630.7 I26b no. 165 cop. 8 •^ UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN AGRICULTURE Rl L t - CIRCULATING C Contents of Bulletin No. 165 PART I. THE STATISTICAL INTERPRETATION OF FEEDING EXPERIMENTS 1. Introduction. — Value of feeding experiments. Difficulties of interpre- tation. Factors producing gains in weight. The problem to be studied. Gen- eral method of solution Pages 4G3 to 4G7 2. The Frequency Distribution and the Average. — The frequency distri- bution. The normal frequency curve. The average as a type. The average as a descriptive value. Pages 467 to 472 3. Variation and its Measurement. — Th? range of observations. The standard deviation Pages 472 to 473 4. The Significance of an Aver.\ge and its Probable Error. — The stand- ard deviation of an average. The frequency distribution of an average. The probable error of an average. The limits of practical certainty. Pages 473 to 478 5. Illustrations of the Use of the Probable Error. Pages 478 to 480 6. A Probability Method for Small Lots of Animals Page 480 PART II. A STATISTICAL STUDY OF VARIATION IN THE GAINS IN WEIGHT OF FARM ANIMALS UNDER LIKE CONDITIONS 7. Introduction. — ^J. B. Lawes on variation. The coefficient of variation. Pages 481 to 483 8. Coefficients of Variation Ordinarily Obtained in Feeding Experi- ments. — Results of Wood and Stratton. Results of Robinson and Hainan. Re- sults obtained from American experiments. Results obtained at Woburn and Rothamsted. Discrepancies among coefficients. Meaning of such discrepancies. Pages 483 to 487 9. Number of Animals per Lot Required in Feeding Experiments. — Ad- vantages of large lots of animals Pages 487 to 489 10. Size of Gains and Their Variability. — Evidence for sheep, swine, steers, and poultry. Summary of evidence Pages 490 to 492 11. Reduction of the Experimental Error in Feeding Experiments. (a) Importance of reducing the experimental error.. .Pages 493 to 496 (b) Selection of animals as regards age, breed, sex, and previous treatment. Conclusion Pages 496 to 507 (c) Changes in variability of gains during the course of a feeding ex- periment. Rothamsted experiments with sheep. Woburn experiments with sheep. Iowa experiment with pigs. Michigan experiment with pigs. Illinois experiment with steers. Canadian experiment with steers. Sum- . mary of evidence Pages 507 to 512 (d) Effect of change of ration on variability of gains. Illinois ex- periment with sheep. Wisconsin experiments with swine. Henry's experi- ments at Wisconsin with pigs. Wisconsin experiment with lambs. Penn- sylvania experiments with steers. Woburn experiment with sheep. Con- clusions Pages 512 to 526 (e) Physiological selection of animals for feeding experiments. Theo- retical considerations. Does physiological selection eliminate poor gain- ers? Does physiological selection reduce the experimental error? Con- clusions Pages 526 to 534 (f ) Summary of methods of reducing experimental error - • rages 534 to 535 12. Repetition of Feeding Experiments. — Henry's experiments at Wisconsin with pigs. Wyoming experiments with sheep. Montana experiments with sheep. Minnesota and Pennsylvania experiments with steers. Difficulties of repetition. Probable explanation of these difficulties Pages 535 to 540 13. Variability in the Composition of Feedstuffs. — Corn. Wheat. Grains in general. Roughages. Commercial concentrates. Conclusions Pages 540 to 548 PART III. SUMMARY AND CONCLUSIONS APPENDIX 14. Statistical Data Concerning the Rate of Growth of Sheep, Swine, Steers, and Poultry Pages 558 to 570 15. Number of Animals to Include in an Experimental Lot, Derivation of Formula Pages 571 to 572 16. Change in Variability of Gains in Weight as Related to Feed Con- sumption. (Additional Data.) Pages 573 to 577 17. Bibliography Pages 578 to 579 ■A THE ELEMENT OF UNCERTAINTY IN THE INTERPRETATION OF FEEDING EXPERIMENTS By H. H. MITCHELL, Assistant Chemist, and H. S. GRINDLEY, Chief in Animai. Chemistry PART I. THE STATISTICAL INTERPRETATION OF FEEDING EXPERIMENTS Introduction Value of Feeding Experiments. — The purpose of the type of feeding experiment considered in this bulletin is the comparison of the fattening value of two or more systems of treatment of farm animals or of the fattening qualities of two or more groups of animals differing in age, breed, type, condition, or other particular. This comparison is made on the basis of the gains in weight recorded, the feed consumption, the results of the block test, and the economic considerations involved. Such an experi- ment is the most direct means of attacking many of the problems confronting the live-stock farmer. Our knowledge of the prin- ciples of animal nutrition is too fragmentary to enable us to fore- tell with certainty, except when greatly dissimilar, which of two rations, for instance, will produce the more rapid or the more economical gains in weight for a particular kind of farm animal, no matter how clearly defined or completely analyzed the rations may be. Actual experiment with those particular rations is gen- erally essential to a satisfactory solution of the problem. How- ever, the information thus obtained has at best a very limited ap- plication to other rations or other conditions, so that such feeding experiments ordinarily contribute little of fundamental importance to the science of animal nutrition. Difficulties of Interpretation. — The plan of the ordinary feed- ing experiment, such as defined in the preceding paragraph, is simple, but when completed its results are often of ambiguous sig- nificance, '^nd the problem of their rational interpretation is in any case worthy of the most careful attention. This is peculiarly tru oi experiment station work, upon which recommendations to the farming community are made. The difficulty of interpreting 463 464 Bulletin Xo. 1G.> [July, the results of the feeding experiment may be diminished to a con- siderable extent by taking great care in the selection of experi- mental animals and by controlling experimental conditions as carefully as possible or practicable; but even after such precautions have been taken, a certain degree of ambiguity still attaches to the experimental results. The ambiguity inherent in feeding experiments, and in fact in all experiments concerned with the functional activity of living or- ganisms, is due to the impossibility of foretelling with certainty the precise result that would be obtained if the experiment were repeated as carefully as possible upon other similar animals or even upon the same lots of animals. This element of uncertainty in the interpretation of feeding trials is the more pronounced, of course, when attention is directed to the results that would be obtained by the practical farmer in following the recommenda- tions of the experimentalist based upon an investigation conducted by the latter, because of the fact that the farmer in many cases cannot impose the precise experimental conditions required. Thus, an experiment station must be doubly cautious in advising its farm- ing community as to the systems of feeding that are best to em- ])loy, since, with the most careful attention to details, a greater or less degree of uncertainty always exists as to whether essentially the same result would appear on repetition of the experiment. Furthermore, this uncertainty is always enhanced by the certaintv that the farming community in general often cannot in practice follow instructions to the letter. The publication of the results of a feeding experiment may be confined to a description of the experimental animals, the rations fed, and all other experimental conditions, and to a statem.ent of the gains in live weight detained, the changes in condition of the animals, the financial gains or losses, etc. "Such a statement is in itself valuable and not void of interest because it contains the description of a fact, but as long as this fact is not connected with other facts its statement is not so much knowledge as the material for the future acquisition of knowledge. On this ground one even cannot conclude that under similar conditions results will be ob- tained which resemble those of the first series of observations. It is, indeed, out of the question to reproduce exactly the same con- ditions, and, since one does not know anything about the conditions which necessitate the result, one cannot positively say that only the observed conditions are of importance and one n 'ist resign the hope to foretell future results. But the main interest of all investigations is to know whether the same, or at least sneillar results will be obtained in a future repetition of the observation. Before such a statement can be made it is necessary to forni one's I /pij] Uncertainty in Interpretation of Feeding Experiments 465 views about the causes which were at work to produce the first result."^ Factors Producing Gains in Weight. — In deahng with experi- mental observations on hving org'anisms, such as observations on rates of gain in Hve weight of farm animals, one forms the hy- pothesis that the experimental results, i. e., the gains in weight actually obtained, are due, in the first place, to a complex of con- ditions definitely imposed upon the subjects of the experiment and under relatively perfect control. These conditions consist, for in- stance, of the rations fed, the preparation of the rations, the method and times of feeding, the method of sheltering, weighing, and exercising the animals, the season in which the experiment is run, etc. If the feeding experiment be repeated, it is this com- plex of conditions that it is possible to maintain constant. In the second place, the gains in weight obtained must be considered as being influenced also by another group of conditions not under control. These conditions may be considered as consisting of the temperaments of the animals as evidenced in their differential physi- cal activity, their feeding capacities, their physiological peculiari- ties, and all of the functional characteristics that render one animal distinct from another and are known collectively as its individuality . Besides the individualities of the experimental animals, there must be included in this second group of causal conditions the environ- mental conditions not under control, such as the weather, and even the personality of the attendant. Such uncontrolled conditions can- not be kept constant, of course, from one experiment to another, but are necessarily variable. Therefore, they constitute the ele- ment of uncertainty in the full interpretation of the results of feeding experiments. In (^rder to deal with these variable condi- tions in foretelling the result of repeating such an experiment, we merely assume that their influence on the rate of gain in live weight is perfectly random, showing no recognizable law or regularity ex- cept in a large series of observations. As Urban aptly says, "We base our expectation that a repetition of the series of experiments will give similar results on the identity of the conditions which we know and on the supposition of the random character of the influences which we do not know." The Problem to be Studied. — It is the purpose of this sec- tion of the bulletin to consider the element of uncertainty in the interpretation of the results of feeding experiments due to these variable, uncontrolled, and largely unknown experimental condi- tions, and to propose methods of dealing with the question in a systematic and rational manner, so that the sphere of uncertainty 'F. M. Urban, Exp. Stud, in Psych, and Ped., Ill, "The Application of Sta- tistical Methods to the Problems of Psychophysics," p. 19. Phila., 1908. 466 Bulletin No. 165 [/»^y, surrounding the conclusions based on experimental results will be reduced to a minimum and be defined as clearly as possible. The methods proposed have been employed in other and closely related fields of research and, in fact, have already been applied in a brief manner to one of the many problems connected with feed- ing- experiments, by Wood and Stratton,^ and later by Robinson and Hainan,'' of Cambridge University. A feeding experiment involves not only a record and interpre- tation of the feed consumption and the gains of each lot, but also a statement of the cost of the experimental animals and of the feeds consumed as compared with a statement of the price real- ized on the animals of each lot when sold. The question of the relative emphasis to be placed on these two subdivisions of the sub- ject matter of a feeding experiment is of importance. In view of the fact that the feed consumption and the resulting gains of a given lot of animals on a given ration determine the final condi- tion of the animals and afford the basis for the economic considera- tions involved in a feeding experiment, and in view of the fact that "conditions as to market price of feeding and fat cattle and cost of feeds have ne\er been identical during any two consecutive years and seldom more than similar at irregular intervals,"*^ it is ob\'ious that the most valuable data of a feeding experiment are the data concerning the feed consumption and the rapidity of gain of the class of farm animals from which the experimental ani- mals were drawn. B. E. Carmichael takes substantially the same position in the following quotation : "The author is thoroughly convinced that too important a place is often given to the cost of gains when discussing the results of a feeding experiment, thus rendering more probable a wrong understanding by the student or feeder. When feeders and experimenters think, reckon, and write concerning feeding experiments with amount of feed and rate and extent of gain in live weight, rather than zcith cost cff feed, animals, and gains and net proUt from the opera- tion as the factors for comparisons, it will be reasonable to expect more intel- ligent selection of rations and consequently fewer failures to secure satisfactory returns for feed and labor required to conduct feeding operations. "The writer would not be understood as saying that a financial statement is of no value or that nothing should be said concerning the cost of gains. On the contrary, each has a value, but it is believed that in either case the value is far less important than is the matter of the amount of feed required to produce a given gain, on account of the sudden and wide variation in price that may occur."" The feed consumption in feeding experiments, according to the ordinary practice, is determined for the entire lot rather than for the individual animals, and such total data are not susceptible •Journ. Agr. Sci., vol. 3, pp. 417-440. 1908-10. See also T. B. Wood, Journ. Board Agr., London, Sup. 7, 1911, Nov., pp. 32-37. "Journ. Agr. Sci.. vol. 5, pp. 48-51. October, 1912. 'Herbert W. Mumford, 111. Agr. Exp. Sta., Bui. 90, p. 203, "Ohio .Agr. Exp. Sta., Bui. 187. pp. 18 and 19. ^913] UXCERTAIXTY IX IXTERPRETATIOX OF FeEDIXG ExpERIMEXTS 467 to treatment by the methods to be outlined below. In this bulletin, therefore, attention is conhned to the gains in weights obtained in feeding experiments and to the methods of comparing ade- quately the gains of two or more lots of animals, since in many experiments individual gains are reported. General Method of Solution. — The problem of the feeding ex- periment that is considered in this section of the bulletin is the comparison of a number of gains in weight made by animals in one lot, treated alike as far as practicable, with a number of gains in weight made by animals in another lot, treated alike, but in one particular treated differently from the animals of the first lot, the object of the comparison being to determine whether the one difference in treatment between the two lots has produced a dift'erence in the rate of gain in weight. This comparison may be most effectually made by considering the two series of gains sepa- rately at first, with the idea of describing each adequately, but with as few terms as possible, and then comparing the two ab- breviated descriptions. The Frequexcy Distributiox and the Average^ In describing the gains made by a group of animals, the total gain of the group is often taken, but for comparative purposes it is almost universallv considered that it is better to reduce this total gain to a per capita basis, and hence it is generally the case that the common average or arithmetic mean of the individual gains of a lot is the one value taken as descriptive of the lot. When a chemist runs a series of atomic W' eight determinations upon a chemical element, and subsequently takes the average of his results, this average has a perfectly definite physical signifi- cance, f. e.^ it is the best approximation obtainable to the actual atomic weight of the element. However, the case is quite different when the investigator in animal nutrition averages the gains in weight made by a group of similarly treated animals during the same period of time. Strictly speaking, there is nothing here that can be called a "true value" to be obtained from a set of values diverging from it as the result of errors of observation. The distinction between the two cases is well brought out by Edgeworth when he says that observations, such as those of the chemist, "are different copies of the same original," while statistics, such as those of gains in wei^lit of a lot of animals, "are different origi- nals affording one generic portrait." The meaning of the average of a set of statistics may be con- sidered in the following way : It is conceivable that the aggregate 'In the following discussion, "average" refers to the arithmetic mean. I 468 Bulletin No. 165 [July, of the direct ly imposed experimental conditions under which the gains in weight were made, operated in the production of a typical gain, from which the individual gains diverge as the result of the casual or random sources of variation, and the average gain may be considered as the best approximation to this type. The Frequency Distribution. — In considering this conception of the average gain in w-eight of farm animals treated in a similar manner, it is necessary to investigate the frequency distribution of such gains. Suppose a large number of animals, say several hun- dred, were treated alike as far as possible with regard to feed, shelter, etc., and suppose the average daily gain in weight for each animal be determined for a considerable period of time, say one hundred days. Suppose the daily gains thus obtained be grouped into class intervals of o. i lb. and the number of gains occurring within each class interval be noted. The series of numbers thus obtained is known as a frequency distribution, since it gives the frequency with which gains of any given magnitude occur. There are, of course, no data in existence of the gains in weight made by several hundred animals treated alike at the same place and during the same time. Therefore, in obtaining such frequency distribu- tions an indirect method has been employed. In obtaining the numbers upon which Fig. i of the chart is based — this figure being a graphical representation of the frequency distribution of the daily gains in weight of 498 sheep — the average daily gains made by 46 lots of sheep were taken, the lots varying in size from 8 to 16 sheep. The lots were treated in different ways, of course, and at different stations during different periods of time. In combining the dift'erent gains for the purpose of forming one distribution, it was desired to eliminate the variation due to dif- ferences in feed and other definite factors, and to retain only that variation due entirely to the casual factors, such as individuality and imperfectly controlled conditions. In accomplishing this ob- ject, the average daily gain in weight of the entire group of 498 sheep was obtained and found to he 0.3485 lb. Xext, the gains in each of the 46 lots were changed, or transmuted, by addition or subtraction so that the average daily gains of the various lots were made identical and approximately equal to 0.3485 lb. Thus, the first lot so treated consisted of 10 sheep with an average daily gain of 0.283 lb. By adding to each of the ten gains in the lot the difference between 0.283 lb. and 0.348 lb., which is equal to 0.065 lb., the desired change was accomplished, ^or another lot of ten animals with an average daily gain of 0.413 lb., 0.065 lb. was sub- tracted from each of the individual gains. Ry thus making the average gains of the 46 lots identical without disturbing the vari- ation within the lots, it was believed that the intluence of the dif- voiua* .10 .15 .20 Z5 .30 .35 40 4S .SO .55 .60 Tr.- 3 7 27 44 92 129 121 48 21 5 1 . 1 Class value* 75 1.00 1.25 1.50 175 2.00 Z Fr«- qu»ncici 1 2 7 20 48 63 FIG. I FIG. 3 Frec^uency Distribution of the Daily Frequency Distribution ( Gains in V/eight of 498 Shee}^. Gains in Weight of 241 1.75 48 63 2.O0I225 59 ZZ a502.75 13 3.00325 Cltiss values .60 .70 .80 li .90 14.5 26 1.00 1. 10 57 sas 97 1.20 1.30 1.40 1.50 1.60 76.5 45532.5 1.70 7.5 1.80 2.5 i.90i2oda I I 1.3 FIG. 3 FIG. 5 ibution of the Dail\( Frecjuerjcy Distribution of the Dailv/ Gains in of 241 Steers. Weight of 461 Pigs. 82.15 84.75 87.35 89.95 92.55 93.15 97.75 FIG. 7 Frequency Distribution of the Coefficients of Digestibility of the Protein of a Normal Mixed Diet, Obtained from 1153 Observations on 23 Men. FIG. 8 Frequency Distribution of the Dail^ Excretion of Indican in the Urine, Obtained from 814 Observations on U Men. jQij] Uncertainty ix iNTERPkETATiox ok 1'eeuing Ilxpekimems 469 ferent experimental conditions among the lots was eliminated as far as possible, while the inlluence of the casual factors represented by the variation within the lots was preserved intact. The 498 gains thus transmuted were used in obtaining Fig. i. In this distribution the class interxal chosen was 0.05 lb., as is indicated by the first row of figures at the base of the diagram. The second row of figures gives the frequencies of the different classes. Thus, the number 129 in the middle compartment indi- cates that 129 of the 498 daily gains in weight (transmuted ac- cording to the alx)ve scheme) fell within the interval 0.325 to 0.375 lb., the number directly above, i. e., 0.35, being the mid- value of this class. The frequency distribution represented by this second row of numbers is graphically illustrated by the superim- posed diagram, which is known as a histogram. Along the base of this histogram, equal spaces are marked ofif representing the equal class divisions. On each space a rectangle is erected, the height of which is proportional to the frequency of the respective class; or, preferably, since the gains must be considered as being continuously distributed along the base line or scale and merely summated at ecjual, convenient, arbitrary intervals, the areas of the rectangles should be considered as representing frequency. Figs. 3 and 5 represent the frequency distributions of the daily gains in weight of 241 steers and 461 pigs, respectively, and have been con- structed from transmuted values in the same way as Fig. i. The Normal freqiieuey Curve. — It will be seen from all three distributions that the frequencies start at zero, rise rather regu- larly to a maximum, and decrease regularly to zero again, the rates of increase and of decrease being appreciably similar. This is well shown in Figs. 2, 4, and 6. In these figures a curve of a definite character, represented by a definite equation, and known as the )ioriiial frequeticy curie, has been fitted to the three distribu- tions. The base lines of these figures have been divided into the same equal divisions representing the same classes as the figures directly above. The closeness of the fit is indicated graphically by the circled points placed at distances above the centers of the class intervals proportionate to their actual frequencies as given in the figures immediately above, and numerically by the frequency values given in the second row of figures below. These frequency values give areas beneath the curve between ordinates erected at the class limits. Thus, the value 122.7 ^^i the second row below the center of Fig. 2 gives the area bounded by the curve, the base line, and the ordinates erected at 0.325 and 0.375 ^^^- "'"• the hori- zontal scale, and corresponds exactly to the value 129 given in Fig. I. The closeness of fit of these three curves is apparently as satisfactorv as could be desired. 