436 Mr Engledow and Mr Udny Yule, The determination of 
The determination of the best value of the coupling-ratio from a 
given set of data. By F. L. ENGLEpow, B.A., St John’s College, 
and G. Upny YutE, M.A., St John’s College, University Lecturer — 
in Statistics. 
[Read 9 March 1914.] 
Many workers in Mendelism who have come across cases im | 
which coupling or repulsion occurred must have felt the necessity — 
for some general method by which to determine from their data _ 
the best value to assign to the coupling-ratio, apart from any — 
theory as to the ratios that are possible. Mr G. N. Collins (Am. | 
Nat., vol. XLVL, 1912) is, so far as we are aware, the only writer who | 
has suggested any such method. He worked out the value ofa | 
coefficient of association for the whole series of possible ratios, | 
1:1:1:1, 2:1:1: 2, etc, and then used the observed value of | 
the same coefficient to decide which ratio gave the best agreement | 
with the facts. While this method is very simple and convenient, 
it does not seem to lead to the most advantageous value for the 
ratio. 
The test to be used for the closeness of agreement between the 
theoretical and observed frequencies seems clearly to be that 
developed by Professor Pearson (Phil. Mag., vol. L., 1900). If 
FF ,F,F, etc. are a set of theoretical or expected frequencies and 
F/F.F. F{ ete. are those observed, and if . 
(F— FY 
F >} 
v= > 
the probability P that in random sampling deviation-systems of _ 
equal or greater improbability will arise is a function of y? which — 
decreases continuously as x? increases. The values of this function | 
for any number of frequencies from 3 to 30 have been tabulated — 
by Mr Palin Elderton (Biometrika, vol. 1.). In order to measure | 
the closeness of agreement between an observed set of the four 
| 
| 
frequencies for any pair of characters, and the expectation based 
on any assumed ratio, it is only necessary to work out the value of 
x? and turn up in Mr Elderton’s table the column headed n’=4 | 
where the probability that an equally bad or worse set of deviations 
might arise in sampling will be found. If P is high, the agree- 
ment is good; if low, it is bad. That value of the ratio, then, 
which gives the most satisfactory agreement with the data is the 
value which makes the probability P a maximum or x? a minimum. 
The value of P is not accurate if any frequencies are small, as a | 
