MISCELLANEA. 
Table of Ordinates of the Normal Curve for each 
Permille of Frequency. 
EDITORIAL. 
The need of a table such as the present has long been felt. A table of the abscissae of the 
normal curve for every permille of frequency was calculated by Sheppard at Galton's suggestion*) 
but the cori'esponding table for the ordinates to five significant figures, which is equally needful, 
has not hitherto lieen provided. 
When we have data classified in broad categories, whether we are dealing with one or two 
variates it is most desirable to exhibit the results in some graphical form in order that the lazy 
reader may have an appreciation of the matter under discussion. In the present state of our 
knowledge, whatever more or less justifiable criticisms may be raised, we do learn something by 
exhibiting the data with the marginal frequencies represented on the normal scale. The main 
answer to such criticisms is that such a representation is better than none at all. It will not 
give an association of the wrong sense, and when the correlation is at all sensible will for many 
practical purposes indicate its drift. The chief source of error lies, of course, when the number 
of categories is few, in treating the arrays of a second variable for a broad category of the first 
as if the.se arrays were themselves normal distributions. Even if the whole distribution were 
truly normal this would not 1 )e the case, but only the case approximately if the categories ceased 
to be ' broad,' and were replaced hy small subranges. In ordinary practice the chief difi&culty 
of the assumption arises from the fact that the centroid of the whole system for the second 
variate may not be in adequate numerical agreement with the centroid of the centroids of the 
arrays obtained on the normal assumption. It is true that if we deal with the whole system as 
normal and confine ourselves to a tripartite division of the arrays and marginal totals of the 
second variate we can Ijy aid of the tetrachoric functions obtain better appreciations to the 
centroids of the arrays ; not only does this involve a previous knowledge of the correlation 
coefficient, but the whole process, especially if it has to be applied to a large number of con- 
tingency tables, is very laborious. We are in using it in fact overlooking the main point that 
all we need is a rough diagrammatic exhibition of the general drift of the association. Those 
who have had occasion to plot large numbers of contingency tables to normal scales will be the 
first to admit the weaknesses of the method, but the last to assert that such diagrams are 
without value. They indicate quite effectively to the casual reader that here the association is 
of no practical importance, and that there it is an essential feature of the characters under 
investigation. 
In reducing data given in broad categories to a normal scale, all abscissae (.r) are measured 
in terms of the unknown standard deviation, all ordinates (s) in terms of unit frequency and 
unit standard deviation, i.e. £ = — L= e"^''"*- 
v2n- 
Now it is well known that the abscissa of the centroid .Tss- of a broad category lying between 
and .^j. is numerically given by 
where iigg' represents the proportional frequency between and sCg'. In most cases it will be, for 
diagram purposes, quite adequate to obtain the abscissae and ordinates of the normal scale 
from the propoi-tional frequencies to three places of decimals. Hence Table I of Tables for 
Statisticians and the present table give the required values in a few minutes, while the old 
process of determining s from Table II or, where it permitted, Table III was much more 
lengthy. 
Illustration. Find the boundaries and means of the following system of broad categories on 
the as.sumption that it corresponds to a normal frequency distributiont. 
* Bioinetrika, Vol. v. p. 405. It is reissued as Table I of the Tables for Statisticians and Bio- 
metricians, t The coutinuous variate may be looked upon as physiological fitness for life. 