470 Bulletin No. 165 [July, Perhaps the most important fact disclosed by these frequency distributions is that variations in rate of gain in weight due to un- controlled experimental conditions, while exhibiting no regularity and rfo conformance to law as regards frequency of occurrence in the small experiment, actually do exhibit a regularity in the long run and actually do conform to a law that may be considered as being approximately represented by the mathematical definition of the frequency curve satisfactorily fitting the distribution, i. e., the normal law of frequency in the cases under discussion. This tendency of the casual variations in gain in weight observed within a lot of similarly treated animals to exhibit frequencies of occur- rence in the long run in conformity to a mathematically defined law is at the basis of all attempts to predict the results of future repe- tition of feeding experiments by finding the probability that an average lot gain or the difference between two average lot gains will lie between any assigned limits. The law defining the frequency of occurrence of casual variations is simply a mathematical expres- sion by which the probability of the occurrence of a given gain in weight is obtained by finding the extent of its deviation from the average gain. The Average as a Type. — Returning to the conception of the average gain in weight as a type which tends to be set up by the definite experimental conditions deliberately imposed, and which is only incompletely realized by reason of the numerous casual factors which are beyond control, it seems that in the frequency distributions such a type would be the position on the horizontal scale of the ordinate passing thru the summit of the frequency curve. This is the value of greatest frequency, the value more often realized than any other under conditions of like control. In Figs. 2, 4, and 6, ordinates are erected at points on the base lines corresponding to the arithmetic means of the gains in weight, and it will be seen that the means of the distributions may be regarded as actually being the points of greatest frequency, or at least very good approximations to such points. From a study of these distributions, it may be considered that a typical gain in weight exists within the lot, and that the aritli- metic mean of the individual gains is as good an approximation to this t^-pe as can be readily obtained. In defining this typical gain to which the arithmetic mean approximates, we may say that it is the gain that would be realized by each animal in the lot if no such thing as individuality existed and if all experimental con- ditions were under complete control and were kept constant for all animals. It may be said in passing, however, that the conception of the arithmetic mean as an approximation to a type does not apply to /p/j] Uncertainty in Interpretation of Feeding Experiments 471 all distributions. Thus, Fig-. 7 gives the frequency curve of the coefficients of digestibiHty of the protein of a mixed normal diet from 1 1 53 observations on 23 men.^" In this case the mean is dis- tinctly situated to the left of the maximum ordinate, due to the peculiar asymmetry of the curve. This condition may be consid- ered as existing in all cases of distribution of percentages where the limiting percent is 100 and the typical or modal percent is very near this limit. Whatever considerable variation occurs in the per- centage, therefore, must naturally draw out the distribution to a greater extent below the type, or, as it is technically caWtd, the mode, than above it. In Fig. 8 is shown another case of asymmetry. This figure gives the frequency curve of the daily excretion of indican in the urine, the data for which were obtained by Folin from 814 observations on 1 1 men during the course of his experiment to de- termine the physiological effect of saccharin.^ Here the lower physical limit of the distribution is, of course, zero, and since the typical value, or the mode, occurs near this limit, and since the variability is rather extreme, the distribution is drawn out above the mode. The Average as a Descriptive J'aliie. — A second conception of the arithmetic mean of a series of gains in weight made by a lot of uniformly treated animals, is merely that of a discriptive value, a representative gain used in place of the whole series of gains, the best representative perhaps of the series. Edgeworth describes it as: "that quantity which, if we must in practice put one quantity for many, minimizes the error unavoidably attending such prac- tice."'^ It must be admitted, however, that if the best that can be said of a mean is that it is merely a descriptive value, it lacks much that is desirable. It has no physical meaning such as is pos- sessed by an average that coincides with the mode. It can be defined only by reciting its method of calculation, and not by de- scribing any characteristics that it necessarily possesses. It must be regarded simply as the result of a mathematical calculation leading to a value occupying an intermediate position in the series, whose principal claim to consideration is that it is easily obtained and is almost universally used, rightly or otherwise. Furthermore, the calculation of the arithmetic mean, leading as it does to a value such that the sum of the differences between it and all values below it is equal to the sum of the differences between it and all values above it, is most significant only when the value desired is the mid- value of a symmetrical distribution, and therefore where asym- metry distinctly exists, the arithmetic mean cannot but suffer a loss "These data were obtained by the Division of Animal Nutrition of the De- partment of Animal Husbandry of this station. "U. S. Department of Agjriculture, Report No. 94. 1911. 'Edgeworth, Trans. Cambridge Phil. Soc, vol. 14. 472 Bulletin No. 1G5 [^"0- of significance. If, for instance, a chemical method of analysis were such that errors in defect of the true value were distinctly and decidedly more frequent and more important than errors in excess, it is evident that the process of taking an arithmetic mean of a number of results obtained by such a method would necessarily be looked upon as leading more often than not to a result less than the desired value. V.VRIATION AND ITS ^^IEASUREMENT In calculating the average gain exhibited by a lot of similarly treated animals, a more or less satisfactory measure is obtained of the influence of the deliberately imposed conditions upon the rate of growth. The incidental and uncontrolled experimental conditions, constituting all individual and environmental factors that have not been kept constant thruout the lot, find direct and complete expression in the variation, or dispersion, of the indi- vidual gains. Hence a measure of the variation of the gains with- in the lot is a measure of the influence of the uncontrolled factors in the experiment, which always render more or less ambiguous the conclusions ultimately deduced. Hence, also, such a measure is another value descriptive of a series of gains in weight ob- tained under similar conditions. In fact, so far as rate of growth is concerned, the average lot gain, and a good measure of the variation of gains within the lot, sufficiently describe for all ordi- nary comparative purposes the response of the animals in the lot to the experimental conditions. In obtaining a measure of the variation or the dispersion with- in the lot, it is obviously necessary to have at hand the gains of the individual animals. The frequent practice, in weighing up lots, of obtaining only the total weight per lot, so that only the total gain per lot for the experiment is finally available, renders all study of dispersion within the lots impossible; for, while the arithmetic mean of the individual gains in weight is obtainable directly from the total gain, any measure of dispersion must take into considera- tion the individual gains and any adequate measure of dispersion must take into consideration all of the individual gains. The Range of Observations. — The simplest measure of disper- sion and the one most commonly used is the range of observations actually obtained, such range being the difference between the minimum and the maximum values. However, it has very little to commend it, in spite of its rather general use, aside from the ease of its calculation. Obviously, one of the properties of a good measure of dispersion is that it have as high a degree of staliility as possible as we pass from one lot of animals to other and other similarly treated lots. Consider, for instance, a large num- ber of lots of steers that have been similarlv treated for the same jp/j] Uncertainty in Interpretation of Feeding Experiments 473 period of time, each lot, say, containing ten steers. It seems evi- dent that that measure of the dispersion of the individual gains in weight is best which is the most constant from lot to lot, since the same uncontrolled factors to which are due the variation within the lots have influenced each and every lot. But the range from the minimum to the maximum gain in a lot is directly affected by the extreme and unusual gains which may have been obtained, the very gains whose influence should be minimized because of their in- frequent occurrence and non-typical character. Furthennore, sup- posing these lots of steers that are under consideration are not of the same size, it is evident that the range of dispersion within the lots will in general increase with the size of the lot, since the steers ex- hibiting extremely high or extremely low gains will be found more frequently in the larger than in the smaller lots. Thus, the range between the highest and the lowest gains in a lot is of little value for comparative purposes, since, like the total gain, it depends in part upon the size of the lot; but, unlike the total gain, the in- fluence of the size of the lot cannot be eliminated by a simple division by the number in the lot, or in fact, by any other reason- ably simple mathematical process The Standard Dcination. — Obviously, a good measure of dis- persion must take into consideration each and every individual gain obtained; otherwise it really is not a characteristic of the whole series of gains and is unduly influenced by the extreme gains. Perhaps the first method that occurs to one of involving all gains in a measure of their dispersion is to take the average deviation of each of the gains from their mean, paying no regard, of course, to the position of the gain — whether above or below the mean. In fact, this is an excellent measure of dispersion that is sometimes used and is known as the az'erage dci'iation. The measure of dispersion in most common use, however, is obtained, by squaring all deviations of individual gains from the average, adding, dividing by the number of gains, and extracting the square root of the quotient. This is known as the standard dcination, or the root-nican-sqnare deviation from the mean. While the standard deviation is much more difticult of calculation, it possesses several advantages over the average deviation,^ and is in more general use. The Significance of an Average and its Probable Error The possession of an adequate measure of variation at once leads to the problem of determining to what extent variation with- 'For a very good discussion of the average deviation and the standard devi- ation, see G. U. Yule : "An Introduction to the Theory of Statistics," chap, viii. London : Chas. Griffin & Co., Ltd. 1911. 474 Bulletin No. 165 [July, m the lot vitiates conclusions based upon averag-e lot gains. The averag-e lot gain and the standard deviation of the individual gains sufficiently describe the lot for all ordinary comparative purposes, and the question now at issue is how these two descriptive terms can be used to render any subsequent comparison the most efficacious. As a matter of fact, the average gain or the total gain of a similarly treated lot of animals is a very deceptive quantity unless its exact significance is quantitatively defined by some additional term. An average gain should be thought of, not so much as an isolated point in the scale of measurement, but rather as the mid-value of an interval such that there is a definite probability that upon repe- tition of the experiment the average gain so obtained will fall within it. Such an interval is defined by the probable error of the average; and the probability that repetition of the experiment will yield an average within the limits of this probable error is exactly one-half. Raymond Pearl, of the Maine Experiment Station, who is applying biometric methods to problems of agricultural science, insists that "an experiment which takes no account of the 'prob- able error' of the results reached is inadequate and as likely as not to lead to incorrect conclusions."* Similarly, \\'ood and Stratton emphasize strongly the advis- ability and, in fact, the necessity of allowing for errors of sampling incurred in the selection of animals for experiment. The follow- ing quotation is especially significant : "With the great growth of interest among the farming community and the increasing ten- dency of the farmer to take note of the work of the experimental- ist and to act upon it, it is becoming increasingly important that due caution should be exercised by experimenters in interpreting their results before laying them before the agricultural public."'* Since an attempt to allow for experimental error in the inter- pretation of aA-erage lot gains is not effective unless the individual gains have been obtained, it is obviously important, in conducting a feeding trial, to ascertain individual behavior — the reaction of each animal to the experimental conditions imposed. Important as this condition is, it is too frequently disregarded in experiment station work. The collection and publication of individual data is too often thought to have little or no bearing on the problem of the experiment and conse(|uently to be a waste of energy and space; and yet by the neglect of this one condition, the investigator throws away the only opportunity of adequately analyzing his data. The Standard Deznation of an Average. — The element of un- certainty in the interpretation of an average gain in weight for a "Scientia, vol. 10 (1911), p. 106. "Journ. Agr. Sci., vol. 3, pp. 417-440. 1908-10. jp/j] Uncertainty in Interpretation of Feeding Experiments 475 lot of animals is due to the fact that successive lots treated simi- larly for a given period will necessarily give different average gains in weight. An arithmetic mean of a series of gains in weight must be considered as possessing a variability, just as is the case with the individual gains, due to uncontrolled experi- mental conditions ; and, since these uncontrolled experimental con- ditions find direct expression in the variability of gains within the lot, it follows that the variability of an average gain bears a defin- ite relation to the variability of individual gains. Obviously, the variability of an average gain decreases as the size of the lot in- creases, the main reason for increasing the size of a lot being, in fact, to render the average gain more significant. It may be shown, however, that the variability of an average gain does not decrease directly as the number of animals in the lot increases, but only as the square root of this nimiber increases.^ In other words, tJie presumptive standard deviation of an average gain of a lot is equal to the standard deviation of the gains zi'ithin the lot divided by the square root of their number. The Frequency Distribution of an Average. — It may further be shown that the variation to which a mean gain in weight is sub- jected as successive samples of animals are taken and treated experi- mentally is such that the distribution of means tends to assume the normal form, definable by the normal frequency curved such as that shown in Fig. 6 of the chart. In fact, whether the original values from which a mean is derived are so distributed or not, it may be shown that the distribution of means tends strongly to as- sume the normal fomi. Xow, in conceiving of the frequency dis- tribution which would be exhibited by a particular average gain obtained experimentally if tb.e experiment should be repeated a large number of times, obviously the best value to assume for the maximum point in the distribution, the point of greatest fre- quency, is the actual average gain obtained, since the one experi- ment actually performed has indicated that this is the most prob- able value that would be obtained upon repetition. The Probable Error of an Average. — Let the normal curve in •Fig. 6 of the chart represent the presumptive distribution of the average gain of a given lot of animals. The mid-ordinate of the curve we will assume to be located at this mean value. Now, in defining the significance of such a mean value, the following pro- cedure is the customary and perhaps the most natural one to pur- sue. Divide the area under the curve into two equal parts, one part symmetrically including the maximum ordinate of the curve. "Yule : "Theon- of Statistics." p. 340. •"See Henderson : "Frequency Curves and ^foments." Trans. Actuarial Soc. of Amer., vol. S, pp. 30-42. 476 Bulletin No. 165 [July, This has been clone in the figure, and that half of the area situated at the center of the distribution is indicated by cross-hatching. Now, since, as explained above, areas under a frequency curve represent frequencies, it may be said that upon continued repetition of the experiment, as many average lot gains will fall within the shaded area as without. Expressed in other terms, the odds in favor of obtaining a second average gain within the shaded area, or without, for that matter, are i to i. The distance on the hori- zontal scale from the center of the distribution to the ordinate on either side defining the shaded area is known as the prob- able error of the mean, so that the probable error may be said to define an ititerral, syiniuetrically including the average, such that the. odds are exactly even that a second average resulting upon repetition of the experiment will fall itnfhin it. One of the prop- erties of the normal frequency cun^e is that the probable error of the mean is obtainable directly from the standard deviation of the mean by simply multiplying by the factor 0.6745,^ from which it follows that the probable error of an az'erage lot gain in weight is equal to the standard deviation of the average multiplied by 0.6745, or is equal to the standard deviation of the indiz'idnal gains tmthin the lot divided by the square root of their ntunber and midtiplied by the factor 0.6745}^ A very good statement of the relation between the three statis- tical constants thus far discussed is given by H. L. Rietz in his Appendix to Eugene Davenport's "Principles of Breeding." Rietz says: "In describing a frequency distribution, the average gives absolutely no idea as to whether deviations are large or small, — nothing in regard to the spread of the distribution. It is the ob- ject of the 'standard deviation' to be descriptive of this variabilit}^, and it is the object of the so-called 'probable error' to indicate what confidence is to be placed in statistical results." The descrip- tion of a series of gains made by a lot of similarly treated ex- perimental animals should be thought of as a more or less com- plete and satisfactory description of the frequency distribution of gains of which the particular series experimentally obtained is a random sample. The Limits of Practical Certainty. — The ordinates situated at distances from the center of the curve of two and three times the probable error are also indicated in Fig. 6. The first pair of or- dinates include nine-elevenths of the area of the curve, or the ratio of the area within the ordinates to the area without is 4.5 to i ; "Yule : "Theory of Statistics," pp. 305-307. "•The probable error of the mean may be expressed mathematically by the formula Em =0.6745 ;=■ where a is the standard deviation of the original observations and n is their number. lyis] Unckktaixtv in Interpretatiun of Feeding Experiments 477 from whicli it follows that the odds of obtaining a second average gain within a distance of twice the probable error from the aver- age actually obtained are 4.5 to i. Similarly, for a distance of three times the probable error, the odds are about 21 to i, for four times the probable error, 142 to i, for five times the probable er- ror, 13 10 to I, etc.^ Since there are no definite limits to a dis- tribution of this kind, the occurrence of average gains upon repe- tition of the experiment extremely removed from the average gain actually obtained, which is represented by the mid-ordinate of the curs'e, cannot be said to be impossible, but only extremely improbable. It becomes necessary, therefore, in assigning the sig- nificance of an average lot gain, to decide upon some value which, when added to and subtracted from the average gain, defines an interval such that the average gain obtained upon repeating the experiment is practically certain to fall within it. Wood and Strat- ton have recommended that for data obtained from agricultural experiments that pair of ordinates situated equidistant from the mid-ordinate of the frequency curve and removed from it to such a distance that the area between them and the curve constitutes 30/31 of the total area under the curve, are good limiting values for use. This merely amounts to assuming that when the odds are 30 or more to i that an event will happen, we are practically certain that it will happen. For a normal distribution, which, as has been seen, an average lot gain tends to assume, a value 3.17 times the probable error, or, roughly, 3 times the probable error, constitutes the limiting value recommended by \\'ood and Stratton. The requirement of odds of at least 30 to i that a feeding ex- periment upon repetition will duplicate the results actuallv ob- tained, before definite conclusions be drawn from it and definite recommendations be made to the farmer, seems reasonable and, judging from the current practice of the investigators in various fields employing these methods, is not by any means severe. Thus, Davenport and Rietz, in Bulletin 119 of this station, say: "It will be noticed that by the time we have made an allowance of three or four times the probable error we have reached a chance which amounts to practical certainty and even 21 to i involves far less chance than is inz'oh'cd in most business transactions/' The merit of such methods as these for the interpretation of feeding trials consists largely of the fact Ci) that thev are per- fectly systematic, (2) that the argument leading from the origi- nal individual data to the resulting conclusions is unbroken and capable of being expressed definitely, and (3) that after it is de- °C. B. Davenport: "Statistical Aiethods." p. 14. New York. John Wiley & Sons, 2nd rev. ed. 1904. 478 Bulletin No. 165 [July, cided that the methods are applicable, the personal judginent of the investigator, which is so liable to introduce bias into the interpre- tation, is practically eliminated. Illustrations of the Use of the Probable Error In illustrating the use of the probable error we will first con- sider an experiment published in Bulletin 71 of the South Dakota Station, the object of which w^as to compare the value of speltz and barley as a single grain ration for fattening sheep. The two lots consisted of 12 animals each. Lot I, fed speltz, made an average gain during the 105 days of the experiment of 25.0 lbs. per sheep. The standard deviation, of the gains in this lot was 9.44 lbs. From these figures, the best estimate we can make of the standard deviation that would be exhibited by the average gain if the experiment were repeated a large number of times is 9.44 lbs, divided by 1/T2 (there being 12 sheep in the lot), which is equal to 2.73 lbs. Since the distribution of such a series of average gains would be of the normal type, the probable error of the average gain obtained in this experiment is equal to its stand- ard deviation, 2.73 lbs., multiplied by the factor 0.6745, the re- quired product being 1.8 lbs. The average g'ain with its probable error is ordinarily written 25.0 ± 1.8 lbs., and the whole expres- sion means that the odds are exactly even that if the experiment were repeated with 12 other sheep fed a grain ration of speltz and treated in all other ways as far as possible the same iis were the sheep in this experiment, the mean gain for the lot wovild fall with- in the interval 25.0-1.8 lbs.^23.2 lbs., and 25.0+1.8 lbs.=26.8 lbs. Similarly the odds are 30 to i that this second average gain would fall within the interval 25.01^(3.17X1.8) lbs., that is, some- where between 19.3 lbs and 30.7 lbs. Thus, while the average gain actually obtained was 25.0 lbs., and while this is the most probable average gain that would be obtained upon repetition of the experi- ment, we can say witli reasonable certainty only that a second aver- age gain would fall somewhere between 19.3 and 30.7 lbs. Thus, the element of uncertainty resulting from the meaningless fluctuations in the gains of the individual sheep due to the individuality of the animals and other uncontrolled experimental conditions, has been fairly definitely and reasonably defined for this lot of animals. Lot II, fed a grain ration of barley, yielded an average gain of 37.9 lbs., the standard deviation of the individual gains being 8.23 lbs. Proceeding as above, the probable error of this average gain will be found to be 1.6 lbs., so that we are practically certain that a second average gain which would result from repeating the ex- periment on other sheep, would fall within the interval 37.9^1 iQjj] Uncertainty in Interpretation of Feeding Experiments 479 (3.17X1.6) lbs., namely, between the limits 32.8 lbs. and 43.0 lbs. Therefore, since we are practically certain that any random sample of 12 sheep selected as were the sheep of this ex[x;riment and treated as was Lot I, wonld exhibit an average gain in 105 days between 19.3 and 30.7 lbs., and that any similarly selected sample of 12 sheep treated as was Lot II wonld show an average gain in 105 days between 32.8 lbs. and 43.0 lbs., it is obvious that we may feel sure that the one deliberate difference in treatment between Lots I and II, i. e., the difference in grain ration, does influence the gain in weight of sheep, barley tending to produce a better gain than speltz under the feeding conditions of this experi- ment. A more systematic way of settling the question, however, is to take the difference in average gain between the two lots, i.e., 12.9 lbs., and find its probable error. It may be shown that the presumptive variability or standard deviation that would be exhibited by a difference between, two averages if the experiment were repeated over and over again, is equal to the square root of the sum of the squares of the standard de\iations of both averages,^ and consequently the probalile error of a difference bears a like re- lation to the probable errors of the two averages. According to this formula, the probable error of the difference under considera- tion is 2.4 lbs., so that we may feel certain that upon repetition, the excess of gain of the barley lot over that of the speltz lot would be within the limits 12.91^(3.17X2.4) lbs., that is, between 5.3 and 20.5 lbs. The average difference, 12.9 lbs., is 5.4 times its probable error, and the odds that the excess average gain of Lot II over that of Lot I would fall between o and 25.8 lbs. are over 70CXD to I. In Bulletin 64 of tlie Pennsylvania Station is reported an ex- periment on steers, one of the purposes of which was to compare the gains made by steers fed during the winter in a barn with those made by steers fed in an open shed adjoining an open yard. The lots contained 12 steers each and were treated alike except as re- gards shelter. Lot I, fed in a barn, showed an average gain in 126 days of 267.71^8.8 lbs., and Lot II, fed in an open shed, a gain of 247.71^7.4 lbs. The difference in gain between the two lots was 19.0=^11.5 lbs. Since this difference is less than twice its probable error, it may well have resulted from the casual factors producing variation within the lot. In the i6th Annual Report of the Wisconsin Station, the re- sults of an experiment to determine the comparative value of rape and clover for growing young pigs is reported. Each lot of pigs contained 21 animals. During an experimental period of 56 days, *Yule: "Theory of Statistics," pp. 207-208. 