338 
NALOKE 
[AucusT 10, 1899 
In the continuations of these Floras we have the same 
standard of excellence to which preceding volumes have 
accustomed us, and which we therefore look for in pub- 
lications coming from Kew. Their issue will be a boon 
not only to the professed botanical world, but also to all 
those who are interested in the many plants now known 
in, and still coming into cultivation from, Africa ; and 
they should give a great stimulus to the further invest- 
igation of the vegetation of Africa and to the introduction 
of interesting and beautiful plants to the horticulture of 
the world. 
No one looking at these volumes can fail to notice that 
their production at Kew, where a collection of living 
plants in a garden is associated with one of dried 
specimens in a herbarium, gives additional value to 
them. The necessity of the latter as a guide to the 
accurate determination of the nomenclature in a scientific 
garden is apparent ; the service of the former as an 
adjunct to the herbarium by affording means for the 
study of the living plants in cases where the dried 
specimen can seldom be satisfactory is clearly brought 
out in the account of the groups of succulent mono- 
cotyledons treated of in these Floras. If all descriptive 
botanists were able, as is possible at Kew, to look at the 
dry bones of the plants with which they deal with some 
consideration of the form that clad them when alive, we 
should be spared much of that prolific synonymy which 
is the bane of the systematist. It is the possession of 
the finest collection of living plants along with a like one 
of dried specimens, through which it can contribute as 
in these Floras to the advance of our knowledge of the 
vegetation of the globe, that gives Kew an absolutely 
unique position as the leading botanical institution of 
the world, a position it has achieved in little over fifty 
years through the scientific ability and remarkable 
administrative capacity of its successive Directors, Sir 
William Hooker, Sir Joseph Hooker, and Sir William 
Thiselton-Dyer. 
STATISTICAL METHODS APPLIED TO 
: BIOLOGY. 
Die Methode der Variationsstatistik. Von Georg 
Duncker. Pp. 75, with 8 figures in text. (Leipzig: 
Wilhelm Engelmann.) 
HIS pamphlet, a reprint from the Archiv fur 
Entwickelungsmechanik, is an attempt to render 
the formule and results of the statistical method some- 
what more accessible to German biologists than they 
are, for example, in Prof. Karl Pearson’s original papers. 
In the first part a complete outline is given of the fitting 
of frequency curves, normal or skew, to observed statistics, 
and in the second part a similar outline of the theory of 
correlation. The whole of this extensive ground is covered, 
however, in some sixty octavo pages, necessitating a 
degree of compression too great for satisfactory results. 
Proofs are necessarily almost wholly omitted, several 
difficulties likely to occur to beginners are slurred over, 
and there is more than one absolute blunder. 
If y = (#) be any frequency curve, the frequency of 
deviations lying between + — }c¢ and x + 3c is given by 
the integral of the frequency-function ¢ (x) between those 
limits. If, and only if, c be very small, we may replace 
NO. 1554, VOL. 60] 
this integral by the product y.c to a close degree of ap- 
proximation. Hence, in any practical case of recording 
the distribution of frequency where we have to choose an 
arbitrary unit of grouping c¢, this should always be made 
as small as possible. If this be done, and if the number 
of observations be large, the observed frequency polygon 
closely approximates to a continuous curve, and the ele- 
ment of area round any ordinate differs very slightly from 
y.c. Moreover, the process of obtaining the moments of 
the observed frequency polygon by treating the observed 
frequencies as isolated loads, then differs very slightly in © 
result from the process of continuous integration by 
which the moments of the theoretical curve were cal- 
culated. But if the element of grouping be not small, 
the element of area round y may differ very sensibly from 
y.c, and the process of calculating moments by treating 
group frequencies as isolated loads is not even a rough 
approximation to continuous integration. Hence Prof. 
Pearson’s original preference of the moments of the 
trapezia system (PAz/. Trans. A, 1895, “On Skew Vari- 
ation,” &c.), and Mr. W. F. Sheppard’s papers on moment 
calculation (Proc. Lond. Math. Soc., vol. 29, and Journal 
Roy. Statistical Soc., vol. 60,1897). This difficulty, due to 
the grouping, is entirely passed over by Herr Duncker. A 
series of observations giving only five base elements ¢ is 
fitted without remark to a normal curve (Fig. 2). In every 
case the ordinates of the fitted curve are calculated only 
for the abscissz of the observed ordinates, their tops 
are joined up, and the polygon so obtained called the 
“theoretical frequency polygon,” as in Figs. 1, 2, 3—a 
procedure of somewhat dubious use in any case, and 
quite illegitimate where the elements are as large as in 
Fig. 2. If the author had not missed this fundamental 
point he would not, perhaps, have been so puzzled by 
Prof. Pearson’s use of first one and then another method 
of calculating moments. It is a pity that all the arith- 
metical examples given of fitting frequency curves refer to 
cases of discontinuous variation, as these are naturally 
the material least suitable for representation by con- 
tinuous curves. 
There seems a corresponding lack of clearness in 
some fundamental points of the theory of correlation. 
The various formule for correlation coefficient, regres- 
sions, &c., are given, but the author nowhere clearly 
points out their meanings and limitations in cases of non- 
normal correlation. It is not noted that for complete 
independence the condition 7 = o is, in general, neces- 
sary but of sufficient. The values of partial correlations 
are given, but it is not noted that only in normal cor- 
relation (so far as we know) is the partial correlation 
the same for every array. The correction given on p. 48 
for reducing the product sum about an arbitrary pair of 
axes to the product sum about the mean is surely 
absolutely wrong; the different sums given are all of 
different dimensions. If S, be the value of the sum for 
axes through the mean, S, its value for the arbitrary axes 
SSNS 7, 
where N is the number of observations and «7 are the 
coordinates of the mean with reference to the arbitrary 
axes. Of course this expression, and method of getting 
Sp, is quite well known, not novel as Herr Duncker seems 
to think. 
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