By Student 
357 
III. 
IV. 
Mean =1-80 : /i2 = l"96 : ju3 = 2-529. 
0 1 2 
Actual 75 103 121 
Calculated 66 119 107 
Whence x- = 9-03 and /'=-25. 
Best fitting binomial (1-0889- •0889)- 
Mean =4-68 
3 
54 
64 
4 
30 
29 
5 
13 
10 
0.2473x400 for which P = -^l. 
Actual 
Calculated 
: /i2 = 'l-46 
0 1 
0 20 
4 17 
;U3 = 4-98. 
2 3 
43 53 
41 63 
4 
86 
74 
."J 
70 
70 
6 
54 
54 
7 
37 
36 
8 
18 
21 
9 
10 
11 
10 
5 
5 
11 
2 
2 
12 
2 
1 
Whence x^=972 and P=-64. 
Best fitting binomial (-9525+ •0475)'-'8-53 x 400 for which P=-68. 
These results are given graphically in Diagram II. on the next page. 
It is possible to fit a point binomial from the mean and the 2nd moment 
according to the two equations ^ = nq, fi.^ = npq and these point binomials fit 
the observations better than the exponential series, but the constants have no 
physical meaning except that nq = m. And since the exponential series is a 
particular form of the point binomial and is fitted from one constant, while two 
are used for the "ad hoc" binomial, this better fit was only to be expected. 
It will be noticed that in both I and III the 2nd moment is greater than the 
mean, due to an excess over the calculated among the high numbers in the tail of 
the distribution. As was pointed out before, the budding of the yeast cell increases 
these high numbers, and there is also probably a tendency to stick together in 
groups which was not altogether abolished even by vigorous shaking. 
In any case, the probabilities '04, -68, "25 and -64, though not particularly high, 
are not at all unlikely in four trials, supposing our theoretical law to hold, and we 
are not likely to be very far wrong in assuming it to do so. 
Let us now apply it to a practical problem : for some purposes it is customary 
to estimate the concentration of cells and then dilute so that each two drops of the 
liquid contain on an average one cell. Different flasks are then seeded with one 
drop of the liquid in each, and then " most of those flasks which show growths are 
pure cultures." 
The exact distribution is given by 
/^2 2! 
3! 
+ 
+ 
which is 
No. of Yeast cells 
0 
1 
2 
3 
Jf 
Percentage Frequency 
60-65 
30-33 
7-58 
1-26 
-16 
or approximately three-quarters of those which show growth are pure cultures. 
Bionietiika v 46 