480 Bulletin No. 105 [Jtdy, Lot I, which was pastured on rape, gained yi.o±.i.4 lbs., and Lot II, pastured on clover, 68.3±:i.3 lbs., the difference in favor of Lot I being 2.7=1=1.9 lbs. The odds are only 2 to i that upon re- petition of the experiment the lot pastured on rape would exhibit a gain between o and 5.4 lbs. above that of the lot pastured on clover, and it may be shown by taking the ratio of the difference in gain to its standard deviation and using* tables of the normal probability integral,^ that the odds are only 5 to i that under the conditions of this experiment rape-pastured pigs would again ex- hibit a greater average gain than clover-pastured pigs. Thus, the data when analyzed by the method under discussion hardly warrant a definite conclusion, A Probability Method for Small Lots of Animals \Miile the calculation of probable errors and the use of tables of normal probability integrals is the best method available, and is undoubtedly a good method for precise definition of the element of uncertainty inherent in the interpretation of averages when the number of animals per lot is large, when the number is ten or less, a probability table compiled by "Student" and published in Biometrika^ for 1908 may better be used for this purpose. In the article in which the table occurs, "Student" considers the distribu- tion of means of small samples and finds certain irregularities Avhich gradually disappear as the size of the sample increases. These discrepancies between the theory of large samples and the theory of small samples are such that by the application of the or- dinary theor}- which has been described above, to small samples, i. e., samples of ten or less, the odds obtained that repetition will result in a certain way are greater than the data actually justify. The methods of analysis described in this article should commend themselves highly to the investigator who is compelled for practical reasons to employ small lots of animals. »C. B. Davenport : "Statistical Methods," p. 119. "Page 1. JQIS] Uncertainty in Interpretation of Feeding Experiments 481 PART II. A STATISTICAL STUDY OF VARIATION IN THE GAINS IN WEIGHT OE FARM ANIMALS UNDER LIKE CONDITIONS Introduction In the preceding section of this bulletin, it is shown that an important factor contributing to the element of uncertainty in the interpretation of feeding trials consists of individual differences in the reaction of experimental animals to environmental conditions and of the unavoidable differences in environmental conditions to which the different experimental animals are subjected. It is further shown that such a factor of uncertainty can be handled satisfactorily by the ordinary statistical methods, — standard devia- tions and probable errors, as well as average gains in weight, being calculated for the different lots of animals in a feeding experiment. With the advent of an adequate quantitative measure of varia- tion in the gains in weight of animals upon like rations and under similar experimental conditions, the possibility presents itself of solving many problems intimately concerned with the methods of conducting feeding experiments and with the improvement of such methods. Other problems possessing a more general significance are also brought within reach of definite solution by the use of statistical measures of variation. This section of the bulletin treats of the extent of variation in gain in weight within the lot and upon what this variation depends. Also, the cjuestion of the reduction of such variation receives attention. Finally, consideration is given to the possibility that other than casual sources of variation in gain in weight are con- cerned in the ambiguity attaching to experimental conclusions as ordinarily formulated. The material for the following investigation was gathered largely from experiment station work in this country, tho some valuable assistance was received from similar work in Canada and England. In thus utilizing experimental results collected by many different investigators at widely varying localities for the purpose of solving diverse problems in live-stock feeding, many difficulties , were encountered in adapting such a heterogeneous mass of data to the solution of a few related problems, the existence of which was in no case recognized when the experiments were planned and undertaken. The facts or suggestions finally elicited, however, are perhaps the more valuable because of the richness and hetero- geneity of the results upon which they are based. /. B. Lazves on Variation. — The existence of extreme varia- tion among the gains in weight obtained within similarly treated 482 ButLETiN Xo. 165 [July, lots of animals has ver}^ frequently been the occasion for comment in experiment station literature. One of the best discussions on this subject that we have been able to find is that of J. B. Lawes, occurring in the course of a report of investigations on the com- parative fattening qualities of sheep conducted at the Rothamsted Station, England, about sixty years ago. Speaking of the selection of the 40 Hampshire and 40 Sussex wethers under investigation, Lawes says : "It is perhaps seldom that animals have been drawn for purposes of ex- periment with more care than in the instances of which the foregoing tables [giving the weights and gains of the sheep] record the results, yet we have scarcely a sheep in either breed which does not give twice, thrice, or more times as great an increase in gross live weight at one period, as at another of equal length; whilst taking the entire period of the experiment, we have nearly double the increase with some animals as with others by their side, and having ostensibly the same description and qualities of food provided. "The variation in the apparent rate of gain of the same animal at different times, is largely; due to the difference in the amounts of the matters of the food retained within the animal at the different times of weighing, and to obviate error from this cause we have only to extend our experiments over a sufficient length of time, and to be careful, as far^ as possible, always to weigh the ani- mals at the same period of the day, and under similar circumstances as regards their hours of feeding. "With respect to the difference of result shown by different animals, hav- ing professedly the same allowance of food, much of it is doubtless due to distinct constitutional tendency to fatten or otherwise; yet in some cases it no doubt depends upon a real difference in the food consumed by individual ani- mals, for it is impossible to secure for each its due share of the several foods supplied; and wherever there are many animals kept and fed together, there are always some who exercise a kind of mastery over the rest, and if they do not eat more food altogether than is allotted to them, they will at least take more of the best of it than is their share, and thus reduce the fair allowance to all the rest. By this cause, indeed, it is not improbalde that the proper feeding and increase of some animals well adapted for it may be prevented ; though in so far as these differences are really due to the quantities of feed consumed by different individuals, it is obvious that the true relation of food to increase will be less misstated by the gross numerical results of feeding experiments, than would be the case were the irregularites entirely owing to varying consti- tutional capabilities of the different animals to grow or fatten upon the same food. "But whatever be the causes of these variations, the figures in the tables show that, notwithstanding the careful selection of the animals, we have among the Hampshire sheep a difference in their average weekly gain of from about 3M lbs. to little more than 2 lbs.; and among the forty Sussex sheep, of from little more than 2i< lbs. to less than 1'^ lbs. Indeed, the tenor of all published results on feeding seems to show that these fluctuations and variations are the rule and not the exception ; and the fact of them, therefore, should lead us to 'great caution in drawing nice conclusions from experiments made with but a small number of animals, and extending only over a short period of time."* The Coefficient of Variation. — As is shown in the first section of this bulletin, the standard deviation, or root-mean-s(|uare deviation from the arithmetic mean, is a good measure of variation for some purpose, e.g., for gauging the value of the arithmetic mean as an approximation to the typical gain in live weight under certain defi- 'Journ. Roy.'Agr. Soc. of England, vol. 12, pp. 419-420. 1851. w ■fp^J] Uncertainty in Interpretation of Feeding Experiments 483 nite experimental conditions, or, as some prefer to consider it, for gauging the value of the mean as a quantity descriptive of a given series of gains in weight obtained under similar conditions, or, again, for measuring the significance of a mean gain in weight. For extensive comparison, however, the standard deviation is inadequate, since, in the first place, it depends upon the units of weight employed, and, in the second place, it depends in some measure upon the mean value itself. Thus, a lot of 19 pigs gained an average of 35.74 lbs. in four weeks, and of 77.11 IIds. in eight weeks. The standard deviation of the 19 individual gains at the end of four weeks was 5.31 lbs., and at the end of eight weeks, 11.28 lbs. In view of the great disparity between the correspond- ing average gains, the question whether the 19 pigs exhibited gains more variable at the end of four weeks than at the end of eight weeks, cannot be settled in fairness by comparing simply the two standard deviations. For the fairest comparison it is customary to convert the standard deviations into percentages based upon their respective averages. For example, 5.31 constitutes 14.86 percent of 35.74, and 11.28 constitutes 14.63 percent of 77.11 ; from which it follows that the variability for the two periods figured in this manner was practically identical. The percentages thus obtained, i.e., 14.86 and 14.63, are known as coefficients of variation, or coefficients of variability. Again, consider a comparison as to variability of gain among lots of different species of animals. Consider, for instance, (i) a lot of 9 cockerels, (2) a lot of 16 sheep, (3) a lot of 21 pigs, and (4) a lot of 15 steers, concerning which the following statistical data have been collected : Lot Average daily gain Standard deviation Coefficient of variation 1 2 3 4 .589 oz. .3.50 lb. 1 . 22 lb. 2.53 lb. .119 oz. .0451 lb. .157 lb. .242 lb. 20.20 12. P9 12.86 9.56 In such cases as the above, the only feasible method of com- parison is to consider the coefficients of variation. Coefficients of Variation Ordinarily Obtained in Feeding Experiments Results of Wood and Stratton. — It is a matter of some interest to study the variation in gain in weight, or the experimental error, ordinarily existing within the lot for the different kinds of farm animals. The only published investigations of this nature that we 484 Bulletin No. 1G5 [Jidy, are aware of are those of Wood and Stratton and of Robinson and Hainan referred to at the beginning of this bulletin. As the result of nine experiments on the fattening of cattle performed at Cam- bridge and involving 90 animals, Wood and Stratton found an average coefficient of variation of 21.20. Five similar experiments performed in Scotland and involving 50 animals gave an average coefficient of 20.75, while two cattle-feeding experiments performed in this country, involving 40 animals, yielded an average coefficient of variation of 20.31. Finally, seven experiments performed at Norfolk on the fattening of sheep, involving 100 animals, gave an average coefficient of 21.21. These four coefficients, three ob- tained with cattle and one with sheep, exhibit a remarkable agree- ment and would seem to indicate that for these two kinds of animals the percentage variability as regards gain in weight for animals within the lot is substantially the same. Results of Rohmson and Hainan. — As the result of a statistical analysis of three feeding experiments, Robinson and Hainan con- clude that "the probable error of one animal in a pig-feeding ex- periment is in the region of 10 percent of the average live-weight increase."^ This is equivalent to asserting that the coefficient of variation of gains in weight in pig-feeding experiments is about 15, a value considerably lower than the coefficients of Wood and Stratton for sheep and cattle. Results Obtained from American B.rperiments. — Results which we have obtained from experiment station work performed in this country entirely are slightly different from those just quoted. From the results of sixteen experiments on the feeding of sheep,^ in- volving 803 animals divided into 80 lots of 5 to 16 animals each, we found the average coefficient of variation of the 80 coefficients calculated, to be 21.63, a- figi^ire comparing favorably with the aver- age coefficient of 21.20 obtained by Wood and Stratton for sheep. Eighteen experiments on steers,*^ involving 449 animals divided into 50 lots of 5 to 15 animals each, yielded an average coefficient of variation of 16.73. This is considerably lower than the three aver- ages for steers obtained by Wood and Stratton, i.e., 21.20, 20.75, and 20.31. From seventeen experiments on swine,"^ involving 507 pigs di- vided into 49 lots of 5 to 23 pigs each, an average coefficient of 17.12 was obtained. This coefficient agrees well with that obtained for steers, i. e., 16.73, but is somewhat higher than that found by Robinson and Hainan for swine. "Loc. cit. "See Appendix, pages 558 to 560. 'See Appendix, pages 563 to 564. "See Appendix, pages 567 to 569. ■^p-fi] UlS'CERTAINTY IN IxTERPRETATIOK OF FEEDING EXPERIMENTS 485 The coefficients here reported would appear to indicate that the variabihty of gains in weights for steers and for swine are substan- tially the same, whereas the variability for sheep is distinctly higher*. Results Obtained at IVobiini and Rothamstcd. — Experiments performed at the Woburn Experimental Farm and at the Rotham- sted Station^ tend to substantiate the conclusion that as a general rule sheep give more variable gains than steers. Eight experiments performed on sheep at the Woburn Experimental Farm, involving 375 animals divided into 25 lots of 10 to 24 animals each, gave an average coefficient of variation of 20.80. If live experiments per- formed at the Rothamsted Station, involving 316 sheep divided into 15 lots of 5 to 46 animals each,^ be included, an average coefficient of 20.40 results. Nine experiments on steers performed at \\ oburn, involving 22 lots of 4 steers each and 2 lots of 6 steers each, i. e., a total of 100 steers, gave an average coefficient of variation of 18.15, o^'^r 2 percent lower than the two coefficients for sheep given above. Discrepancies Among Coefficients. — Upon reference to the Ap- pendix, which gives in tabular form all of the data upon which the above discussion is based, it will be seen that the percentage variability of the individual lots varies in a remarkable manner. This is shown by the following frequency distributions of the co- efficients of variation of the various lots of animals, including both English and American experiments. Kind of Class intervals animal 0-5 1 5-10 10-15 23 16 17 15-20 36 13 24 20-25 27 8 13 25-30 12 3 30-35 9 1 35-40 3 1 2 40-45 45-50 Sheep Swine 5 7 12 1 1 Steers Extreme coefficients not included in the above table are : for sheep, 58.21 for a lot of 11 animals, 55.33 for a lot of 10 animals, and 76.9 for a lot of 5 animals; for steers, 51.90 for a lot of 4 animals. The three distributions tend to confirm the conclusion that sheep in general exhibit greater variability as regards fatten- ing qualities than do either steers or swine. It is worthy of remark that this extreme variability exhibited by coefficients calculated from data obtained from many separate lots of animals treated differently at different localities and at dif- ferent times, is to be expected, not only from the heterogeneity of 'See Appendix, pages 561-562 and 565-566. 486 Bulletin No. 165 [July, the data, but also in large part from the mere size of the coefficients obtained. Thus, according to Pearson, the standard deviation of a coefficient of variation C, may be represented by the formula l/^/i ^100/ y2 from which it follows that o; increases as C increases, ti being the number of observations from which C is calculated. Thus, suppose that a lot of 15 sheep exhibits a series of gains in live weight whose variability is measured by a coefficient of 20. Then if successive series of sheep taken 15 to a lot were treated in the same manner, the best estimate we could make of the standard deviation of the coefficients of variation obtained, using only the data of the first series, would be 20 1 + 2 (^) = 3.79. Taking the probable error of C as 0.6745 o; and multiplying by 3.17,'' we define an interval symmetrically including the coefficient 20 such that the odds are 30 to i that a second coefficient obtained from a second lot of 15 sheep would fall within it. Thus, we are practically certain only that a second lot of sheep would exhibit a coefficient falling within the limits 20±8.i, i.e., between 11.9 and 28.1. Meaning of Such Discrepancies. — It is because of the large probable errors attaching to coefficients of 15 to 20 that it is so difficult to demonstrate that a given ration or other system of treatment is capable of producing more (or less) uniform gains than a second ration or other treatment. It is no exaggeration to say that a single experiment with lots of the moderate size ordinari- ly employed can shed practically no light upon a question of this kind, no matter how extreme the difference in variation between lots, except in conjunction with other experiments of a like nature. The point under discussion is worthy of illustration. Consider the results of two experiments conducted by W. L. Carlyle at the Wisconsin Station to determine the relative value of rape and clover pasture for fattening pigs.^ The lots of pigs employed con- tained 19 animals each in the first experiment and 21 animals each in the second. In the first experiment, the coefficient of variation of the gains in weight of Lot I, allowed to run on rape pasture, was 15.50, while that of Lot II, turned out on clover pasture, was "See page 477. "iSth and lOth Annual Reports Wis. Sta. J9I3] UnCF.RT.\1XTY IX IXTERPRETATIOX OF FeEDIXG ExPERIMEXTS 487 28.03. One might conclude from this experiment that rape pasture tended to produce more uniform gains than clover pasture. In the second experiment, however, the lot on rape pasture gave a coeffi- cient of variation of 13.23, while the lot on clover pasture gave a coefficient of only 12.88. Number of Animals per Lot Required ix Feeding Experi- ments Statistical theory is capable of attacking directly a problem of considerable importance to the technic of feeding experiments, i. c, the number of animals that should be included in the lots of a feeding experiment. The calculations upon which Table i, giv- ing the results of a statistical study of this problem, is based are given in the Appendix.^ The number of animals required to dem- onstrate satisfactorily the significance of various percentage dif- ferences in average gain in weight between two lots of animals, for sheep and for pigs and steers, is given in this table, the sup- position being, as the evidence seems to indicate, that in general, in experiments on sheep more animals are required per lot than in experiments on swine and steers. The few data that we have collected concerning the variability of the gains in weight of poul- try are quite comparable with those for swine and steers, indicating that the same number of animals per lot are required for the former as for the latter. It will be seen from Table i that only a moderate number of animals are required per lot except for differences of less than 12.5 Table 1. — Number of Aximals per Lot Required to Demonstrate the Signif- ICAXCE of Various Percextage Differexces Between Average Lot Gains For experiments on sheep For experiments on steers and swine Percentage difference between average lot gains Number of animals per lot required Percentage difference betv.een average lot gains Number of animals per lot required 50 40 30 20 17.5 15 12.5 10 • 7.5 5 2.5 2 2 4 8 10 14 20 31 54 121 482 50 • 40 30 20 17.5 15 12.5 10 7.5 5 1 2 3 5 7 9 13 20 36 80 317 'See pages 571 to 572. 488 Bulletin Xo. iGj [Ji'ly, to 15 percent between lots. For differences of less than 12.5 to 15 percent the number of animals required increases at a very rapid rate. Advantages of Large Lots of Animals. — In order to appreciate the significance of Table 1, it is necessary to form some idea of the percentage differences ordinarily obtained between lots of animals treated differently. In the case of rations markedly different in nutritive value, such as corn meal alone and corn meal sui)plemented by meat meal, shorts, middlings, tankage, etc., in swine experi- ments, differences between average lot gains may run as high as 95 to 100 percent. Experiments comparing the relative efficiency of alfalfa, timothy, and clover hay, or of some of the more com- mon grains, or feeding on pasture and in dry lot, in the pro- duction of gains in weight, may yield differences of 15 to 50 percent between lots. However, such cases as those just cited are exceptional. In the common run of feeding trials, the purpose is to determine the relative efficiency of two rations of approximately e(iual value, so that differences of more than 10 to 15 percent be- tween lots are not to be expected. Consequently, according to the best information available, the lots of animals used should contain at least 10 to 14 animals, if definite information is to be derived from the experiment. In fact, for differences as low as 10 percent between lots, 25 to 30 animals are required.^ Such a large number of animals is rarely used and is perhaps prohibitive for most experiments. However, when working with animals whose feeding capacities and other individual characteristics are so variable, and when, in general, experimental conditions are under such loose control that the standard deviation of gains w ithin the lot averages 17 to 21 percent of the average lot gain, the point to insist upon is that the results of single experiments with four or five animals are in general ])ractically worthless except in conjunction with other experiments performed under the same conditions. This is the conclusion to which Wood and Stratton have come, and it seems to be inevitable, at least until some method of lowering this extreme variability is discovered. In the course of the elaborate exj^eriments perfonned at the Rothamsted Station on the comparative fattening qualities of dif- ferent breeds of sheep, J. B. Lawes again and again calls attention to the variability in fattening qualities exhibited by sheep under 'These estimates of the number of animals pier lot required in order to ob- tain definite information concerning a problem in animal feedinp;. r^te to the feeding experiment as ordinarily run, in which no particular effort is made to reduce the experimental error. When such effort is made in an effective man- ner, perhaps according to the suggestions hereinafter outlined, the above esti- mates may be reduced to a greater or less extent. ^9^3] L'nlektaintv in Interpretation ok Feeding Kxpekimems 489 supposedly like conditions and selected with the utmost care. Thus, in his investigation of the Cotswold breed, he says, in speaking of the table giving the gains in weight per four weeks and for the entire experiment of each of the 46 wethers: "This table brings prominently to our view the point to which we have so often called attention, namely, the great variation in the rate of gain of the same animal during different consecutive periods and of different animals of the same breed, however carefully selected, and having ostensibh- the same description and qualities of food. This point we feel it is important to insist upon so often, as showing the uselessness of comparative experiments on feed- ing, unless both conducted with a large number of animals, and extended over a considerable period of time, so as to eliminate, as far as possible, the effects of the various sources of irregularity which we have before pointed out."