Karl Pearson 
539 
III. 
For X 
For z 
Distribution of 640,000 indices for 800 samples of 25 counts. 
v-i = 55-03056, V, = 222-90230, 
2^; = -024861, I// = -007834, 
^3= 1-36811, m, = 1-74622, 
« = 1-2480, 
Mean = 1 0356, 
Mode = -9330. 
= 18-42551, 
z., =3-68160, 1/2=14-00990, 
I// = -281298, 1/,' = -082055, 
mi = 1-03563, ??i2 = 1 14959, 
/X2= -077063, (r = -2776, 
yU3 = -017943, ;8i = -7060, 
/i4 = -025583, ^2 = 4-3088, 
The distribution is of Type VI* : 
r = -31-25271, = 51-67822, 
a = 1-52075, log ?/„= 26-6813462, 
and the equation to the distribution of the 640,000 indices possible is given by : 
F= antilog 26-6813462 x (Z - l-52075)™i X-'^'^'^^^ (iii). 
The origin is at the opsonic index — 1-4304. 
Diagram III gives this distribution reduced to a total of 1000 indices and set 
against Green-wood and White's experimental curve for 800 increased to 1000. 
This exaggerates the apparent deviations, but enables us to compare with the 
samples of 50 and 100. Considering the relative paucity of Greenwood and White's 
drawings, and the fact that we have fitted our curves to their non-replaced material, 
whereas a random sample would probably be better represented by the replaced 
material, the fits seem fairly close. The actual goodness of fit test was not applied, 
as at the time of writing the paper means of drawing the curves on a large scale 
and mechanically integrating them were not accessible. 
(6) It is as well to look at a combined table of the frequency constants of 
the three distributions given above. 
No. of Counts 
average based ou 
No. in Distribution 
of Means 
Mean 
Mode 
(T 
^1 
ft 
25 
800 
1 -0-356 
•9330 
•2776 
•7060 
4-3088 
50 
400 
1-0205 
-9611 
•2072 
•4837 
4^1022 
100 
200 
1-0116 
•9774 
•1562 
•3123 
3^9472 
It will be seen that these values form a very consistent relatively smoothly 
altering system. But the approach to normality is very slow. Even with counts 
of 100 the distribution of the opsonic index is markedly skew and platykurtic, and 
it would not be safe to treat the distribution as a normal curvef. In all cases the 
* Type VI is adjacent to Type I^V and in this case the curve is almost on the boundary line — 
i.e. Type -V: see Ehind's Diagram, Biometrika, Vol. vii. p. 389. 
t The reduction of /3i and to the Gaussian values 0 and 3 is, of course, not at the same rate as if 
we averaged 2-5, 50 and 100 opsonic indices. What we are doing here is to average the number of 
bacilli in 25, 50 and 100 cells on which the two factors of the index are based — a very different 
process. 
