190 
Miscellanea 
VIII. The Calculation of the Probable Errors of Certain Constants 
of the Normal Curve. 
By RAYMOND PEARL. 
Anything which serves to lighten tlic labour of computation incident to biomet.ric investi- 
gations is very welcome. On this account I venture to call attention to a point in which 
the "Tables for Facilitating the Computation of Probable Erroi'S," recently published by 
Miss Winifred Gibson* will be found very useful. 
It has been shown by Pearson t that in order to determine whether a given frequency 
distribution deviates sensibly from the normal or Gaussian law, it is necessary to know the 
probable errors for the normal curve of the following constants : the skewness, Vft, ft— 3 the 
kurtosis and the "modal divergence." The formula for the probable error of the skewness, 
when 11 is the total number of individuals, is 
p.E.sk= -6744898 
In a recently published note| I pointed out the fact that, after having calculated the value 
of this expression for any given n, in order to obtain the probable errors of V^i, ft -3 and the 
"modal divergence" for the same distribution it was merely necessary to multiply the calculated 
value successively by 2, by 4 and l)y <t (i.e. the standard deviation of the distribution). We 
have then only to find an easy way of getting the value of the expression 
•6744898 
in order to make the whole process of testing any distribution for normality very simple indeed. 
It is clear that we may write 
yS=-<"«™^- -s/l' 
where K= sj\= 1 -224:7 U9 §, 
and xi is t'l^ xi of Miss Gibson's Table I. 
Therefore to test the approach to normality of any distribution we have merely to perform 
the following operations : 
(i) Look out x^ foi' the given n from Miss Gibson's tables and multiply it by the factor 
1-2247449, using as many places of decimals as necessary. This gives the probable error of the 
skewness for the normal curve. 
(ii) Multiply the result by 2, which gives the probable error of v' (ii for the normal curve ; 
multiply this result again by 2, and so obtain the probable error of /Sj — 3 ; finally multii)ly the 
probable error of the skewness by the <t of the distribution and obtain in this way the probable 
error of the modal divergence. 
The relative divergence from zero of the skewness, Vft, ft - 3 and the distance from mean to 
mode in comparison with their probable errors, measures the probability that the given 
distribution does not follow the normal or Gaussian law. 
* Biometrika, Vol. iv. pp. 385—393. 
t Biometrika, Vol. iv. pp. 169 — "212, and elsewhere. 
t Science, N. S. Vol. xxii. p. 802. 
§ From Barlow's Tables. 