* The same warning is given in the investigation of Leicester and crossbred lambs. We wish to emphasize this attitude of Lawes as being assumed over half a century ago by a man of undoubted authority in such matters, as the result of an extensive experience in the fattening of sheep and of other farm animals. It is evi- dently an attitude necessarily assumed by the careful observer in practical animal husbandry, as well as by the statistical investigator after analyzing by methods at present peculiarly his own the wealth of data which experiment stations everywhere have rendered ac- cessible to him. The necessity of employing large lots of animals in demon- strating the relative efficiency of two treatments of approximately equal value, for instance two treatments capable of producing a lo-percent difTerence in gain in live weight between two lots of animals, is capable of illustration in another and perhaps more ef- fective way. Assuming an equal percentage variability of gains in the two lots, each of which contains 10 animals, this percentage variability must be no higher than 12.06 in order to set up odds of at least 30 to i, that is, in order to adequately prove any differ- ence whatever in efficiency between two experimental treatments capable of effecting a lo-percent difference in gain. Considering the American experiments only, of the 80 lots of sheep whose co- efficients of variation were determined, only 11 exhibited a varia- bility as low as this; of the 49 lots of pigs only 8 possessed a coefficient of 12.06 or less; of the 50 lots of steers only 9 gave coefficients as low as or lower than 12.06. With 14 animals to the lot, a coefficient of variation for each lot at least as low as 14.23 is necessary-. Fifteen of the 80 lots of sheep, 19 of the 49 lots of pigs, and 15 of the 50 lots of steers possessed coefficients of variation as low as or lower than 14.23. With 16 animals to the lot, a coefficient of variation for each lot of at most 15.26 is re- quired. Sixteen of the 80 lots of sheep, 24 of the 49 lots of pigs, and 19 of the 50 lots of steers possessed coefficients of variation as low as or lower than this. •Journ. Roy. Agr. Soc. of England, vol. 13, p. 182. 1852. 490 Bulletin No. 165 [July, Size of Gains and Their Variabiuty From inspection of the data given in the Appendix, one receives the impression that in general, for the same feeding experiment, there is a tendency for the variabihty of gains witliin the lot to correlate itself with the average gain, low average gains being in general associated with high variabilities. The detailed data are so heterogeneous as to render any systematic study of this ques- tion impossible. However, confining ourselves to those experi- ments in which the lots consist of at least lo animals and the differences among average lot gains are large, we will consider only those results capable of affording the most decisive evidence either one way or the other. Evidence for Sheep. — Considering the feeding experiments with sheep^ first, Experiment i offers little evidence either one way or the other, the gains for most lots being quite similar. However, Lot I, with the lowest average gain (32.9 lbs.) exhibits the highest coefficient of variation (25.08) ; while Lot IH, wnVa the highest average gain (41.3 lbs.) possesses a coefiicient of variation of only 17.31. The standard deviations of these two lots stand in the same relation to each other. In Experiment 4, Lot I (10 sheep) shows an average gain of 31.3 lbs., a standard deviation of 4.80 lbs., and a coefficient of variation of 15.34; Lot H (10 sheep) shows an average gain of 23.4 lbs., a standard deviation of 7.79 lbs., and a coefficient of variation of 33.29. Thus, in this case the lot giving the lower average gain exhibits the higher absolute and percentage variability, the differences being very marked. In Experiment 5, Lots la, Ila, and Ilia were under experiment in the fall of 1908, while Lots lb, lib, and Illb were under experi- ment in the fall of 1909, the lots designated by the same Roman numeral receiving similar rations. It will be seen from page 558 of the Appendix, that much better gains were obtained in 1908 than in 1909, even after reduction to a daily basis; also, that the variability of gains is greater for the year giving the poorer gains (1909). Furthermore, on comparing Lot Illa with Lots la and Ila, Lot Ilia is seen to have the greatest average gain and the smallest coefficient of variation. Similarly, on comparing Lot Illb with Lots lb and lib. Lot Illb is seen to possess the greatest aver- age gain and the least absolute and percentage variability. In Experiment 6, it will be noted that Lots I and III, with the lowest average gains, exhibit the highest absolute and percentage variability. Lot I, Experiment 7, shows an average gain of 25.0 lbs. in 105 days, and a standard deviation of 9.44, i. e., 2>7-77 P^^" "See Appendix, pages 558 to 562. /p/j] UXCERTAIXTV l.\ INTERPRETATION OF FEEDING EXPERIMENTS 491 cent of the mean. Lot II exhibits an average daily gain of 37-9, a standard deviation of 8.23, and a coefTficient of variation of 21.70, In Experiment 8, Lots la and Ila before weaning exhibit much better average gains than after weaning'. The results in th.e latter case are recorded under Lots lb and lib. The percentage varia- bility before weaning is correspondingly less than that after wean- ing. We shall not attempt an analysis of Experiment 9 because the lots are so small, but from a cursory glance at the results for the four lots before and after weaning it will be seen that they agree admirably with the theory that the better gains are also in general the more uniform gains. As further support for this conclusion, we cite Experiments 13, 17, 18, 20, 23, and 24 of the Appendix. Bindcucc for Szvinc. — The experiments on swine^ do not af- ford very strong confirmation of the theory under consideration, it must be admitted. This is due in large part to the fact that in many of the swine experiments the lots made similar gains, and that in many experiments small lots of animals were employed, — conditions unfaA-orable to the solution of the problem at hand. In Experiment 62, tho the lots were small, the inverse correlation between average gain and variability for the six lots is very evi- dent. In Experiments 63 and 64, with 8 and 9 animals to the lot, the evidence is more or less contradictory. In Experiment 65, the data are very irregular, tho they fall in with the theory after a fashion. Thus, the average coefficient of variability for the three lots giving the three lowest average gains is 37.9; for the three lots giving the next lowest gains, 16.8; for the three lots giving the next lowest gains, 15.8; and for the lot giving the highest gain, 13.5. In Experiments 66 and 69 the evidence is contradictory, while in Experiments 67 and 68, it is favorable to the theory. In Experiment yy, the evidence is contradictory. Ei'idcnce for Steers. — For steers,^ conditions are about the same as for swine. We will not consider the Pennsylvania experiments (35> 36, 37, 38, 39, 41, and 42), since in all of them the two lots gave very similar average gains in weight. The most comprehen- sive single steer experiment the data for which are given in the Appendix, is Xo. 43, an experiment by H. W. Mumford performed at the ^Michigan Station. In this experiment the correlation be- tween average daily gain per lot and the coefficient of variation, while far from perfect, is quite perceptible. The two largest co- efficients obtained are those for Lots IX and X, exhibiting the "See Appendix, pages 567 to 569. "See Appendix, pages 563 to 566, 492 Bulletin No. 165 [July, two smallest average gains, while the smallest coefficient is that of Lot VII, exhibiting the largest average gain. Arranging the lots in groups of two in the order of increasing average lot gains, the average coefficients of variation per group run as follows: 24.15, 16.40, 15.83, 17.00, and 14.84, Evidence for Poultry. — Considering next the poultry experi- ments,^ aside from Experiment 78, in which the results are very irregular, probably because the lots were composed of different breeds, and Experiment 84, in which there was only one lot, all show a greater absolute and percentage variability for the lot ex- hibiting the lower average gain. The unanimity exhibited by these five experiments is quite remarkable. Siinimary of Bvidcnce. — The preponderance of evidence thus favors the conclusion that good gains are in general uniform gains, and that in any experiment involving two or more lots of animals there will be more or less close correlation between average lot gains and the corresponding coefficients of variation, such that large values of the former will in general be associated with small values of the latter. While the evidence that we ha.ve presented in support of this view is more convincing for sheep and poultry than for steers and swine, the distinction is more apparent than real. As explained above, the particular sheep and poultry experi- ments cited are more favorable to a solution of the problem than the steer and swine experiments. The conclusion of this section may be stated in other words, i. e., it appears that experimental conditions favorable to growth and fattening are favorable to uni- formity of individual gains. Re:duction of the Expivrime^ntal Error in Feeding Experiments The question whether the extreme variability of gains in weight ordinarily encountered in feeding experiments, constituting the ex- perimental error of such investigations, can be reduced without diminishing the significance of experiment station work, is a legiti- mate object for discussion and investigation. As the result of a single careful experiment on four steers. Wood and Stratton con- clude that there is no way of surmounting the difficulty caused by the extreme variability of gain in weight of farm animals and that "the requisite precision in feeding trials can only be obtained by increase of numbers, or if that is impossible, repetition of the ex- periment." We are of the opinion that such a sweeping conclusion as this is not based upon sound and sufficient evidence. The results of our studies considered in the following pages are not in harmony with such a conclusion. 'See Appendix, page 570. jp/j] Uncertainty in Interpretation of Feeding Experiments 492 In fact, three experiments on swiiije (76, jy, 78) conducted by one of the Canadian experiment stations seem to present evidence in direct contradiction to the conclusion of Wood and Stratton. These experiments involve i lot of 5 pigs, 5 lots of 6 pigs, and 4 lots of 10 pigs. The coefficients of variation of the gains produced are remarkably, and with one exception, uniformly low, averaging only 10.98. Only one of the 10 lots possesses a coefficient as high or higher than the average for tlie experiments on swine performed in this country, i. e., 17.12. The reports of these experiments are too meager to enable one to tell what feature or features of ex- perimental control are responsible for this low variability, but it seems that here, at least, a relatively high precision in feeding trials has been attained with only moderately large lots of animals. (a) Iiiiportaiicc of Reducing the Experimental Error The importance to the technic of feeding experiments of some method of increasing the uniformity of gains within the lot as the period of observation increases should not be underestimated. The difference between the fattening qualities of two lots of animals selected differently, or between two systems of treatment, is ex- pressed fairly well as "a percentage difference rather than a dif- ference of so many pounds or ounces. The statement that Ration A dift'ers in fattening qualities from Ration B to the extent of x pounds has no meaning whatever; the statement that Ration A differs from Ration B to the extent of x pounds in 3' days, or of X pounds per day, is perfectly definite and involves all necessary information ; but the statement that Ration A is x percent better as regards fattening qualities than Ration B is less cumbersome and more intelligible than the latter statement, while contain- ing all necessary infomiation. It is a perfectly legitimate as- sumption, until proof to the contrary is presented, that the percentage difference between two rations tends to remain con- stant thruout a feeding experiment. Now, it may be shown that the smaller the coefficient of variation of the gains in weight with- in a lot, other things being equal, the smaller the minimum per- centage difference between the average gain for the lot and the average gain for a second lot that can be definitely traced to the difference in treatment or the difference in make-up between the two lots.^ Hence the value of legitimately reducing the coefficient of variation of gains is obvious. We will illustrate the point with the data from one of the Rothamsted experiments given in Table 7, e. g., the data for the lot of 40 Sussex wethers. If this experiment had closed at the end of 4 weeks, the final coefficient of variation of the 40 total "See Appendix, pages 571 to 572. 494 Bulletin No. igo [July, gains in weight would have been 30.32, If another lot of 40 Sus- sex w^ethers had been under observation for the same period of time and had also exhibited a variation of 30.32 percent, on some other ration we will say, then the smallest difference in fattening qualities between the two rations that could be detected with rea- sonable certainty would be a difference of 12.5 percent; that is, a difference between the two average lot gains of 12.5 percent is the smallest difference that could set up odds of 30 to i that the difference in ration was actually concerned in the difference in gain. If this experiment had ended at the end of 8 weeks, this minimal difference between average lot gains would have been reduced to 9.5 percent; at the end of 12 weeks it would have been 7.2 per- cent; at the end of 16 weeks, 6.2 percent; and at the end of 20 weeks, 5.6 percent. We see, therefore, that during the course of this experiment, which was so conducted that the individual gains were becoming more and more uniform, the average lot gain be- came much more efficient as a comparative value and much more representative of the experimental conditions whose influence on the fattening of sheep it is supposed to measure. To illustrate further the great practical value of definite meth- ods of reducing experimental error, we will consider the statisti- cal data of four experiments with poultry performed by F. T. Shutt of the Central Experiment Farm, Canada. Experiments 79, 81, and 82 were performed in 1901-02, and Experiment 83 in 1904-05. As far as the meager reports of the experiments indicate, the feed in all lots was given to the fowls "in such quantity as was immediately consumed." In the first three experiments, no tendency for gains to become more uniform is evident. In fact, in Experiment 81 the contrary tendency may be seen. In Experiment 83, however, the gains in each lot regularly increase in uniformity to a very marked degree. The ration in this experiment was not very dift'erent from that of Experiments 79 and 81. In the latter, the ration consisted of ground oats, 4 parts, ground barley, 3 parts, and meat meal, i part, made into a mash with skim milk. In the former, the ration was ground oats, 3 parts, and ground barley, 2 parts, also mixed with skim milk. It would obviously have been to the advantage of Ex- periments 79, 81, and 82 if they had been conducted as was Experiment 83, tho just wherein Experiment 83 dift'ered essentially from the others, one cannot discover from the report. Perhaps the difference would have been evident only upon careful investi- gation, for instance of the quantities of feed consumed during each week of the experiment. The comment of Shutt upon the variability of the gains ob- ser\-ed in his several lots of fowls is of interest: "What we may term individualism is as strone amone fowls as in other classes I9i3] ^ ■ — < O -a H O u X ><
  • ■*-' O C co-c n! -3 rt o c f3 ;n C U U5 ■)-> "^ c o o '^ OJ « a; O > :S u^ ^ c (/I c rrt *w c M J^.S .— < •o - C^ C-- o re > -tJ (U *■*-. LO-^ r^ ni n '>J ^^ ra ^« o Oj lU t/] ■^ r-*. m j: o _, 13 "a] S3 C U <: 6C O M t~ o r~- !C ;c M C» O t^ tC t^ ■* o c^ ■T r^ -^ J- tT C-. <-< CO c r- C-^ -5- O O t- C5 in ;d 00 t~ >n O i- O N O C M t- iH O OS T- ro ■>*<■<*< «5 Tf en M -H C5 00 cc w C! O C- ■* rl C >-o tx; X t~ a c> IX) y-l t~ t- . c/1 wa t/5 e/3 cfi ^ ^ ^ j>; ^ ^ o o a; i; aj aj a^ a; aj ■; ^ ^ S ^ is "-I W CC M< >n tc = c n c c c Cl, c ^-N -a o u >-- >n O OC W 50 ■* Tt «0 t~- en !■} Tt ■>)■ O C O O «S 00 C-. C-. C. -^ ■* N T-l ^ t; !- '-^ 00 t- w ^ !- c »n 1-^ o W — C> M ■* ^ . c/1 c/3 tn (/3 t/; ^ ^ ^ ^ -:i ^ Q^ ^ O CJ O tL> u aj tu a; a^ u ^ ^ ^ ? js & ,-j CQ CO Tjl lO to c n c c c c •a ^- P. P w >, •w CLi c/l •o aj a; t. u
  • -~ c: ^t -n >.-; 05 CC ■* ;C T- C O CT. in Tf iC t- c c -t c o o •* c n- C X Tf 00 c m en C IT! «D ■<*• m tc m 00 C! o ■ t/] en t/) CD (/) ^ ^ ^ ^ ^ a: ij o a^ aj aj a.; is ^ -s ^ ^ s: !-< N en •* in tc c c c n c c >> « Ph « CU aj aj 1- ^ a u CO o 00 495 en r: t- t^ m (M >- I- tC CO C OC T-< C5 «c c. -^ t- en M- a> CJ c-* aj Qj a> a; 1^ ^ ^ ^ ^ ^ 1-^ CQ en -^ m c n n c c 496 Bulletin No. 165 [July, of live stock. Vitality, constitutional vigor, and ability to digest and assimilate food are not rneted out alike to all, and tho there is no apparent cause, lack of thrift is not uncommonly to be ob- served in some members of a hatch." This belief, which is strik- ingly confirmed by the statistical study we have made of some of Shutt's experiments, is undoubtedly at the basis of the general practice of the chemistry and poultry divisions of the Central Ex- perimental Farm, of presenting individual data in all feeding ex- periments on fowls and in all experiments to determine the effect of different methods of poultry management on egg production. It is to be regretted that this practice is so unusual among investi- gators of the problems of poultry management, since it has so much in its favor in rendering the results of experiments more intelligi- ble and less ambiguous. (b) Selection of Anhnals as Regards Age, Breed and Type, Sex, and Previous Treatment In securing the greatest possible uniformity of gains within the lot in feeding experiments, obviously the first care should be in the selection of the animals. Age. — That animals at different ages exhibit different fattening qualities, needs no demonstration, and the necessity of including only animals of approximately the same age in an experimental lot is pretty generally recognized. Breed and Type. — That different breeds of the same species of animals behave differently on the same rations is undisputed in some cases, while in all cases it is a possibility, if not a probability, in the absence of definite evidence to the contrary. These dif- ferences are to be expected more especially when the breeds differ in general type. Thus, in the case of steers, the dairy and beef types, in the case of swine, the lard and bacon types, and in the case of sheep, the mutton and wool types, may be supposed to differ most markedly in fattening qualities. In the case of sheep, the extensive breed tests conducted at the Iowa Station by \\'ilson and Curtiss^ and the extensive experiments of J. B. Lawes at Rothamsted (27, 28, 29) leave no doubt that breed differences as regards rate of growth do exist. In the case of steers, the evidence for the existence of breed difference is apparently not so convincing, or at least not so gen- erally recognized. Thus, W. A. Henry says : "So far as the data go, we have no evidence that beef-bred animals make more rapid growth than do others."^ H. P. Armsby is inclined to the same "Iowa Agr. Exp. Sta., Ruls. 33 and 35. "Feeds and Feeding, nth ed., 1011, p. 320. /p/j]- U.\ CERTAINTY IN INTERPRETATION OF FEEDING EXPERIMENTS 497 opinion.^ ^^'hile we have not made an extensive study of the lit- erature, some experiments that we have reviewed indicate in no uncertain fashion that different breeds, especially when of differ- ent types, may exhibit different fattening qualities under the same conditions. An extensive experiment by H. \\'. Mumford of this station^ presents indisputable evidence to this effect. The object of the investigation was a comparison of the six standard grades of feeding steers as regards their fattening qualities. Each lot con- sisted of i6 steers of the same grade. A very complete description of the lots is given in the original bulletin. However, we shall give only a brief resume, more especially of the characteristics of the lots as regards their breeding. Of Lot I, containing the fancy selected feeders, Mumford says: "They contained nearly loo percent of the blood of the improved beef breeds. The dams were high-grade Shorthorn cows and the sire a registered Hereford." Lot 2, containing choice feeders, were high-grade Shorthorns. In Lot 3, containing the good feeders, beef blood still predominated. Concerning Lot 4, the medium feeders, the author says : "It should be said that this lot did not contain a steer that failed to show evidence of improved beef blood, altho the predominating blood seemed to be native or unimproved, with occasionally a dash of the blood of some one of the dairy breeds." Lot 5, the common feeders, "showed but a ver}- small percentage of beef blood. Xative and unimproved blood predomi- nated." Lot 6, the inferior feeders, "showed no evidences of beef blood and every evidence of being scrubs." During a feeding period of 179 days, these lots exhibited the following average daily gains in weight : Lot i, 2.570 lbs. ; Lot 2, 2.543 lbs.; Lot 3, 2.341 lbs.; Lot 4, 2.128 lbs.; Lot 5, 2.207 lbs.; and Lot 6, 1.950 lbs. While complete individual data are not given, thus precluding a complete analysis of the significance of average lot differences, there can be no reasonable doubt, from a study of these averages, that the infusion of beef blood tended strongly to accelerate the rate of gain of the better grade steers. While the data of the above experiment indicate that breeds of different general types may dift'er in fattening qualities, some data presented by Curtiss before the Ames Graduate School during the summer session of 1910,*^ indicate that decided, tho slight, dif- ferences in rapidity of gains exist even among breeds of the general beef type. •U. S. Dept. Agr.. Bur. An. Ind.. Bui. lOS. pp. 29 and 44. 1908. "III. Agr. Exp. Sta., Bui. 90. 'See E. Harrison and J. A. S. Watson : "Correlations between Conforma- tion and the Production of Beef in Beef Cattle, etc." Thesis for the M.S. degree, Iowa State College, 1911. \9S Bulletin No. 165 [July, In the case of swine, the evidence is unmistakable that at least some breeds can be differentiated from each other as regards fattening qualities. Here again breed differences are the more marked when accompanied by differences in general type. Appar- ently these differences are not at all constant, but vary with the rations fed or with the conditions of feeding; that is, in some ex- periments one breed may show a marked superiority over another, while in another experiment, in which other rations are used or other conditions obtain, the relation found in the first case may be reversed. As an illustration of this point, we shall first cite an experiment by W. A. Henry of the Wisconsin Station on two lots of 12 pigs each.^ The total gains made by the individual pigs at the end of 12 weeks on a ration of corn and wheat middlings, and the breed and sex to which each pig belonged, are summarized in Table 3. Table 3. — Total Gains in Weight of Two Lots of Pigs, with Breed and Sex OF Individuals No. of Lot I Lot II pig Breed Sex Gain 92 Breed Sex barrow Gain 1 Grade Berkshire barrow Grade Berkshire 133 2 Grade Berkshire sow 77 Grade Berkshire sow 95 3 Poland-China barrow 29 Poland-China sow 55 4 Grade Berkshire sow 103 Grade Berkshire barrow 98 5 Berkshire barrow 60 Poland-China barrow 64 6 Grade Berkshire barrow 80 Grade Berkshire sow ' 113 7 Yorkshire sow 80 Poland-China barrow 30 8 Berk, razorback sow 71 Pol.-Chin. raz'b. sow 98 9 Pol.-Chin. raz'b. barrow 84 Berk, razorback sow 75 10 Berkshire sow 87 Yorkshire sow 81 11 Poland-China sow 71 Berk, razorback barrow 109 12 Poland-China barrow 59. Berkshire sow 87 It will be noticed, especially in the case of Lot II, that the Po- land-China pigs did very poorly. As Henry says : "The Poland- China hogs proved unsatisfactory feeders, showing losses at the weighing period on several occasions. Towards the last they be- came lame and their conditions may be characterized as 'broken down.' As they had received the same treatment at all times as the others, we cannot offer any explanation excepting that they were weaker animals generally." An Iowa experiment on the feeding of corn and supplementary feeds to pigs (65) affords data concerning the differential fatten- ing qualities of different breeds of pigs. Each lot consisted of 9 or 10 pigs and of representatives of 4 or 5 breeds. The gains of the "18th Annual Report of the Wisconsin Station. 