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Correlation and Application of Statistics to Problems of Heredity 43

Now I think this does not involve all the tricolour ancestry of the four tricolour grandparents, for 0.21 of the great grandparents are non-tricolour,

andtherewillbe(0.21) x (0.56) x (0.5)3x(0.79)x(0.5)2ofthegreat great grand

parents tricolour. At each stage a non-tricolour branch will split off, showing in the next ascending generation some tricolour. It appears to me that Galton has overlooked the sum of all these ancestral tricolour contributions in estimating the tricolour in a0. They may be considerably less than those retained, but I do not think they can be disregarded without justification.

"By a similar process," Galton writes, "the average tricolour contribution from the ancestry of each non-tricolour grandparent is found to be 0.0243." (p. 406.)

It would seem that this is obtained from

(0.5)3 x (0.56) {1 +(0.5) x (0.56)+(0.5)2 x (0.56)2+etc.}= •0972, for one-fourth of this is 0.0243.

But the above expression is not, I think, correct, for after the great grandparental 0.56 of tricolour we must surely use not 0.56 but 0.79 to pass from tricolour to tricoloured ancestry. Thus the result should be

(0.5)3 x (0.56){1 +(0.5) x (0.79)+(0.5)2x(0.79)2+etc.}= •1157,

of which the fourth part is •0289.

Here as before the non-tricoloured ancestors of earlier generations who would themselves have tricoloured parents, etc., are neglected.

Taking Galton's illustration (p. 406) of both parents tricolour, three grandparents tricolour, and one lemon and white, Galton's factor of •8342 is only changed to •8388 b_y the above correction, but this gives 100 tricolour hounds out of a total of 119 offspring in this category, while the observed tricolours were 101, a remarkably close accordance.

I illustrate the sort of accordance obtained in the following examples

Number of Tricolour Grandparents | |||||||||||

Both Parents Tricolour |
|||||||||||

4 |
3 |
2 |
1 |
Totals | |||||||

Tricolour Offspring |
|||||||||||

Observed |
... |
106 |
(119) |
101 |
(119) |
24 |
(28) |
8(11) |
239 (277) | ||

Calculated |
... |
108 |
100 |
21 |
8 |
237 |

Both Parents and three Grandparents Tricolour |
Number of Tricolour Great Grandparents | |||||

8 |
7 |
6 |
5 |
4 |
Totals | |

Tricolour Offspring |
||||||

Observed ... |
- |
17 (18) |
19 (21) |
14 (16) |
6 (6) |
56 (61) |

Calculated ... |
- |
16 |
18 |
13 |
5 |
52 |

The numbers in brackets denote total offspring.

6-2

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