 
the following way : it is not a matter of practical significance in ordinary statistical work that 
the raw moments be used to calculate the probable error of a standard deviation which has 
itself been deduced from a moment to which Sheppard's correction has been applied. The 
error made by using equation (iii) instead of equation (iv) is not likely in the great bulk of 
ordinary biometrical work ever to reach the magnitude where it will be of any practical con- 
sequence whatever. One should, of course, always keep in mind the existence of this source 
of error, and if very fine, close work is to be done take pains to eliminate it. It is obvious, 
however, that the occasions where it will have to be taken into account will be extremely rare. 
Putting all our results together we may summarily state the practical conclusion as follows : 
In calculating the probable error of the standard deviation of frequency distributions which 
deviate sensibly from the normal curve, great caution must be exercised in using the customarily 
applied formula for this probable error (•67449 -^). The use of this approximate formula 
\ V 2?i/ 
is not unlikely to leiid to serious error when standard deviations are compared on the basis of 
their probable errors. It will be safest, in the absence of special investigation of the point, 
for each curve or group of curves under treatment, to calculate the probable error of o- by the 
formula -67449 
11. In using this formula for the probable error it is practically 
immaterial whether the moments used have or have not been corrected according to Sheppard's 
formulae. 
\^Note. The standard deviation of the standard deviation o- is known to be— ^^^^ i i 
v2;i - '' 
where jy is the kurtosis = /32 — 3. For 7 small the multijilying factor is l + and sujiposing 
we may neglect for practical purposes an error of 5°/„ in a probable error, we conclude that the 
ordinary foi'mula may be applied as long as the kurtosis lies between + -2 or for (32 = 2-8 to 3-2. 
This agrees with Dr Pearl's Table II, where only the curve of Type III has its kurtosis within 
this range. Thus the kurtosis alone, independently of the type, settles whether the ordinary 
formula is sufficient or not. A further point not yet discussed but of some importance is the 
significance of the "probable error" at all when we are taking the standard deviation of standard 
deviations in the case of very skew material. How far does the distribution of standard devia- 
tions follow a normal curve 1 The answer de^^ends largely, I think, on the size of the sample, 
and the deduction of conclusions from the "probable error" of o- may lead to greater error 
than even neglecting unless we are fairly certain that our a follows a normal distribution. 
See the paper by "Student" in the present number of this Journal. K. P.] 
VIII. Addendum to Memoir: " Split-Hand and Split-Foot Deformities." 
Biometrika, Vol. vi. pp. 26 — 58. 
Since the final proof sheets of the above paper were sent to press, several papers have 
appeared to which reference is necessary. 
The evidence of the Bateson School to show that Mendelism is applicable to deformities in 
man was criticised in the memoir. At that time stress had been laid on three family trees, that 
of Farabee, an instance of hypophalangia, and those of Nettleship, instances of congenital 
cataract. The tendency for the deformity to abate in successive generations was noted, for 
in these families the last generation in each is the only one which shows the undeformed out- 