1901. I9I3] UXCERTAIXTV IX IXTERPRETATIOX OF FeEDIXG ExPERIMEXTS 499 pigs of the various breeds as regards their position above or below the mean gain of their respective lots are given in Table 4. The data in this table are of interest, not so much by reason of what thev prove as regards the relative fattening qualities of dif- ferent breeds of swine, but by reason of what they suggest. In Table 4. — Data Coxcerxixg the Gaixs ix Weight of Tex" Lots of Pigs axd Their Relatiox to the Breed of the Pigs Average No. of gain for Lot ration lot lot Pigs giving gains above the respect- ive lot average Pigs giving gains belozc the respect- ive lot average I 103.4 Corn meal. timothy pasture 3 York-Durocs 1 Poland-China 2 Berkshires 1 Yorkshire 1 Poland-China 2 Berkshires II 123.5 Corn meal 2 pts., shorts 1 pt.. timothy pasture 2 York-Durocs 1 Poland-China 2 Berkshires 1 Yorkshire 1 York-Duroc 2 Poland-Chinas 1 Berkshire III 133.2 Corn meal 1 pt., shorts 1 pt., timothy pasture 1 York-Duroc 1 Berkshire 3 Poland-Chinas I Yorkshire 2 York-Durocs 2 Berkshires IV 140.9 Corn meal 3 pts., meat meal 1 pt.. timothy pasture 1 3 York-Durocs 2 Poland-Chinas 1 Yorkshire 3 Berkshires V 133.9 Corn meal 5 pts., tankage 1 pt.. timothy pasture 2 York-Durocs 2 Poland-Chinas 1 Yorkshire 1 York-Duroc 1 Poland-China 3 Berkshires VI 133.7 Corn meal, clover pasture 3 York-Durocs 1 Poland-China 2 Berkshires 1 Yorkshire 1 York-Duroc 1 Poland-China 1 Berkshire VII 90.9 Corn meal 2 pts., shorts 1 pt., in dry lot 1 York-Duroc 1 Poland-China 2 Berkshires 2 York-Durocs 1 Poland-China 2 Berkshires 1 Yorkshire VIII 100.2 Corn meal 1 pt., shorts 1 pt, in dry lot 1 Poland-China 1 Berkshire 3 York-Durocs 1 Poland-China 3 Berkshires 1 Yorkshire IX 121.8 Corn meal 5 pts., meat meal 1 pt., in dry lot 2 York-Durocs 2 Poland-Chinas 1 York-Duroc 3 Berkshires 1 Yorkshire 1 Tamworth X 102..') Corn meal ■> pts.. tankage 1 pt.. in dry lot 3 York-Durocs 3 Poland-Chinas 1 Tamworth 3 Berkshires 1 Yorkshire 500 Bulletin No. 165 [July, some cases the suggestion is accompanied by a considerable prob- ability, tho with such few and heterogeneous data that whatever interpretation is attempted must be couched in very moderate lan- guage. It will be noticed that each lot contained a Yorkshire pig. In the first six lots, the Yorkshire pigs exhibit gains above, and gen- erally considerably above, their respective lot averages. In' the last four lots, however, the Yorkshire pigs exhibit gains far below their respective lot averages. It will be noticed that the first six lots were turned out on pasture, while the last four lots were confined in dry lots. The evidence is very suggestive, therefore, that the advantage of pasture over dry lot is much more marked in the case of Yorkshire pigs, as representatives of the bacon type perhaps, than in the case of the other breeds experimented upon. It will also be noticed that the Berkshire pigs exhibit gains both above and below the average in Lots I, II, III, VI, VII, and VIII. In Lots IV, V, IX, and X, however, the Berkshires con- sistently show gains below the average, and, as the original data indicate, far below the average ; in short, the Berkshire gains in Lots IV, IX, and X are the lowest gains in the lots, and in Lot V, the three Berkshires exhibit the two lowest and the fourth lowest gains in the lot. These; four lots are the lots in wiiich the corn meal was supplemented by meat meal and tankage, two of the lots being turned out on pasture and two being confined in dry lots. The behavior of the Berkshire pigs seems to be specific and to distinguish these representatives of the Berkshire breed sharply from the representatives of the other breeds. An extensive breed test on swine, extending over three years, was. conducted at the Iowa station by Curtiss and Craig.'" The rations in the three experiments differed to a greater or less ex- tent. A summary of the results obtained after the pigs were weaned is given in Table 5. Table 5. — Breed Test at the Iowa Experiment Station I I ] First experiment : 92 days Breed Ay. daily Rain 10 Duroc-Jerseys 6 Yorkshires 7 Tamworths 10 Chester-Whites 7 Crossbreds r^ Poland-Chinas 10 Berkshires .90 .80 .77 .74 .73 .72 .RS Second experiment : 153 days Breed Av. daily gain 9 Yorkshires 9 Duroc-Jerseys 10 Berkshires 10 Chester-Whites 10 Tamworths 8 Poland-Chinas 16 10 03 01 00 1.00 Third experiment: 165 days Breed I' 5 Yorkshires 10 Berkshires 8 Tamworths 10 Poland-Chinas 10 Duroc-Jerseys 9 Chester-Whites dail ^- 1.16 •■| .91 "Iowa Agr. Exp. Sta., Bui. 48. 1900. ip-fj] Uncektaintv in Inteki'Retatiun ok Feeding Experiments 501 The rank of the different breeds as regards average daily gain IS quite different in the three experiments, possibly because of the different rations used. The experiments agree, however, in several particulars, e. y., in attributing to the Yorkshire breed a general superiority, and to the Chester-White and Poland-China breeds a general inferiority. It may be shown that with a variability in the Duroc-Jersey and Berkshire lots as high as 33 percent, the odds are 30 to i that the former breed possesses greater fattening .powers than the latter under the particular conditions that obtained in the first experiment. For other experimental data on breed tests with swine, the reader is referred to Bulletin 47 of the U. S. Dept. of Agr., Bu- reau of Animal Industry, by Rommel. As further evidence on the question under discussion, we wish to cite a few experiments in which each lot consists of a separate litter. Some statistical data on these experiments are given in Ta- ble 6. Table- G. — Vartatiti.ity of Gains for Lots of Pigs, Each Lot Consisting of a Single Litter Reference No. of pigs in litter Length of exper- ment in days Statistical data of total in weight gains Mean Standard deviation C oefficient of ariation "Pigs observed from birth. ""Pigs observed from weaning Wis. 7th Ann. Rpt.^' 7 119 64.33 7.09 11.02 8 119 86.74 13.73 15.83 7 119 70.91 9.43 13.30 7 119 65 . 06 6.69 10.29 Mich. Bui. 138" 8 119 104.60 9.77 9.34 9 119 83.55 4.11 4.92 Wis. Bui. 104" 10 56 53.3 6.25 11.73 10 56 24.0 4.58 19.08 5 .56 37.6 3.44 9.15 6 56 45.3 7.25 16.00 8 56 50.5 8.56 16.95 9 56 33.0 6.32 19.15 7 56 51.3 5.00 9.75 6 56 38.0 2.77 7.30 8 56 46.0 6.76 14.70 5 56 49.2 7.11 14.45 5 56 31.4 6.25 19.90 .1 56 57.6 6.67 11.58 Total . ■. 130 Average 13.00 time. The coefficients of variation of the gains made by these 18 lit- ters are in general comparati^•ely low. In only 3 litters is the coefficient of variation greater than the average coefficient for pigs 502 Bulletin No. 165 [July, as indicated by the American feeding experiments reviewed, i. e., 17.12. It is interesting to note, in view of what has been said al)ove as regards the relation between the size of gains and their variabihty, that the three Htters exhibiting the three highest co- efficients of variation of the 18 coefficients exhibit also the three lowest average lot gains obtained in the Wisconsin experiment. The average coefficient of variation for the 18 litters is more than 4 percent lower than the general average for American swine ex- periments, indicating with a high degree of probability the ad- vantageous effect of the rigorous selection of experimental animals as regards breed, type, age, and ancestry. As regards the relative fattening qualities of the different breeds of poultry, we shall simply refer the reader to an extensive investi- gation of this question conducted at the Central Experimental Farm and reported by Frank T. Shutt.^ Nine different breeds were under investigation, and while some were quite similar in fattening qualities, some were either markedly superior or markedly inferior to others. It is difficult to reconcile such unequivocal evidence as ap- pears to exist as regards the differential fattening qualities of dif- ferent breeds of animals with the cautious statements often made by authorities on the fattening of farm animals. We believe the explanation lies in the method ordinarily used in collecting data for the solution of the question. The ordinary method of solving the question of whether breed is a factor in determining the rate of growth is open to considerable objection. The indiscriminate averaging together of a large niunber of experiments cannot be expected to bring out any differential effect of breed. It may, in fact, actually obscure all such effects, since it is highly probable (in some cases actually demonstrated) that the effect of breed on growth and fattening is a function of the ration fed as well as of other experimental conditions. The Kansas Station has shown, for instance, that scrub or native steers do much better than Short- horn steers when turned out on poor pasture, because of their bet- ter foraging ability, which is a distinctive characteristic of the unimproved cattle of Kansas; whereas, in the fattening pen, the Shorthorn steers possess a more or less distinct advantage.'' Simi- larly, the typical bacon hog ordinarily has the advantage over the lard hog when turned out on pasture, while when fed in dry lot his more restless temperament and his ability to get around better put him at a disadvantage. An interesting illustration of this fact is the behavior of the Yorkshire pigs in the Iowa experiment dis- cussed above (page 500). Therefore, the averaging of the results "Canadian Experimental Farms, Report for lOOf?, np. 210-222. "Kan. Agr. Exp. Sta., Bui. 51. 1895. iy;j] Uncektainty in Interpretation of Feeding Experiments 502 of experiments in which the conditions are not strictly comparable tends to obscure any effect of breed, some experiments com- pensating for the advantage or disadvantage given to certain breeds by other experiments. The best method of attacking the problem, therefore, is the detailed study of individual experiments. It may be concluded on first thought that while different breeds may react differently to any given ration, the disadvantage accru- ing from this fact may be counteracted by balancing lots carefully, i.e., by including the same number of each breed in each lot. Fur- thermore, it seems to be the opinion of some that by including dif-. ferent breeds in the same lot the experiment acquires a more general significance and the conclusions of the experiment have a more general application. As a matter of fact, as we shall show, this greater generality does not at all result from a loose selection of animals. Such selection simply renders the results of the ex- periment more ambiguous. In demonstrating this fact, we shall first consider an experiment by Henry of the Wisconsin Station on 2 lots of 12 pigs each, the data of which are given in Table 3. The pigs of Lot II gained, on an average per head, over 22 lbs. more than the pigs of Lot I. It may be supposed, on first thought, that the conclusion that corn meal, which was fed to Lot II, is better for fattening swnne than whole corn, which was fed to Lot I, applies to all the breeds of pigs experimented upon, i. c, grade Berkshires, Poland-Chinas, Berkshires, Berkshire razorbacks, Poland-China razorbacks, and Yorkshires. An analysis of the individual data reveals a very dif- ferent state of affairs. The four grade Berkshires of Lot I gained, on an average, 86.5 lbs., and the four grade Berkshires of Lot II, 109.8 lbs.; the three Poland-Chinas of Lot I gained 53.0 lbs., on an average, while those of Lot II gained only 49.7 lbs. ; the two Berkshires of Lot I gained 60 and 87 lbs. respectively, while the one Berkshire of Lot II gained 87 lbs. ; the one Berkshire razor- back of Lot I gained 71 lbs., and the two pigs of the same breeding in Lot II gained 75 and 109 lbs., respectively; the Yorkshire pig in Lot I gained 80 lbs., and the Yorkshire in Lot II, 81 lbs.; the Poland-China razorback of Lot I gained 84 lbs., and that of Lot II, 98 lbs. In summing up such evidence as this, no certain conclusion applying to any one breed of animals can he deduced. A fairly high degree of probability has been established that for grade Berk- shires corn meal is better than whole corn, tho one grade Berk- shire in Lot I, fed v.diole corn, gained more than two in Lot II, fed corn meal. For Poland-China pigs, however, the opposite conclu- sion is more applicable. The Berkshire pig in Lot II exhibited a gain identical with that of one of the Berkshires in Lot I. The Yorkshire in Lot II gained only i lb. more than the Yorkshire in 504 Bulletin Xo. 1G5 [July, Lot I. The Berkshire razorback of Lot I gained only 4 lbs. less than one of the Berkshire razorbacks of Lot il, while the difference between the gains of the two Poland-China razorbacks was much less than half the ditiference between the gains exhibited by the two Berkshire razorbacks of Lot II fed on the same ration. Evidently the generality of the conclusion deduced from such heterogeneous experimental results has not been extended in the slightest by in- cluding such different breeds in the same lot. Only a confusing ambiguity has resulted, so that one is not by any means certain that the conclusion applies to any of the breeds. Many instances of the marked disadvantages of including sev- eral breeds in the same lot may be seen in the Iowa experiment of Kennedy and Robbins, the data of which are given in Table 4. Thus, consider Lots IX and X, the average gains for which were 121. 8 lbs. and 102.5 lbs., respectively. The three Yorkshire-Durocs of Lot IX gained 116.7, 122.0, and 155.0 lbs., respectively, and the three pigs of the same breed in Lot X gained 129.3, 108.3, ^"d 123.0 lbs. It would be difficult indeed to differentiate these two lots of Yorkshire-Duroc pigs. The two Poland-Chinas of Lot IX gained 152.7 and 125.7 lbs., while the two Poland-Chinas of Lot X gained 165.0 and 140.0 lbs. Here also differentiation is impos- sible. The Berkshires of Lot IX made gains of 113.7, 101.7, and lOi.o lbs., and the Berkshires of Lot X made gains of 35.0, 26.7, and 61.7 lbs. Apparently with this breed there is a sharp differen- tiation between lots. The Yorkshire pig of Lot IX gained 107.7 lbs., and the Yorkshire of Lot X, 64.3 lbs. The Tamworth of Lot IX gained 12 1.7 lbs., and the Tamworth of Lot X, 171.7 lbs. On such detailed analysis of Lots IX and X as the above, the avoidable am- biguity due to differential breed characteristics is plainly revealed. We. are firmly of the opinion that froDi every standpoint the inclusion of different breeds and types in the same lot of experi- mental animals is a had practice, possessing no redeeming feature. Sex. — The evidence concerning the effect of sex on fattening qualities seems to indicate quite clearly that the castrated male gains faster than the female of the same species and breed. In the case of sheep, the evidence for this statement is very convincing. Carmichaer found in 3 lots of sheep, each containing 22 wethers and 22 ewes, that the wethers, on an average, made 10 percent greater gains than the ewes. Thus, the average daily gains at the end of 117 days were 0.218 and 0.233 lb. for the ewes and wethers, respectively, of Lot I; 0.210 and 0.231 lb. for the ewes and wethers of Lot II; and 0.212 and 0.239 lb. for the ewes and wethers of Lot IV. Curtiss and Wilson^ found the average daily ■Ohio Agr. Exp. Sta., Bui. 1S7. 1907. "Iowa Agr. Exp. Sta.. Bui. 35. 1897. I ^9^3] Uncertainty in Interpretation of Feeding Experiments 505 t> gains for a lot of 9 Shropshire wethers during a feeding period of 106 days to be as follows: 0.43 lb. from September 16 to 30, 0.44 lb. for October, 0.30 lb. for November, and 0.28 lb. for De- cember. A lot of 10 Shropshire ewes on the same ration made the following average daily gains for the same periods : 0.48, 0.32, 0.25, and 0.26 lb., respectively. The lambs in each lot were fed to their full capacity of the grain mixture used, and of roots and hay. The ewe lambs, however, were the lighter eaters. They took on fat rapidly and were more nearly finished during the latter part of the experiment than the other lots, which consisted of wether lambs. According to the authors, "This distinction between the sexes has been observed in all of the experiments made at this station, including both cattle and sheep." J. B, Lawes, in an ex- periment covering a period of 140 days (29), found an average gain of 44.50 lbs. for a lot of 40 crossbred wether lambs, and an average gain of 42.50 lbs. for a lot of 40 crossbred ewe lambs, the breeding being the same. The ration was oil meal, chaffed clover hay, and roots, and was fed to the two lots in the same quantities per 100 lbs. live weight. The fact that the difference between the gains of ewe and of wether lambs in this experiment is small is probably due to the method of apportioning the ration. W. L. Carlyle, experimenting with two lots of lambs before and after weaning (8), reports the individual gains. In Lot I, the average gain of the 10 ewes during the 10 weeks before weaning was 33.50 lbs., and that of the 7 wethers during the same period, 39.12 lbs. During the 10 weeks after w^eaning, the average gain of the ewes was 23.40 lbs., and that of the wethers, 26.86 lbs. In Lot II, before weaning, the average gain of the 13 ewes was 35.70 lbs., and that of the 4 wethers, 40.00 lbs. After weaning, the aver- age gain of the ewes w^as 22.77 ^^^s., and that of the w^ethers, 16.00 lbs. W. C. Coffey (3) experimented on three lots of lambs of different ages, there being 10 lambs to the lot. The lambs were under observation for 98 days. In Lot I, the 5 ewes gained an average of 25.1 lbs., and the 5 wethers, an average of 31.5 lbs. In Lot II, the 6 ewes gained an average of 25.1 lbs., and the 4 wethers, an average of 38.4 lbs. In Lot III, the 5 ewes gained an average of 29.5 lbs., and the 5 wethers, an average of 33.7 lbs. As regards pigs, the evidence on the wdiole favors the view that barrows gain faster than sows under the same conditions. Table 7 contains data pertinent to the question at issue. Carlyle's experiments present contradictory evidence. His lots contained various breeds of pigs. A possible explanation, there- fore, is that there were more sows than barrows of the breeds that gained the faster. His experiment, however, is not reported in sufficient detail to test the correctness of this view. Nevertheless, the table supports the view that barrows are in general better gainers than sows. 506 Bllleti.v Xo. 105 [Jieriment. The statistical data of this investigation are given in Table 5 of the Appendix.'' IllinGis Bxpcrinicnt zcith Steers. — In Bulletin 103 of the Illi- nois Station, H. W. Mumford reports the results of an experi- ment on 10 lots of high-grade Shorthorn steers, the lots consisting of 10 or 15 steers each. The statistical data for the first 16 weeks and the total 22 weeks are given in Table 12. In this table it will be noticed that the average daily gain per steer, instead of the aver- age total gain per steer, is considered. It may be easily shown that T.\BLE 12. — Ch.^nce in Variability of Aver.vge Daily Gains in Weight of Ten Lots OF Shorthorn Steers (43) (All weights expressed in pounds) Average Statistical data of daily gains in weight daily gain per steer Mean Standard deviation Coeffi- cient of variation Mean Standard deviation Coeffi- cient of variation Lot L 10 Steers Lot IV. 15 Steers In 16 weeks. . . In 22 weeks. . . 2.481 2 . 486 .512 . 3,54 20.64 14.24 2.321 2.412 .473 .383 20.38 . 15.96 Lot IT. 15 Steers Lot VII. 15 Steers In 16 weeks. . . In 22 weeks.. . 2.533 .636 25.09 2.499 .503 20.13 2.353 2 . 534 .524 1 22.27 .242 1 9.55 Lot III. 15 Steers Lot VITI. 10 Steers In 16 weeks. . . In 22 weeks. . . 2.072 .497 \ 23.99 2.181 .415 19.03 1.072 .350 17.75 2.106 .200 13.77 Lot TV. 15 Steers Lot IX. 10 Steers In 16 weeks. . ., In 22 weeks. . . 2.353 1 .437 18.58 2.473 .489 1 19.77 2.026 1 .652 2.025 .431 32.18 21.28 Lot V. 15 Steers Lot X. 10 Steers In 1 6 weeks . . . In 22 weeks. . . 2.266 .375 | 16.55 2.470 .388 ' 15.71 1.920 1.995 .640 1 33.33 .539 ' 27.02 'Mich. .\gr. Exp. Sta.. Riil. 138. 1896. ''See page 573. 512 Bulletin No. 16j [J»ly, the coefficient of variation of gains reduced to the daily hasis is the same as the coetiicient of variation of the original total gains. In each lot, with the exception of Lot IV, the gains for the 22- week period were more uniform than the gains for the 16-week period. The quantities of feed consumed by the lots were changed as the experiment progressed, the concentrates in general being increased and the roughages decreased. Canadian Experiment iSth Steers. — At the Xappan, Xova Sco- tia, Station, R. Robertson ran an experiment on 3 lots of 8 Short- horn steers for 135 days. The statistical data for periods of 15 days are given in Table 13. In each of the above three lots of steers, it is evident that there was a pronounced tendency for the gains to become more uniform as the experiment progressed. It is of interest to note that the high coefficients for all lots for the first 15-day period are probably con- nected with the fact that in this experiment, contrary to ordinary practice, a preliminary feeding period was not included. All lots were fed alike, as far as possible, for the entire experiment. The concentrates in the ration (mixed meals) were regularly increased from start to finish, while the hay (or straw) and succulent feeds (roots and ensilage) were decreased. Summary of the Bi'idence. — The experimental data presented in Tables 8 to 13 inclusive are sufficient to show that in some cases, at least, feeding experiments may be so conducted that the percent- age variation of gains within the lot will decrease as the period of observation increases. As to whether this decrease would continue indefinitely or would stop with some minimum coefficient, the data cited can offer no conclusive verdict. It is evident, however, that the increase in the uniformity of lot gains is ordinarily the most rapid in the early periods of the experiment, while in the closing periods of the experiment in most cases the change in the coefficient of variation is gradual. In fact, in some of the cases cited, the co- efficient was practically constrnt for the last two or three periods. From such facts we are inclined to believe, therefore, that the co- efficient of variation of lot gains cannot be reduced beyond a cer- tain mininmm characteristic of the experimental conditions and of the sample of animals under observation. ((/) Bffect of Change of Ration on J^ariability of Gains Illinois Experiment icith Sheep. — Many of the feeding experi- ments whose results we have submitted to a statistical analysis either have failed utterly to exhibit any progressive change in the coefficient of variation of gains within lots, or have exhibited only slight reductions, generally in the fore part of the experiment. We shall consider, first, in this connection, some unpublished data on IQU] Uncertainty in Interpretation of Feeding Experiments 513 w U H to OS o S H Si O « in o n ►J M ^ 'I O S o < Sh H < S < > a o < X u Coeffi- cient of variation 1 v> u (LI u O u O J= t/) 00 LO ^ rt ^ lO M ^ rH CO ■O00't rt u i^; ^O00t-«0«0>0«0t-l c-:oc5cc'.omt^ioao r-l 1-1 W Cl u a^ a> c u O O CO cc c O OJ W Cvt 1-1 tH tH 1-1 T-H •H.2 rt > -<-*MooinM>nwic fO C5 O i> C! O CI J- CI CO O t^ CI C5 t~ Tf X C! lMi-((MCOOJ(MCOrOC>t c/5 — iNrti-li-lTfJ>i-^0 O O Mf-3fO00CO'*000C>- 'i'OOrtioooi— ^ooo ^ 1-1 rt (M C\! N W ^2| aj c " u c u O u C CAl GC o 000 0->*-*-<*>i-lO>0»0000 TJl (M 1-1 1-1 iH a! LO-O M O O W I> O •* 1-1 to OOt-t-CO^t-(M(M CO r0t-O-*'t>Tfco«oi-i«5coc£r: 1-1 T-l CQ W IM CO CO C C H c3cJrSc3n!r3r!c;!cTj lo O >o o "O o 'O o >n i-iCOitiOt^OlOCiCO ccccccccc 514 Bulletin No. 105 [July. lamb feeding collected by this station. Three lots of 7 lambs each were nnder observation for 24 weeks. A statistical resume of the experiment, as far as the gains in weight and the feed consumed are concerned, is given in Table 14. fi Table 14. — Chance in Variability of Total Gains in Weight of Three Lots OF Lambs, and Total Feed Consumption per Lot per Period" (Gains in weight expressed in pounds: feed consumed expressed in kilograms) Total gain Statistical data of total gains in weight Mean Standard deviation Coeffi- cient of variation Total consumption of feed per lot per 4-week period Alfalfa hay Corn meal Oil meal Lot I. 7 Lambs In 4 weeks. . . . 10.93 2.162 19.78 156.4 82.5 5.41 In 8 weeks. . . . 20.00 2.000 10.00 155.5 96.0 5.06 In 12 weeks. . . . 29.86 1.619 5.42 144.6 108.7 5.72 In 16 weeks. . . . 38 . 50 1.615 4.19 125.2 118.8 6.25 In 20 weeks. . . . 45.53 3.963 8.70 119.2 120.4 6.33 In 24 weeks. . . . ,54.00 3 . 093 5 . 73 117.6 117.2 6.16 Lot II. 7 Lambs In 4 weeks. . . . 13.64 3 . 248 23. SI 166.7 66.2 23 . 5 In 8 weeks. .. . 22.79 3 . 853 16.91 173.8 79.1 26.6 In 12 weeks. . . . 33.86 4.307 12.72 159.7 88.8 29.8 In 16 weeks. . . . 42.29 6.702 15.85 138.3 97.8 32.5 In 20 weeks. . . . 49.39 8.371 16.95 119.3 102.8 34.3 In 24 weeks. . . . 60.50 9.464 15. C4 124.1 103.0 34.3 Lot III 7 Lambs In 4 weeks 1 2 . 79 1.870 14.62 177.3 48.4 48.4 In 8 weeks. .. . 23 . 29 2.801 12.03 182.6 56.2 56.2 In 12 weeks. . . . 32.07 2.692 8.39 163.8 62.3 62.3 In 16 weeks. .. . 42.93 2.744 6.39 150.9 68.4 68.4 In 20 weeks. .. . 50.40 4.045 8.03 139.1 73.0 73.0 In 24 weeks. . . . 1 . .'