158 
and charted, it is Been that these will permit of the formation of a curve, deviating 
from the mode, which is called the frequency curve. There are two features of this 
curve discussed by 1'eaisou, which will be referred t<>. First, the mode is not neces- 
sarily the same thing as the mean. He says: "Suppose we Bet about counting butter- 
cup petals, then we should find that 5 petals occur most frequently, hut that there 
are buttercups to he found with 10 and even more petals. The mean will he found to 
lie Dearer to i; than 5 petals, and after Belection and cultivation, may even differ by 
as many as 2 petals from the mode. The amount by which the mean differs from 
the mode gives us a conception of the amount of asymmetry or skewness of the fre- 
quency cm \i — the greater length of tail, so to speak, on the mean side of the mode." 
Second, experience soon shows that very large deviations are not frequent, most of 
the frequency occurring in a limited range about mode and mean, and by calculation 
a standard deviation is adopted as applying to this. There is a course variation in 
the standard deviation, according to the subject under record. The frequency classes 
can he plotted off on diagram sheets, having the various classes located along a base 
line and drawing perpendiculars at these points proportional in length to the fre- 
quency. When the tops of the perpendiculars are joined by a line, the so-called fre- 
quency polygon results. This polygon may be symmetrical or unsymmetrical. The 
condition of skewness depends upon the symmetry of this polygon. The more 
symmetrical it is, the less variation may be expected in the subject studied. 
Experimental work of this kind, which has thus far had its greatest application in 
biology in general rather than as applied to farm animals, is worked out through 
tin' use of mathematical computation in which logarithms play an important part. 
It certainly is a line of investigation suited only to those familiar with calculus and 
advanced mathematics. 
Again quoting Pearson, a he says: ''The reader will be curious, however, to learn 
what frequency curves, deducted from coin tossing and dice experiments, have to do 
with mortality. The answer is this: If the laws of frequency we are here dealing 
with hold very generally for the distribution of artificial frequency in cases where 
we have no knowledge how the individual instance will turn out, but only statistics 
of what happens in the mass, may we not reasonably assume that they are essentially 
the laws of all large numbers, and that even the frequency of death, its distribution 
with age, will obey the same laws?" 
Davenport has published a little work on this subject,'^ which is really a laboratory 
manual with many mathematical tables. In this he lays special emphasis on the 
necessity for quantitative study as applied to the laws of variation, causes of variati< >n, 
selection, etc. Touching a matter quite direct in its application to animal breeding, 
lie says: " Quantitative studies in heredity will give definite information on prepo- 
tency of sex or race. By examining hybrids quantitatively and comparing them 
with their parents, we shall unravel the laws of inheritance in crossbreeding, and 
the principles of mixing character in biparental inheritance." 
While human subjects have furnished a considerable field for investigation in this 
line of inheritance and breeding, commencing with Galton twenty-five years ago or 
so, no doubt future investigators will give greater attention to its application to farm 
animals than has heretofore been the case. As mathematics is regarded as the basis 
of all exact science, we should promote as fully as possible its reasonable use in 
those investigations bearing on the breeding of farm animals. Most certainly it is 
true that repetition in observation and extensive duplication of record will do much 
to assist in reducing our knowledge to a more exact basis, and permit of drawing 
more stable conclusions. 
At the present time there is being expended in the United States by various agri- 
cultural experiment stations something in the neighborhood of 81,000,000 per year. 
A large percentage of the stations using this money are more or less engaged in con- 
ducting feeding experiments with farm animals. If you will consult the records 
generally available on the work of the stations, with the exception already given, 
you will find almost nothing relative to breeding experiments. Some work in grad- 
ing or crossing has been attempted as, for example, grading up sheep, or feeding 
crossbred sheep or swine, but these have really been feeding experiments rather 
than a study on breeding problems. It has not been so much of a case as to how a 
breed might be improved as, having a given type or form of animal, how will it feed? 
Farmers over the United States have studied long and industriously over feeding 
tables, and have anxiously inquired for information on what to give or buy to feed, 
and how to feed to secure desired results, w it bout giving any serious consideration to 
"The ( lhances of heath, etc., p. L8. 
''Statistical Methods with Special Reference to Biological Variation. By ('. B. 
Davenport, L899, pp. 59. 