{6 3.758 6.12 137.0 72.6 72.6 "Unpublished data from the 111. Agr. Exp. Sta. H. S. Grindley. W. C. Coffev, and A. D. Emmett, with the co-operation of W. E. Carroll and Sleeter Bull. It will be noticed that in all three lots a decrease in the coeffi- cient of variation occurred for the first three or four periods only. An examination of the quantities of feed consumed per period is, we believe, suggestive of an explanation. We have tabulated this information in the last three columns of the table, the weights of feeds being given in kilograms. It will be seen that the alfalfa hay consumed in general decreased for all lots from the first to the last period, the most notable ex- ceptions being slight increases for Lots II and III from the first to the second 4-week periods. The corn meal and oil meal in general were increased rather steadily from the first to the fourth period. jp/j] Uncertainty in Interpretation of Feeding Experiments 515 From the fourth to the fifth period only a shght increase in the consumption of these feeds occurred in Lots I and II, tho in Lot III there was a more marked increase. During the fifth and sixth periods the consumption of all feeds was practically constant. Comparing these changes in feed consumption with the cor- responding changes in the percentage variability of gains in weight, one is inclined to the opinion that an increasing feed consumption is in general accompanied by an increasing uniformity of gains within the lot as measured by the coefficient of variation, while a constant or decreasing feed consumption is in general accompanied by a constant or decreasing uniformity of gains. This conckision is not incompatible with the experiments above cited, tho in the latter, at times, a constant feed consumption was accompanied by an increasing uniformity of gains. Also, the conclusion reached above, that within any experiment the lots making the best gains generally exhil)it the most uniform gains, falls in line with this conception, which may be restated in the proposition that conditions favorable to good gains are also favorable to uniform gains. It is probable that other factors than change in ration enter into the ([uestion. Possibly the relation of the feed consumption to the bodily requirements is also concerned in the changes in variability of the gains in weight, a liberal feed consumption, possibly, being conducive to an increasing uniformity of gains. Other factors, such as changes in the weather conditions, very probably exert an etfect (see page 523). Wisconsin Experiments with Swine. — We wish to consider next two experiments conducted at the Wisconsin Station by W. L. Carlyle in 1897 and 1898. The object was to compare rape and clover pasture for fattening swine. Corn meal and shorts or mid- dlings were given as supplementary feeds. The average total gains, standard deviations, and coefficients of variation, as well as the total consumption of supplementary feeds per lot per period, are given in Table 15. In the experiment of 1897, the consumption of corn meal and shorts varied only within narrow limits for 8 weeks and was the same for the two lots. However, in the case of Lot I the coefficient of variation decreases almost continuously from period to period, while in the case of Lot II no consistent change is evident. In kK)king for an explanation of this difference between the two lots, the description given of the rape and clover pastures is suggestive: "The rape was quite immature when the experiment began and as a consequence it steadily improved in quality, while the clover was better when the experiment began than it was later." Apparently, for Lot I the pasturage w^as more palatable and was more vora- ciously eaten as the experiment progressed, while for Lot II, the 516 Bulletin No. 165 [July. Table 15.— Chance in Variability of Total Gains in Weight of Four Lots OF Pigs, and Total Feed Consumption per Lot per Period (.58, 60) (All weights expressed in pounds.) Total gain Statistical data of total gains in weight Mean Standard deviation Coeffi- cient of variation Feed consumption per period Corn meal Shorts or middlings Lot I. 19 Pigs on Rape Pasture (1S9T ) In ii weeks 12. 2G 3.242 26.44 590 295 In 4 weeks 22.21 3.915 17.63 560 280 In 6 weeks 35.26 6.640 18.83 627 313 In 8 weeks 47.11 7 . 933 16.84 630 315 In 9 weeks 54 . S9 8.509 15.. 50 315 157 Lot 11. 10 Pigs on Clover Pasture (1897) Lot I. 21 Pigs on Rape Pasture (1898) In 2 weeks 10.11 3.837 37.95 590 295 In 4 weeks 19.95 7.937 39 . 78 560 280 In 6 weeks 38.05 10.831 28.46 627 313 In 8 weeks 45.47 14.711 32 . 35 630 315 In 9 weeks 49 . 53 13.885 28.03 315 157 In 2 weeks 20.43 3.320 16.25 650 325 In 4 weeks 37.62 4.402 11.70 770 385 In 6 weeks 54.47 6.751 12.39 910 455 In 8 weeks 7 1 . 05 9.400 13.23 980 490 Lot 11. 21 Pigs or Clover Pasture (1 808) In 2 weeks. . . 16.52 2.570 15. 56 650 325 In 4 weeks . . . 33.43 5.114 15.30 770 385 In 6 weeks . . . 50.48 6.814 13.50 910 455 Jn 8 weeks. . . 68 . 33 8.800 12.88 980 490 reverse was true ; for this reason, probably, there was an increasing uniformity of gains in Lot I but not in Lot IL The experiment of 1898 affords an interesting confirmation of this view. Here both lots received the same quantities of corn meal and middlings, the consumption of these concentrates increas- ing as the experiment progressed. In this case, however, the clover- pasture lot (II) exhibited a regularly increasing uniformity of gains, while the rape-pasture lot, except for an initial increase from the first to the second period, exhibited a decreasing uniformity of gains. Again referring to the description of the pasturage, we find that "When the pigs were first put on the rape it was in prime condition for them, whereas later it became more ripe and woody and they did not relish nor eat it as they did at first. The clover, on the contrary, was somewhat parched and dry when the experi- ment began, but grew very succulent and tender as the fall rains came on." Henry's Expcr'unoiis at Wisconsin li'itJi Pigs. — \\'e shall cite next several experiments perfonned at the \Msconsin Station by W. A. Henry and associates. The experiments are representative of Jp/J] Uncert-M-ntv in Inteki'Ket.vtio.n or Feeding F.xperiments 517 a series extending over ten years, the purpose of which was to determine the vahie of whole corn as compared with corn meal as the main portion of the ration for fattening pigs. The statistical data of the first experiment which we shall consider are given in Table i6. T.ABLE 16. — Change in Variability of Total Gains in Weight of Two Lots OF Pigs, and Total Feed Consumption per Lot per Week (59) (All weights expressed in pounds) Total gain Statistical data of total gains in weight Mean Standard deviation Coeffi- cient of variation Feed consumption per week Corn Wheat middlings Lot L 19 Pi gs (Whole Corn) lu 1 week 8.37 20.05 2.65 2.72 31.66 13.57 490 560 245 In 2 weeks 280 In 3 weeks 30.26 5.24 17.32 630 315 In 4 weeks 35.74 5.31 14.87 560 280 In 5 weeks 45.89 6.77 14.75 560 280 In 6 weeks 58.16 8.74 15.03 616 308 In 7 weeks 69.63 11.88 17.06 616 308 In 8 weeks 77.11 11.28 14.62 616 308 In 9 weeks 87.05 12.26 14.09 616 308 In 10 weeks 96.26 14.82 15.40 616 308 In 11 weeks 102.40 14.38 14.05 616 308 In 12 weeks 112.41) 16.02 14.25 588 294 Lot II. 19 Pigs (Corn Meal) In In In In In In In In In In In In week. . weeks, weeks, weeks, weeks . weeks, weeks. 8 weeks. 9 weeks. 10 weeks. 11 weeks. 12 weeks. 8.37 19.26 28.95 35.16 44.53 56.47 67.37 76.42 89.21 97.16 101.30 112.20 Z. i.} 4.41 4.15 5.70 7.01 8.14 8.87 10.63 10.18 11.61 13.99 14.82 32 . 90 22 . 90 14.33 16.23 15.73 14.41 13.17 13.91 11.41 11.95 13.81 13.21 490 560 630 560 560 616 616 616 644 644 644 616 245 280 315 280 280 308 308 308 322 322 322 308 In Lot I, from the 6th week to the I2tli the ration was prac- tically constant and consequently the coefficient of variation of the total gains in weight produced shows no tendency to consistently decrease as the period of observation increases. In fact, from the 2d to the 1 2th week the coefficients vary within relatively narrow limits. The coefficients of variation of gains in weight for Lot TI con- form more closely to the changes in the quantity of feed consumed. The continuous increase in the latter during the first 3 weeks is associated with a continuous decrease in the coefficient of variation. } 513 Bulletin No. 165 [July. The decrease in feeds during the 4th week produced an increase in the coefficient. Xo change in feed during the 5th week accom- panied a shght decrease in the coefficient. A large increase in feed intake at the beginning of the 6th week produced a reduction in the coefficient. A constant intake for the next two weeks was ac- companied by a further decrease in the coefficient, followed by an increase. An increase in feed intake at the beginning of the 9th week occasioned a marked decrease in the coefficient. The con- stant feed intake of the next two weeks apparently effected first a gradual increase and then a marked increase in the coefficient of variation. The decrease in feed intake at the beginning of the 1 2th week was accompanied by a slight decrease in the coefficient of variation, this effect being anomalous. The second experiment of this series, which we have subjected to a statistical study, gave the data contained in Table 17. Table 17.— Change in Variability of Total Gains in Weight of Two Lots OF Pigs, and Total Feed Consumption per Lot per Week" (All weights expressed in pounds) Total gain Statistical data of total gains in weight Mean Standard ' deviation Coeffi- cient of variation Total feed con- sumption per week Corn Middlings Lot L 12 Pigs (Whole Corn) In In In In In In In In In week. . weeks, weeks, weeks . weeks, weeks, weeks . weeks . weeks. In 10 weeks. In 11 weeks. In 12 weeks. 14.92 19.67 28.58 34.33 46.67 49.58 53.00 57.58 62.83 66.08 69.33 74.42 ,19 .50 ,27 ,44 6.86 9.85 8.20 9.06 11.97 15.42 16.25 18.20 34.70 22. SG 14.94 15.84 14.70 19.87 15.47 15.73 19.05 23.34 23.44 24.45 2S5 320 326 340 325 308 290 288 284 252 260 226 142 160 163 170 163 154 145 144 142 126 130 113 Lot II. 12 Pigs (Corn Meal) In In In In In In In In In In 10 In n In 12 week . . weeks, weeks, weeks, weeks, weeks, weeks, weeks, weeks, weeks, weeks, weeks. 14.00 2.45 17.50 18.50 3.55 19.17 29.12 4.40 15.12 39.83 6.22 15.61 50.58 8.67 17.15 58.42 10.50 17.97 61.00 10.93 17.90 66.00 13.63 20.65 71.17 14.27 20.05 76.75 18.18 23.69 80.67 22.21 27.53 86.50 26.71 30 . 88 293 340 350 360 361 356 317 312 302 302 308 230 146 170 175 178 181 178 159 156 151 151 154 115 *Wis. Agr. Exp. Sta., 18th Annual Report. 1901. 1913] Uncertainty in Interpretation of Feeding Experiments 519 After an initial increase for the first 4 or 5 weeks, the ration decreased fairly regularly to the end of the experiment, in both lots, the increase in ration was accompanied by a decrease in per- centage variability of gains in weight, while the decrease in ration from the 4th or 5th week occasioned an increase in variability that, in Lot II at least, became more and more pronounced as the ex- periment progressed, so that in Lot I the variability of the gains for the entire experiment was larger than the variability of the gains for the first 2 weeks, while in Lot II the unique condition of much more variable gains at the end than at the beginning of the experiment, resulted. Four more of the experiments 01 Henry and associates have been subjected to an analysis similar to the above. The statistical data for these are included in Tables 6 to 9, inclusive, of the Ap- pendix.* In some of the lots in these investigations the correlation between change in ration and change in the percentage variability of gains is very evident. Of these experiments, only Experiment 32 calls for special com- ment. In this experiment, altho the ration increased more or less regularly from the beginning to the end, the coefficient of variation of the gains in weight of both lots decreased to a minimum and then abruptly increased and remained at the higher level for the rest of the experiment. It is obvious that this is hardly to be ex- pected from the experiments thus far reviewed and from the con- clusions we have thus far developed of the effect of change of ration upon change of variability of gains. The calculations contained in Table 18 afford a more or less satisfactory explanation of these exceptional changes in the coefficient of variation. For the purpose of measuring most effectively the changes in- stituted in the rations, the quantities of the different feeds con- sumed per w'eek have been converted into Scandinavian feed units. One feed unit, according to this system, is equal to i lb. of standard grain, such as corn, or its equivalent in feeding value. According to this system, in the case of pigs, i lb. of corn is equivalent to 1.2 lbs. of shorts or middlings, and to 6.0 lbs. of skim milk. On reference to Table 9^ of the Appendix it will be seen that in Lot II the coefficient of variation decreases for 5 wrecks and then abruptly increases during the 6th week. In Lot I, the decrease continues for 4 weeks, but an increase occurs during the 5th as well as the 6th week. Referring now to Table 18, it will be seen that the number of feed units per 100 lbs. live weight remains at a com- paratively high level in both lots for the first 5 weeks, and then "See pages 574 to 577. "See page 577. 520 Bl'Lletin No. 165 [July, Table 18. — Relation of Feed to Body Weight ix Experiment 32 Lot I Lot 11 Total Total Feed Total Total Feed feed weight units per feed weight units per units of 100 lbs. units of 100 lbs. per lot,* live per lot,» • live lot lbs. weight lot lbs. weight 1st week 257 815 31.5 263 822 32.2 2d week 265 891 29.7 273 883 30.9 3d week 314 977 32.1 330 964 34.2 4th week 318 1072 29.7 323 1069 30.2 5th week 359 1172 30.6 373 1177 31.7 6th week 326 1308 24.9 370 1318 28.1 7th week 339 1369 24.8 394 1396 28.2 8th week 369 1494 24.7 416 1528 27.2 9tli week 397 1552 25.6 448 1623 27.6 10th week 437 1673 26.1 470 1743 27.0 11th week 455 1798 25 . 3 500 1862 26.9 12th week 474 1903 24.9 525 2011 26.1 'That is, the total weight of the lot at the beginning of the week. assumes a lower level rather suddenly during the 6th week. Fur- thermore, in Lot I this lower level is maintained till the end of the experiment, while in Lot II further decreases occur. Turn- ing again to Table 9 of the Appendix, it will be seen that in Lot I, from the 6th week to the end of the experiment, the coefficient of variation maintains practically the same level, while in Lot II, a notable increase occurs at the beginning of the 8th week, coinci- dent with a decrease of one unit in the number of feed units per 100 lbs. live weight, establishing a higher percentage variability for the rest of the experiment. Whether this correlation is really significant or is simply a more or less remarkable coincidence, we are not prepared to say definitely. It is at least highly suggestive. If significant, it would indicate that the effect of change of ration on change of variability of gains is modified by the relation between ration and body weight or ra- tion and nutritive recjuirements. JVisconsin Experiment li'iih Lambs. — An experiment performed at the \\'isconsin Station by \V. L. Carlyle is of considerable in- terest in this connection. Two lots of lambs, 17 in a lot, were fed for 10 weeks before and 10 weeks after weaning. The lambs in Lot I were fed coarsely ground corn, while those in Lot II re- ceived coarsely ground peas. Before w'eaning, the lambs in both lots had all the grain they would eat. After weaning, they were limited to about J4 lb. per capita per day. The statistical data con- cerning the total gains in weight every two weeks thruout the ex- periment, and the total quantities of grain consumed per lot per period of two weeks, are given in Table 19. Jpi3] Uncertainty in Interpretation of Feeding Experiments 521 OS M o ^-« H CO d < Ci O •J < O Q < fc. c o fc 2 o « , w o es < c < O H [I. o >< n < < > U < u I u pa < - ^ S •- Ji O ^ I* ^ o N •a 1- > n u o u C u O o K n ■u r, r* > o r CO — Oi 00 O K -f O OS O lO C! ^B O u. c r3 • - ci > C f^ c> «o or o o w o CO -T o o l^ o i.*^ w o ■^ fir o w Cv> C-. GC ^ '-' CO c ^ w CO -* »o >n Ci 00 v.*? l^ -f o to r~ .-o o c CO X T-t T— t ^- br *— 03 O oi OS Tj< t^ 00 -* T— ( CO to ro Urn o ^^ c c _ n Oi rH ^^ T— 1 *+- Oj C X r^ on o "* to ■ ,_ •— o r- re r- c t- i> Tf CO c ►J " r-t c^ CO ■* r- ^^ 1— < y~4 to 00 ■^ OT c Tt> o 00 w 00 hJ 00 to CO ^ >o tH w CO CO ID o 1-1 1-H Tf to t- 00 >n t> CO co CO Tf CO 1-1 iH 1-H C! Tf LO 00 ^ CO (M l:~ C5 CO Tf I— © CM CO »o to t- 00 oo' l^ t- 00 «0 to c/1 c/1 c/: U-, tA ^ ^ ^ ^ is Cl ■* to X c IT, ^. 'X. ir. w^ j£ ^ ^ ^ ^ to K ^^ Ph < !/l ■•"^^ P o en o o P:j H b O b b W m < c c -j: ^ .2 "O C C/2 -a a; -u rt O C -^ m 2 5 > M r^ tr> OS 00 ■«< ci CO CO w r-l i-l 1— 1 ■* r- -^ c •*".': c «c •.'; t^ — ct c! o « ir: c o o r- t- CO N Ct CO •* m 1.0 t- in >i^ I- in i> (M m (M ca «o o «> (M O m o N iH T-l N Ca CO M" ».T 5D on CO ^ r- w CR ca «l ■* w T— ( 00 r- W ^ CO (N w T-l r-l IH •* -* O c> C -* W CO oc X tr: « c >n C C! CO C ■* ■* t- 1.0 w CO ■<*< ■* •* ■«*> CC r^ .0 ■* CO rt N cj CO Ti< 1.0 «r CO ^ cs t^ 1.0 r- oc CO O N CO X t- o O t- ?> O -f «c C! r— r-" OJ C> CO Tf ,n c/1 CO c/) in in ty. n qj qj CJ (U c 4^ a! >»■* 00 W «> O IM I- i-l 1-1 N CJ c c n c c c c 'p • — ■ — •- •- •- ■- in in in in t/i in .- C C _C .Z^ _C u ni rs rt rt c! C3 PLiOOOOCJOl 534 Bulletin No. 165 [July, periment of its generality or its practical availability will be of value, of course, in rendering the feeding experiment more efficient as an instrument of research. (/) Summary of Methods of Reducing Experimental Error We have shown in this section of the bulletin that according to the evidence available for study, the necessary precision in the determination of the relative fattening powers of two rations or other systems of treatment of farm animals, or of the relative fat- tening qualities of two different lots of animals, not greatly dis- similar in character, is to be obtained in several ways. It is de- sirable, first, to select the experimental animals carefully. If we are to determine the relative fattening value of two rations, or two methods of shelter, or of confinement, or of such environmental factors as these, the animals in both lots should be of the same breed and type, sex, age, and, as far as possible, should have been under the same treatment for some time previous. Disregard of such requirements as these does not, as might at first be supposed, give to the conclusions of the experiment a more general applicability, but simply attaches to them an additional and entirely avoidable ele- ment of uncertainty, for the probability always exists that the dif- ferent breeds, sexes, etc., react differently to the experimental con- ditions imposed. If we are testing the fattening qualities of two breeds of animals, or two lots at different ages, the animals within the lots should be homogeneous and the only difference between the lots should be that of breeding, or age, i. e., the factor under investigation. In the second place, the lots of animals employed should be fairly large. It seems unwise to use less than lo or 15 animals to the lot, and wherever possible the lots should be of more generous proportions, for increasing the size of the lots is the most certain method of rendering the conclusions of the experiment more sig- nificant and less ambiguous. Large lots of animals, however, offer no excuse for a poor selection, because the objections attendant upon poor selection are not removed by increasing the number of animals experimented upon, except in so far as one may sub- divide the lots into larger and larger groups of the requisite homo- geneity. In the third place, the experiment may be conducted in such a way that the experimental error, /. e., the effect of individuality and unequal conditions within the lot, will continuously decrease and the experiment will become more and more efficient as an instru- ment for the solution of the problem at hand. We venture to say that this is perhaps one of the most important of the conclusions of ip/j] Uncertainty in Interpretation of Feeding Exi'ERIMENts S3S this bulletin, it is universally conceded that the larger the size of the lots in a feeding' experiment, the better. Some investigators, at least, fully appreciate the fact that the more homogeneous the lots as regards age, breed and type, sex, and previous treatment, the less ambiguous will be the experimental results obtained. It is also the general opinion that within certain limits peculiar to the animals under investigation, the longer the feeding period, the better for the solution of the problem at hand, but this opinion is founded upon the conviction that the animals must become thoroly adapted to the experimental conditions, and not upon any theorem concern- ing the experimental error. Our results afford a basis for the general proposition that conditions favorable for fattening are fa- vorable for uniform gains. In conducting feeding experiments it appears that if experimental conditions, such as the prescribed rations, are constantly or increasingly favorable to good gains, the percentage variability of the gains and the experimental error will become less and less. Repetition of Feeding Experiments Aside from the aliove method of decreasing the experimental error in feeding trials, the necessary precision in the solution of problems of live-stock raising may be secured by the repetition of experiments that by themselves do not settle the point at issue. In an attempt to determine, by consulting experiment station litera- ture, the efficacy of repetition of experiments in furnishing con- firmatory evidence, a striking condition of affairs and one of vital importance, it would seem, to experiment station work, was dis- covered. After reviewing the large amount of material available for such a study, it was found that frequently when the same sta- tion, and in fact the same investigator, attempted to confirm the results of previous experiments that apparently pointed to very definite conclusions, entirely different results were obtained. This constitutes no reproach or criticism against the particular station or investigator who thus failed to duplicate results. It does indi- cate, however, some defect in the ordinary method of controlling the conditions in feeding experiments, which is worthy of investi- gation and remedy. Henry's B.vpcrii)iciits at JViscoiisiii zcith Pigs. — It was with no difficulty whatever that illustrations of the frequent failure of in- vestigators to duplicate their own results were found. We shall first consider Henry's experiments at Wisconsin, extending over ten years and involving 280 pigs. The object of this extensive investigation was to test the value of feeding whole corn as corn- pared with corn meal as the main portion of the ration for fatten- ing pigs. Eighteen feeding trials were performed, and in fourteen 536 Bulletin No. 1G5 [July, of these trials the corn-meal lots made greater average gains in weight than the shelled-corn lots. The percentage differences in average gain in weight between lots varied from 31.22 to 0.19, six of the trials exhibiting percentages above 20 and six below 10. The number of pigs per lot in the eighteen trials is shown in the following table : Lots Pigs Trials per trial per lot 4 2 3 1 2 4 2 2 5 1 2 6 1 2 7 2 2 8 2 2 9 1 2 10 2 2 12 1 2 14 1 p 19 For the experiment of 1899 on two lots of 19 pigs each the percentage difference between average lot gains was 0.19 in favor of the shelled-corn lot, while in the experiment of 1900 on two lots of 14 pigs each the percentage difference between average lot gains was 20.92 in favor of the corn-meal lot. The average initial weight of the pigs in the latter trial was about 175 lbs., and in the former trial, about 186 lbs. In the first trial, there were 10 pure-bred Po- land-Chinas and 28 crossbred Poland-China-Berkshires, and in the second trial, 21 pure-bred Poland-Chinas and 7 crossbred Poland- China-Berkshires, divided between lots as equally as possible. The first trial contained 18 sows and 20 barrows; the second trial, 11 sows and 18 barrows. The same ration of ^ shelled corn or corn meal to ^ wheat middlings was fed in both trials. The methods of feeding the pigs were practically identical, the main difference, apparently, being that in the 1899 trial the shelled corn and mid- dlings were fed separately to Lot I, while in the 1900 trial they were fed together. The 1899 ^^^^^ extended over 12 weeks and the 1900 trial over 14 weeks. This brief comparative description of the two trials plainly shows their substantial identity as regards the planning and execution of the experiment and the known experi- mental conditions and does not in the least prepare one for the widely divergent results. In the first trial, the shelled-corn lot gained, on an average, 1.338 lbs. per day, and the corn-meal lot, 1.336 lbs. In the second trial, the shelled-corn lot gained, on an average, 1. 145 lbs. per day, while the corn-meal lot gained 1.4 13 lbs. An analysis of this lat- ter experiment by the methods explained in Part I of this bulletin jp/j] Un'CERTAINTY IX IXTERPRETATION' OF FeEDIXG ExPERIMEXTS 537 shows that under the conditions of the trial the odds are only i in about 7700 that on repetition the shelled-corn lot would give a greater average gain than the corn-meal lot. The fact that such a contradictory result was obtained in the preceding year indicates beyond all reasonable doubt that for some cause the tw^o trials were not duplicates; that is, that there was some experimental con- dition or combination of conditions not under control and not de- fined, and yet oi)erating in one trial but not in the other, or operat- ing very unequally in the two trials, which created the discrepancy in the results obtained. Further analysis of the data of the 1900 trial indicates that for some reason which the report does not specify, the shelled-corn lot ate considerably less corn and middlings than the corn-meal lot. Thus, Lot I consumed, on an average, 4.27 lbs. of shelled corn and 2.13 lbs. of wheat middlings per head per day, while Lot II con- sumed 4.51 lbs. of shelled corn and 2.26 lbs. of wheat middlings. In the 1899 trial, this marked difference between lots did not exist, Lot I consuming 4.44 lbs. of shelled corn and 2.22 lbs. of w'heat middlings per day, on an average, and Lot II consuming 4.51 lbs. of com meal and 2.25 lbs. of wheat middlings. Thus, the condi- tion or conditions causing the discrepancy between these two sup- posedly duplicate trials were probably involved in the composition or the preparation of the rations fed, or possibly in the selection of animals that had been subjected to radically different treatment just previous to the experiment. Wyoming Experiments zdth Sheep. — In Bulletins 81 and 85 of the Wyoming Experiment Station, A. D. Faville reports presum- ably duplicate feeding trials undertaken with the idea of testing the value of Wyoming-grown grain for fattening sheep. In the first test, performed in 1908-09, the 34 sheep constituting Lot III con- sumed, on an average, 2.83 lbs. of hay and 0.83 lb. of barley per day, and made an average daily gain in 91 days of 0.33 lb. The 35 sheep constituting Lot I consumed 2.72 lbs. of hay and 0.81 lb. of corn, and made an average daily gain of 0.30 lb. In the second trial, performed the following year. Lot II, consisting of 41 sheep, consumed 2.22 lbs. of hay and 0.89 lb. of barley per day, and made an average daily gain of 0.28 lb. ; while Lot I, also consisting of 41 sheep, consumed less hay per day than Lot II, and 0.89 lb. of corn, and made an average daily gain of 0.35 lb. The average initial weights of the barley and corn lots in the first trial were 60.5 and 59.2 lbs., respectively, and in the second trial, 64.5 and 63.9 lbs. The sheep used in each trial represented various breeds, types, and sizes, divided between lots as evenly as possible. The individ- ual data of these experiments are not given and a complete analy- sis is therefore impossible. Assuming, however, a variability of 538 Bulletin No. 165 [July, about 21 percent in all lots, we find that for the first trial the odds are I2 to i in favor of the barley ration, and, for the second trial, over 100,000 to I in favor of the corn ration. In the latter ex- periment it may be shown that even if the variability of the lots were as high as 79.0 percent, a value extremely improbable, the odds would still be 30 to i in favor of the corn ration. We must con- clude that this is a second illustration of the fact^that the most careful efforts to duplicate experimental conditions in feeding ex- periments as they are ordinarily run often result in failure.^ The same bulletins offer a third illustration of this fact in the relation of the barley to the speltz lots in the two investigations. Montana Experiments with Sheep. — F. B. Linfield reports sup- posedly duplicate feeding trials in Bulletins 47 and 59 of the Mon- tana Station, the object being to test the value of local feeds in fattening sheep. In the first experiment, 22 lambs fed mixed grain and clover hay made an average daily gain of 0.286 lb. in 95 days, and a second lot of 22 lambs fed oats and clover hay made an aver- age daily gain of only 0.220 lb. Again assuming a variability of about 21 percent, in the absence of more definite information, the odds are only i to 33,000 that repetition would result in a greater gain for the oats lot than for the mixed-grain lot. With a variabil- ity as high as 46.2 percent, these odds would still be i to 30. Nev- ertheless, in the second trial, performed the following year, 24 lambs fed mixed grain and clover hay made an average daily gain of 0.231 lb., while an equal number of lambs fed oats and clover hay made an average daily gain of 0.246 lb. Other similar exam- ples occur in the same two bulletins, indicating the frequent inabil- ity of experiment station workers to duplicate their own experi- ments. Minnesota and Pennsylvania Experiments zvith Steers. — In Bul- letin 76 of the ^Minnesota Station, Thomas Shaw reports an in- vestigation regarding the relative gains made by steers while be- ing fattened during the winter in the stall and in an open shed. The seven steers fed in the barn made an average daily gain of 1.742 lbs. per steer in 140 days, while the seven steers fed in the open shed made an average daily gain of 2.256 lbs. on the same ration. In this bulletin the gains of the individual steers are given, and, applying biometric methods, we find that the odds are 1561 to I in favor of out-door feeding. Numerous experiments per- formed at the Pennsylvania Station, however, have uniformly "Concerning the second experiment, Faville says : "The test with barley was hardly a fair one, as four of the lambs in this bunch did very poorly. This was through no fault of the grain." This explanation is hardly satisfactory, since (1) poor gains by four of the lambs would not lower tlie average of 41 gains to any marked degree, and (2) no reason is given for supposing that these poor gains were not due to the grain. jQij] Uncertainty in Interpretation of Feeding Experiments 539 failed to show anything approximating a significant difference in rate of gain between lots fattened in a barn and lots fattened in an open shed during the winter. Difficulties of Repetition. — The illustrations cited are sufficient to show the difficulty of truly duplicating feeding experiments as ordinarily planned and executed, that is, the difficulty of keeping constant, in two consecutive experiments, all conditions affecting to an ai)preciable extent the rate of growth of the experimental ani- mals. The conclusion seems to be that the ordinary manner of con- ducting such experiments contains some serious defect. How seri- ous the defect is and how important it is to remedy such defect is evident when one asks the question: If the experimentalist him- self cannot duplicate his own experiments and obtain similar re- sults even when the most careful attention is given to the details of management and of experimental conditions, what are the chances that the practical live-stock farmer, who necessarily cannot duplicate experimental conditions except in a very approximate manner, will duplicate the results obtained by experiment stations and profit by their recommendations? In many cases the chances are probably remarkably small. When such instances of disagreement between two similarly conducted experiments occur, the attempt is often made to explain away and minimize the disagreement, but the fact that such disa- greements occur in spite of all efforts to duplicate experimental conditions, is significant and worthy of serious investigation, since it is intimately concerned with the value to the agricultural com- munity of all experiment station work of the type under discussion. A Probable Explanation of These Difficulties. — The conclusion to be drawn from the occurrence of discrepancies between sup- posedly duplicate feeding trials is that the conditions deliberately imposed upon the experimental animals have not been sufficiently defined, so that if a more complete definition were made the ex- planation of the discrepancies would be revealed. It is conceivable, for example, that the conclusion that barley has a higher fattening value than speltz when fed to lambs applies only when certain grades of the two grains, definable, perhaps, by chemical analysis, are compared. It appears, therefore, that in formulating the conclusions of a feeding experiment, it must always be borne in mind that rations of a definite chemical composition, as well as of a definite qualita- tive description, have been compared, and that the probability al- ways exists that if the rations had been the same as regard? quali- tative description but much different as regards chemical composi- tion, very different results would have been obtained. A chemical analysis of experimental rations may be supposed, therefore, to 540 Bulletin No. 165 [July, yield valuable if not indispensable data to the proper appreciation of feeding experiments. Other conditions of the experiment should also be clearly speci- fied in experiment station bulletins, and the tendency to continually generalize from data of a very specific description should be guarded against. Is the experimentalist in a position to assert, for instance, that his conclusions are not peculiar to the methods of feeding, the times of feeding, the preparation of the feeds, the mode of shelter, the extent of confinement, the breed of animals experimented upon, the particular herds or localities from which the animals were drawn, etc., etc., which he has employed? We ven- ture to suggest that such possibilities are worthy of consideration, and that the determination as to which of two rations is the best for the fattening of animals is not the simple problem that it is frequently supposed to be. Variability in the Composition oi^ Feedstuffs In connection with the question of the advisability of running chemical analyses on the experimental rations of feeding trials, it was thought essential to investigate, if only in a preliminary way, the natural variability to which the composition of some of the more common American feedstuffs is subjected. For it is of course obvious that if this variability is slight, so as to be negligible for all practical purposes, there is no necessity for the analysis of rations in each experiment, the average analyses compiled by the Bureau of Chemistry, for instance, being sufficient ; on the other hand, if this variability is of such size that it cannot properly be disregarded, then, for the full appreciation of feeding experiments, experimental rations must in each case be analyzed. The study of the variability in the composition of feedstuff's, therefore, is undoubtedly of considerable importance, and, in view of the large mass of data available, it could be pursued as exten- sively as the most ardent statistician might desire. The statistical measures of variation above developed are indispensable to such a study. In the preliminary study of this question that has been undertaken and that has yielded the results briefly summarized below, these statistical constants have been employed. The import- ance of a complete study of the natural variability in the composi- tion of American feedstuffs is such that it is hoped to continue the study later. Corn. — Corn has long been recognized as one of the most stable cereals as regards composition. Thus, Richardson says, speaking of American corn : "There is apparently the same amount of ash, oil, and albuminoids in a corn wherever it grows, with the excep- ■fp-fj] Uncertainty in Interpretation of Feeding Experiments 541 tion of the Pacific slope, where there seems to be no facility for obtaining or assimilating nitrogen."^ Hopkins has even doubted the advisability of excepting corn from the Pacific slope, with ap- parent justification.^ However, the statement above quoted may very easily be mis- understood. It means simply that the average analyses for corn from the different states of the union, when compiled from a suf- ficient number of analyses, generally agree within narrow limits. For example, Richardson reports analyses of corn sampled in diff- ferent states from the crop of 1883,*= The average percentages of protein run as follows: 9 samples from New York, 10.54; 20 samples from Illinois, 10.06; 16 samples from Minnesota, 10.07; 15 samples from Dakota, 10.75; ^3 samples from Nebraska, 10.47; and II samples from California, 10.26. The agreement is certainly very close, considering the numbers of samples from each state. The statement that corn has a very stable composition thus means that there are no constant differences in its composition in different localities of the country. It does not mean that its com- position is practically constant in any one locality. Thus, refer- ring again to Richardson's analyses, the 20 samples of dent corn from Illinois exhibit a coefficient of variation of 11.79 ^^ ^^" gards protein content, certainly no inconsiderable variability. Hop- kins^ analyzed three rows of kernels taken lengthwise of the ear, from 163 ears of Burr's white corn grown on the Illinois Experi- ment Station Farm in 1896. The 163 analyses gave the following results : Ash Protein 10.93 1.048 0.58 Fat 4.690 .4232 9.02 Carbohydrates Average Standard deviation Coefficient of variation 1 . 426 .1090 7.64 82.96 1.182 1.42 We have determined the variability of several groups of sam- ples of corn, each group comprising samples from a single state and a single year's crop, tho not always of a single variety or even of a single class. As a matter of fact, the differences in composi- tion between different classes and varieties of corn (excluding sweet corn from consideration) are apparently slight, judging from the data we have studied. From the coefficients of variation ob- tained in each group of samples, we have computed average co- 'U. S. Dept. Agr., Bur. Chem., Bui. 1, p. 67. 1883. "111. Agr. Exp. Sta., Bui. 53, p. 136. 1898. 'U. S. Dept. of Agr., Report for 1884, pp. 84-85. ^111. Agr. Exp. Sta., Bui. 55, pp. 208-9. 1899. 542 . Bulletin No. 165 [July, efficients, proper consideration being given to the size of the groups in combining their coefficients.^ For protein, we have obtained an average coefficient of varia- tion of 9.30, and for ash, an average coefficient of 12.59, these two coefficients involving 233 analyses. The following coefficients in- volve analyses of 154 samples of corn: moisture, 8.94; fat, 9.22; fiber, 16.77; ^"d carbohydrate, 1.93. The c[nestion arises. How are these coefficients to be inter- preted? \A'e know roughly that the great bulk of fluctuations of sampling lie within a range of ±3 times the standard deviation from the mean,^ except when the frequency distribution is of a very abnormal type. It has been our experience that while the distribution of percentages of the constituents of feeds is very often far from being normal, especially in the case of small percentages, such as of fiber or ash in corn (see page 471), they are nevertheless not ordinarily extremely asymmetrical, so that we can set the limits of such distribution at roughly d=3 times the standard deviation from the mean. Thus, if the average percentage of protein in a year's crop in Illinois, for instance, is 10.50, the standard deviation of samples taken thruout the state, as regards their protein con- tent, could be estimated at 9.30 percent of 10.50, or 0.976, and the rough estimate may be made that all such samples would possess protein contents ranging between 10.50+ (3 X 0.976 percent), i.e., between 7.57 and 13.43 percent. The extreme deviations allowed for by these limits, however, would be of rare occurrence, and if we are concerned, for instance, with the range within which any one sample of corn would be practically certain to fall, perhaps fairer limits would be ±2.25 times the standard deviation, that is, between 8.30 and 12.70 percent. Thus, if we applied the average of 10.50 percent of protein to any one sample of Illinois corn for the purpose of determining the protein intake of a lot of animals in a feeding experiment, we might be in error to the extent of 20 to 30 percent of the total intake of protein, and it may be said without exaggeration that errors of 9 to 12 percent must be expected from such an approximate method of determining protein consumption. In the case of the ash intake, a still cruder approximation would result from the use of an average percentage, since an error of 35 to 40 percent might result, while errors of 12 to 15 percent would be of relatively frequent occurrence. On the other hand, an aver- age percentage of carbohydrate could be used with confidence, since, with the most atypical sample of corn, an error of more than 6 percent could hardly result, while the most frequent errors would be those of 2 to 3.5 percent. "Since the standard deviation of the coefficient of variation decreases in- versely as the square root of twice the number of observations (see formula on p. 486), in averaging coefficients it was thought best to weight each with^2^ "Yule, "Theory of Statistics," p. 262. jp/j] Uncertainty in Interpretation of Feeding Experiments 543 The question under consideration may be approached from an- other standpoint. We have calculated the nutritive ratios and pro- duction values of 6 1 samples of flint corn grown in Connecticut in the same year, the analyses of which are given in the Report of the Connecticut (New Haven) Station for 1893. The average nutritive ratio was i : 10.89, '^"^ ^^^ standard de- viation of the second member of the ratio, 1.398, or 12.84 percent. According to the standard adopted, the limits of distribution may be set at the ratios i 16.70 and i : 15.08.^ Ratios of i :8 or 9, and 1:12 or 13 cannot be regarded as extremely improbable of occur- rence. The nutritive ratio has long been considered a valuable factor in indicating the general character of a food and the function it is likely to perform in a ration. The above calculations would ap- pear to indicate that for corn this ratio is extremely variable for different samples of the grain. The average production value of the 61 samples of Connecticut flint corn was 89.38 therms per 100 lbs. of grain, the standard deviation being 1.348 therms, or 1.50 percent of the average. This is a small percentage deviation and it may therefore be concluded that, as regards energy value, different samples of corn do not vary to any appreciable extent,'' or at least to an extent that cannot properly be neglected in practical work. The more important constituents of corn whose variability can- not properly be neglected are the protein, ash, and moisture. Con- cerning the latter constituent, while its variation in grains is not of any particular moment to the nutritive value of the grain as ordinarily considered, it may, and probably does, bear a close rela- tion to the palatability of the grain for farm animals. Wheat. — Sharply contrasted with the stability in the composi- tion of corn in different sections of the country is the extreme variability in the composition of wheat. Richardson"^ gives the average composition of wheat from different sections of the coun- trv as follows : "Of the 61 ratios actually calculated, the lowest was 1:S.52, and the highest 1:15.17, indicating, as would be expected from the discussion on page 471, that deviations of any given extreme magnitude are more frequent above the mean than below, due to the smallness of the numbers in the second members of the ratios and to their extreme variability. The same condition exists in the case of the distribution of percentages of crude fiber, and in a modified form, in the case of ash percentages, while percentages of protein, moisture, and fat exhibit a distribtition more nearly approaching the normal. "Chamberlain's figures indicate that substantially the same is true of other grains. See Bui. 120, Bur. of Chem., U. S. Dept. of Agr. 1909. "V. S. Dept. Agr., Report for 1884, p. 77. 544 Bulletin No. 165 Table 25. — Average Composition of American Wheat [July, Section Number of analyses Water Ash Protein Carbo- hydrates Atlantic and Gulf states Middle states Western states Pacific states 117 91 177 20 10.34 10.61 9.83 10.25 1.77 1.85 2.06 1.87 11.35 12.50 12.74 9.73 76.54 75.04 75.37 78.15 The variation in the percentage of protein is very marked, and is in fact somewhat obscured by considering such large areas as those in the table. Thus, Richardson gives the average protein content of 8 samples of Oregon wheat as 8.6o percent, of 22 sam- ples of North Carolina wheat as 10.43 percent, of 33 samples of Pennsylvania wheat as 11.44 percent, of 106 samples of Colorado wheat as 12.73 percent, cjf 19 samples of Texas wheat as 13.14 percent, of 13 samples of Minnesota wheat as 13.19 percent, and of 12 samples of Dakota wheat as 14.95 percent.^ Not only does wheat ^ary markedly in composition from one section of the country to another, but also from one crop to the suc- ceeding crop in the same locality. The wheat investigations con- ducted by the Washington Station on the Washington crops of 1905-09 inclusive and reported in Bulletins C4, 91, and 100 are of interest in connection with this point. Considering the Bluestem variety only, since this variety was better represented than any other, 22 samples of the 1905 crop Q2y:i an average percentage of moisture of 10.54, of protein, 11.79, ^"^ of ash, 1.93; for the crop of 1906, represented by 24 samples, these percentages were 11.25, 13.75, ^^'^ 2.18 respectively; for 30 samples of the crop of 1907, 10.83, ii-56, and 1.69; for 22 samples of the crop of 1908, 9.20, 13.25, and 1.88; and for 28 samples of the crop of 1909, 8.1 1, 12.15, and 1.74. From such evidence as the above, it seems that wheat is one of the most susceptible of grains to environmental influences. As regards the variability of wheat in any one locality and from any one crop, we have obtained the following average coefficients of variation from data compiled by Richardson : for moisture, 7.10, involving 242 samples; for protein, 9.66, involving 242 sam- ples; for ash, 11.73, involving 242 samples; for fat, 11.34, in- volving 104 samples; and for fiber, 19.49, involving also 104 samples. The coefficients obtained for the carbohydrate constitu- ents were comparable to those obtained for corn. Comparing these average coefficients with those given above for corn, it seems that in the case of the protein content the two grains are about equally variable; as regards moisture, wheat seems the least variable ; in the case of fat, corn is the least variable ; in 'Tn this connection see also Bui. 128, Bur. of Chem., U. S. Dept. Agr., by LeClerc. 1910. jp/j] Uncertainty in Interpretation of Feeding Experiments 545 the case of ash, the difference is sHght and probably of no signifi- cance; and in the case of fiber, both grains exhibit a high varia- bihty, testifying to the general untriistworthiness of average per- centages of fiber in grains. The \\'ashingtoii wheat investigations mentioned above yield coefficients very dift'erent from those obtained from Richardson's data. Of the Washington analyses, we have considered only the data for the three varieties best represented, i.e., the Bluestem, Club, and Turkey Red. Coefficients of variation were computed for each variety for each of the live crops investigated. The fif- teen coefficients thus obtained, representing 247 analyses, were averaged together, each being weighted with the square root of twice the number of analyses from which it was derived (see foot- note, page 542). The fifteen coefficients for the percentage of moist- ure averaged 9.88, and the fifteen coefficients for protein, 13.64. The average coefficient of variation for moisture is thus almost 3 percent higher than the corresponding average from Richard- son's data, while the average coefficient for protein is almost exactly 4 percent higher than the protein coefficient of Richard- son's analyses. This would appear to indicate that in Washing- ton the composition of wheat varies to a much greater extent than elsewhere. In fact, the average coefficient of variation for the protein content of Washington wheat, 13.64, is higher than the protein coeffi.cient obtained for any other single state, the high- est single coefficient for the other states being 12.67, obtained from 61 analyses of Colorado wheat for 1883. From the above study of the variation to which the composition of wheat is subjected as its environment changes with the locality and the year of growth, it is obvious that average percentages cov- ering the entire country, either for one year's crop or for several combined, can have very little if any practical utility, since they are not strictly applicable to the crop of any one state for any one year, and since they are not even approximately applicable to the crops of many of the states. In this respect, wheat is markedly different from corn, for which average analyses seem to be about equally applicable to all sections, tho, even in the case of corn, varia- tions from year to year seem to occur and oftentimes to be of such magnitude that they cannot properly be neglected. Considering only the variation in the composition of wheat for any one state and for any one year's crop, for most sections of the country the evidence would seem to indicate that corn and wheat are not widely dissimilar, being closely comparable especially as regards variation in protein content. Grains in General. — We have made no statistical study of grains other than corn and wheat. However, of these two corn seems 546 Bulletin No. 105 [July, to be regarded as the most stable and wheat as the most labile of the grains. From a close inspection of the analyses collected by Chamberlain in Bulletin 120 of the Bureau of Chemistry, and from the data of Richardson, LeClerc, and Jenkins and Winton,^ we are inclined to believe that all grains may be roughly charac- terized as follows : ( 1 ) the energy value, either total or that available for metabolism or that available for fattening, of one unit weight of dry substance of any grain is approximately con- stant, no matter where or when grown; (2) with the exception of corn, the chemical composition of grains raised in different sections of the country varies decidedly and depends, not so much upon the variety of the grain, but upon the climatic conditions peculiar to the locality of the crop; (3) all grains vary in composition from year to year, the extent of the variation in composition seemingly depend- ing upon the extent of the variation in meteorological conditions, corn being apparently the least and wheat the most susceptible to such changes; (4) the content of moisture, protein, and ash in grains varies considerably, even in the same locality and in crops of the same year, the variation being such that if the average composition for a given locality and a given year be applied to any one sample of grain for that locality and year, an error of 10 and 15 percent would not be improbable, while an error of 30 to 40 percent would not be impossible. Roughages. — We have made no detailed study of the variability in the composition of roughages. Inspection of such data as those compiled by Jenkins and \\'inton^ would seem to indicate that roughages are much more variable in composition than grains. This is to be expected when it is recalled that with these feeds, besides the climatic conditions, the fertilizers used, the time of cutting, the manner and time of curing, etc., are probably of con- siderable importance in modifying the composition of the feed. From 52 analyses of commercial alfalfa meal obtained from several bulletins on the anal3'sis of commercial feedstuffs,^ we found an average percentage of protein of 14.83, a standard devia- tion of 2.172, and a coefficient of variation of 14.65. The latter figure indicates a very considerable variability, a deviation from the mean of 44 percent being possible, while deviations of 15 to 25 percent would be of rather frequent occurrence. Commercial Concentrates. — In obtaining infonnation concern- ing the variability in composition of some of the commoner com- "U. S. Dept. of Agr., Off. Exp. Sta., Bui. 11. 1892. "Bulletins 141 and 152 of the Purdue Station, Bulletin 141 of the Texas Station, Bulletins 316 and 340 of the New York Station at Geneva. Bulletin 147 of the New Hampshire Station, and Bulletins 71, 78, and 120 of the Massa- chusetts Station. 19^3] Uncertainty in Interpretation of Feeding Experiments S47 mercial concentrated feedstuffs, we have utilized the data from a large number of bulletins on feedstuff inspection. The average coefficients obtained are given in Table 26. Table 26. — Variability in the Composition of Commercial Feedstuffs* Feedstuff Protein content Number of an- alyses Average varia- bility Fat content Number of an- alyses Wheat bran Wheat middlings and shorts. Corn chops Cottonseed meal or cake Linseed meal, old process. . . . Gluten feed Beef scraps" Beef scraps" Tankage" Tankage" Blood meal" 678 963 916 722 73 59 24 29 9 19 S 6.87 8.97 8.64 5.29 5.74 10.09 8.47 7.05 6.14 4.08 4.25 352 711 73 59 Average varia- bility 12.01 18.73 15.91 31.84 Assuming that the distribution of the percentages of protein is approximately normal, in appreciating the significance of the above coefficients of variation the following statement may be made : If samples of wheat bran be taken thruout Illinois, for instance, for any one year, and the percentage of protein in each be determined, I sample on an average out of every 7 taken would exhibit a con- tent of protein at least 10 percent greater or less than the average content for all samples, and i sample on an average out of every 32 would exhibit a protein content at least 15 percent removed from the average protein content for all samples. In the case of stand- ard wheat middlings and shorts, i sample out of every 4 would give a protein content 10 percent or more on eitb.er side of the mean, and i out of every 1 1, a content 15 percent or more on either side of the mean. For the other feedstuffs, the following figures would hold approximately : Feedstuff Corn chops Cottonseed meal Linseed meal . . . Gluten feed Beef scraps" Beef scraps* Tankage" Tankage" Blood meal Number of samples 10 percent or more greater or less than the mean 1 out 1 out 1 out 1 out 1 out 1 out 1 out 1 out 1 out of every 4 of every 17 of every 12 of every 3 of every 4 of every 7 of every 10 of every 70 of every 53 Number of samples 15 percent or more greater or less than the mean out of every 12 out of every 214 out of every 110 out of every 8 out of every 13 out of every 30 out of every 68 out of every 4500 out of every 2200 °A11 adulterated samples were left out of the computations contained in this table when adulteration was noted. "Guarantee of about 40 percent protein. ''Guarantee of about 55 percent protein. "Guarantee of about 60 percent protein. "Guarantee of about SO percent protein. 548 Bulletin No. 165 [July, From such considerations it appears that if an average analysis be used in computing the protein intake of experimental animals in a feeding trial, instead of a direct analysis, as far as the above commercial feeding stuffs are concerned an error of lo percent or more would not be infrequent in most cases, and in some cases an error of even 15 percent or more should not occasion surprise. Conclusions. — It is evident from such a preliminary study of the question of the variability in the composition of American feedstuff's, that as regards feeding experiments, the practical utility of average analyses is liinitcd, and in the case of many of the grains and roughages is small indeed. This conclusion is especial!}- to be emphasized in the case of averages supposed to apply to the entire country and to all crops, since it cannot be doubted that marked differences occur in the composition of grains and rough- ages from locality to locality and are even likely to occur in the same locality in different years. These remarks apply, in the case of grains, to the content of moisture, protein, and ash especially; while, in the case of roughages, even the energy value of a unit weight of fresh substance may be subject to marked variation, a problem that we hope to investigate further. The protein content of commercial concentrates is also often subject to marked varia- tion. Even when averages are taken of the composition of any one feed in any one locality for any one year, it has been demon- strated that samples possessing protein, ash, and moisture contents 10 percent, 15 percent, or more, greater or less than the mean con- tents, must be reckoned with. In view of the great variability in the composition of feed- stufifs and of the fact that a proximate analysis of rations can be secured relatively easily, we are inclined to believe that one cannot afford to omit such a precaution in feeding experiments, especially when they are otherwise comprehensively planned and capable of quite definitely settling the problem at hand. ipij^ ■ Uncertainty in Interpretation of Feeding Experiments 549 PART III. SUMMARY AND CONCLUSIONS (i) Difficulties in Interpreting Feeding Bxperiinents. — The simple feeding experiment is of value in the solution of many prob- lems of practical live-stock raising. Under the best conditions, how- ever, the results of the feeding trial do not point unequivocally to one conclusion, but are of more or less ambiguous significance. The cause of this ambiguity is the dissimilarity existing among the gains of individual animals due to what may be termed indiznduality as well as to unequal conditions within the lot of animals. One of the essential problems in the interpretation of a feeding experiment is the comparison of the gains in weight obtained for one lot of animals with the gains in weight obtained for another lot, the purpose of the comparison being to determine whether the difference in treatment accorded the two lots, or the difference in their make-up, as the case may be, has been instrumental in securing a difference in their gaining abilities. If one can assure himself by the proper methods of analysis that the relative position of the average gain of one lot with respect to the average gain of the second lot will remain essentially unaltered if the experiment be repeated on other similar animals under similar conditions, it fol- lows that one is justified in attributing to the essential difference or differences in treatment or make-up between the two lots, an influ- ence on their gaining qualities. If one cannot so assure himself, there remains only the alternative conclusion that whatever differ- ences in gains are observed between the two lots are due entirely to the individualities of the animals and to other uncontrolled factors. (2) The frequency Distribution. — One of the most fruitful conceptions of the biometric method of analysis is that of the fre- quency distribution. A set of data obtained tmder comparable experimental conditions is to be thought of as tending to assume a definite distribution about some typical value, to which value the arithmetic mean, or the common average, is often a good approxi- mation, in spite of the fact that the sources of variation under such conditions act in a random fashion. It is on this tendency of comparable experimental data to assume a definite frequency dis- tribution, expressible by a frequency curve capable of mathematical definition, that all attempts to predict the result of repeating an experiment must be based. (3) Use of Average Results. — Average results should be used with extreme caution. An average is at best only an imperfect description of a series of experimental data, and when used for comparative purposes is often extremely misleading. The calcu- lation of an average should not be considered a reason for not 550 Bulletin No. 165 I [hdy, collecting or reporting- the original data, since only by mi^nce to the original data can its value be determined, and cong^aj^fntly only by publishing original data can experiment station workers criticise and properly appreciate each other's investigations. (4)Variatio)i and Its Measurement. — In adequately comparing the gains exhibited by one lot of animals with those exhibited by a second lot, it is necessary to calculate, not only the average gain of the lot, but also the variation or dispersion of the gains within the lot, a measurement of the latter being a measurement of the influ- ence of the uncontrolled factors in the experiment. A good meas- ure of variation for this purpose is the standard deviation, which may be defined as the square root of the average squared deviation of all individual gains from the average gain for the lot. The average of a series of gains in weight, as well as the in- dividual gains, must be considered as possessing a variability due to the experimental factors that were not under control, and since these uncontrolled factors find direct expression in the variability of gains within the lot, it follows that the variability of an average gain bears a definite relation to the variability of the individual gains within the lot. Obviously, the variability of an average gain decreases as the size of the lot increases, and it may be shown that the relation is such that the presumptive standard deviation of the average gain is equal to the standard deviation -of the original gains divided by the square root of their number. (5) The Probable Error. — In predicting the result of repeating a feeding experiment, on two lots of animals w^e will say, using other animals, but subjecting them to the same conditions that obtained in the given experiment, we first make the assertion that the most probable average lot gains that would be obtained in a second experiment are the average lot gains actually obtained in the^first experiment. Our prediction is very inadequate, however, until we estimate from the data of the first experiment what devia- tions in a second experiment we must expect from these most probable values, since it would be remarkable indeed if exact dupli- cation occurred. It is the purpose of the probable error of these average lot gains to afford this information. The probable error of an average gain is that value which, w^hen added to and subtracted from the average, defines two limiting values such that the odds are .even that a second experiment will give an average gain falling between them. If w^e add to and subtract from an average gain 3.17 times its probable error, there are obtained two limiting values such that the odds are 30 to i that a second experiment will give an average falling between them. Now odds of 30 to I represent a degree of confidence amounting to practical certainty, so that we may feel reasonably certain that ipij] Uncektainty in Interpretation of Feeding Experiments 551 a second experiment will give an average gain for a lot of animals of a specified description and under specified conditions, lying somewhere within an interval defined by adding to or subtracting from the average gain experimentally obtained 3.17 times its prob- able error. The probable error of an average gain is obtained by simply multiplying the standard deviation of the average by 0.6745. It is generally desired, however, to determine the significance not only of average lot gains, but also of differences between aver- age lot gains. The probable error of such a difference may then be calculated by squaring the probable errors of the two averages involved, adding, and extracting the square root of the sum. The probable error of a difference between two average gains defines its significance in exactly the same manner as the probable error of an average gain defines the significance of that average. By the use of such a probability method as that briefly out- lined above, we are able to interpret the results of feeding experi- ments in a fairly satisfactory manner. The element of uncertainty resulting from the meaningless variation existing among individual gains, due to uncontrolled experimental factors, has been definitely and reasonably defined. (6) CoefHciciits of Variation. — For some purposes, the stand- ard deviation is inadequate as a measure of variation, due to the fact that it depends upon the units of measurement employed, and for gains obtained during different periods of time or gains ex- hibited by different kinds of animals, is correlated with the aver- age gain. For extensive comparisons of variation, therefore, the coefficient of variation is used, this coefficient being simply the standard deviation calculated as a percentage of the average. The coefficient of variation of gains within lots is a good measure of the experimental error. From an extensive review of experiment station literature in this country, we have obtained an average coefficient of variation of gains of about 21 for similarly treated lots of sheep. For steers and swine, an average coefficient of about 17 has been ob- tained. From these figures, supplemented by a detailed study of the data, it appears probable that sheep in general exhibit greater variability in gaining qualities than do either steers or swane. The-, small amount of data we have collected concerning the fattening of poultry indicate an average variability of about 16 percent. Apparently poultry may be classed with steers and swine as re- gards variability of gains. Extreme discrepancies were found to exist among individual coefficients of variation. This is doubtless due in part to the het- erogeneity of the data, but it is in large part to be expected from the mere size of the coefficients. A determination of a relation be- tween particular rations or systems of treatment and the variability 1 SS2 Bulletin No. 165 [July, of gains is practically impossible except in extensive or in repeated experiments. • (7) Number of Animals Required per Lot. — Based upon the average coefficients of variation found for sheep and for steers and swine, calculation indicates that experimental lots should con- tain at least 10 to 14 animals, or even 25 to 30 animals wh^i the rations or other conditions under investigation are very similar. The necessity of using at least 10 to 15 animals per lot in feeding trials seems to be well established. Wherever this number can be increased, the better, for this is the surest and most generally rec- ognized means of increasing the significance of experimental re- sults. Again, however, it is well to note that increasing the size of lots is no remedy for a poor selection of experimental animals. Furthermore, increasing the size of lots cannot eliminate individ- uality, but merely reduces its effect on the average. It has been shown that when there are as many as 40 animals to the lot, an appreciable degree of uncertainty still attaches to average lot gains. Also it should be borne in mind that the beneficial effect of increas- ing the size of lots varies not with the number in the lot, but with the square root of this number. Thus, for the same standard deviation of individual gains, a lot of 10 animals will give a prob- able error of the average gain only twice as large as a lot of 40 animals. (8) Uniformity of Gains is Desirable. — Whenever and where- ever possible it is advantageous to reduce the experimental error of feeding trials, i.e., to increase the uniformity of gains within the lots, provided the value of the experiment and its practical availability are not also thereby reduced. The smaller the coeffi- cient of variation of the gains in weight within a lot, other things being equal, the smaller the minimum percentage difference be- tween its average gain and that for a second lot that can be defi- nitely traced to the difference in treatment or to the difference in make-up between the two lots. Hence a reduction of the experi- mental error means a reduction in the coefficient of variation of gains within the lots. (9) Selection of Animals to Insure Uniformity of Gains. — It is well known that animals at different ages exhibit different rates of growth and different fattening qualities. It is also obvious that different breeds of the same species of animals often exhibit similar differences, especially if they are of different general types, and even where it is not obvious that breed differences exist it is not justifiable to assume that they do not exist. The available data indicate with a high degree of certainty that wethers gain faster than ewes, barrows faster than sows, and cockerels faster than pullets, at least at the fattening age. Furthermore, it is beyond dispute that differences in treatment of animals previous to ex- /p/j] UXCERTAIXTV XX IXTERPRETATION OF FEEDING EXPERIMENTS SS3 periment may frequently be the cause of differences in fattening qualities. The careful and intelligent selection of the experimental animals is one of the best methods of reducing the experimental error and thus obtaining more valuable and more significant re- sults without interfering with conditions that the experiment must conform to by reason of the use to which its conclusions are to be put. We cannot over-emphasize the necessity of securing per- fectly homogeneous lots as regards age, breed, type, sex, and pre- vious treatment. The great preponderance of evidence indicates that by thus selecting the animals for experimental purposes, the experimental error will be greatly reduced. The necessity of selecting homogeneous lots of animals is not appreciably dimin- ished by the balancing of heterogeneous lots. (lo) Good Gai)is are Unifonn Gains. — In ^ny experiment in- \T)lving two or more lots of animals, it has in general been found that the lots exhibiting the best average gains also exhibit the more unifonn "gains, and vice versa. (ii) Changes in the Variability of Gains During an Experi- ment. — It has been found from experiments in which the ex- perimental animals have been weighed periodically during the investigation that frequently the coefficient of variation of gains progressively decreases from the l^eginning q| the experiment to the end, the rate of decrease being greater during the early periods than during the later periods of the feeding trial. Apparently this decrease would not, under the best conditions, continue indefi- nitely, but would gradually attain to a minimum coefficient char- acteristic of the particular sample of animals under observation and of the particular experimental conditions. In other experiments, a continuous decrease in the coefficient of variation of gains is not evident. In most cases of this descrip- tion that we hav6 analyzed, a more or less close correlation be- tween changes in ration and changes in variability of gains may be observed, such that an increasing ration is generally accom- panied by a decreasing coefficient of variation, a constant ration by a constant or slightly increasing coefficient, and a decreasing ration by an increasing coefficient. Unfavorable weather condi- tions seem also to be instrumental in producing more variable gains, while in a few instances the correlation between ration and coefficient of variation above defined seems to be complicated or obscured by other factors, such as the relation of food intake to body weight or bodily requirements. \\'hile the evidence adduced does not unanimously point to one explanation of the changes in variability of gains during the course of a feeding trial, consider- able support may be found for the general statement that when conditions are constantly or increasingly favorable to growth and 554 Bulletin No. 165 [July, fattening, an increasing uniformity of gains is generally secured, or in other words, the experimental error is progressively reduced. It seems, therefore, that whenever practicable and whenever the nature of the experiment will permit, the animals should be in- duced to consume an increasing amount of food, that is, a con- stant ration per lOO lbs. live weight. A considerable increase in the ration near the close of the experiment for the purpose of "finishing off" the animals for the market is frec[uently very effi- cacious in securing more uniform gains. (12) Physiological Selection. — Another method of reducing the experimental error of feeding trials that is in vogue in one form or another at different stations, has been investigated. The es- sence of this method is the selection for experiment of only those animals that during* the course of a preliminary feeding period have proved themselves to be functionally similar as regards the rate of growth or fattening". Hence we have called the method physi- ological selection. From theoretical considerations alone, it appears that even if physiological selection is efficacious in accomplishing its purpose of eliminating poor gainers and reducing experimental error, it will so multilate the feeding experiment itself as to render it much less valuable to practical li\"e-stock farming and to limit its applicability and thus reduce its significance. Experimental evidence, however, indicates clearly that physi- ological selection does not eliminate the poor gainers. In fact^it appears that those animals exhibiting the poorest gains in a pre- liminary period are in general no worse than a random sample of the entire group of animals in a subsequent feeding experiment. Furthermore, physiological selection is very inefficient in reducing experimental error, even when conducted along the most rigorous lines. Hence this method is both theoretically faulty and practi- cally incompetent to accomplish its purposes. (13) Repetition of E.vpcriuients. — The necessary precision in feeding trials may be attained by a reduction of the experimental error as above shown or by repetition of the experiment. From a study of the efficacy of repetition, it appears that frequently under the most favorable conditions, feeding experiments cannot be duplicated. Frequently experiment stations have obtained re- sults from feeding trials pointing unequivocally to a certain con- clusion, and yet subsequent attempts to duplicate such experiments have yielded results quite incompatible with the first conclusion. The gravity of such a situation cannot be over-emphasized. Its remedy seems to be, first, the more careful reporting of experimental conditions, including a chemical analysis of rations, and second, the conviction that the conclusions of feeding experiments are more intimately connected with the particular experimental conditions ip/j] UXLEKTAINTV IN INTERPRETATION OF FEEDING EXPERIMENTS 555 that obtained than has heretofore been beheved. The conckision, for instance, that one feed is better for fattening purposes than another may be totally at fault if other samples of the two feeds, possessing quite different compositions, be used, or if other breeds of animals, or animals more (or less) mature, be used, or other methods of preparing the feeds or sheltering the animals be fol- lowed. Such possibilities should always be kept in mind, and the frequent tendency to generalize from data of a very specific de- scription should be carefully guarded against. (14) Variability in the Composition of Feedstuff s. — The ad- visability of submitting experimental rations to a chemical analy- sis is clearly indicated by a study of the variability in the composi- tion of feedstuffs. In the case of grains, this varialnlity is negligible, apparently, as far as the energy value of the feed is concerned, but it is considerable and in many cases extreme in the case of the moisture, protein, and ash content. With roughages, inspection of analytical data w^ould indicate an even greater vari- ability than with grains, apparently involving even the energy value. In the case of commercial concentrates, variation of the protein content is often quite comparable to that in grains, tho in the more highly nitrogenous concentrates, such as blood meal with a protein guarantee of 80 percent, the percentage variability is less evident. (15) Individual Feeding Not Essential. — The simple feeding experiment concerning itself entirely with the gains in weight and the feed consumption of farm animals under certain definite experimental conditions, has served many useful purposes and yielded much valuable information to practical live-stock farming. Its purpose is to yield specific information which must generally be considered in connection with the specific conditions under which it was conducted, as opposed, for instance, to the purpose of the nutrition experiment, which is the securing of more or less general information, not so strictly limited by the conditions under which the experimental data were collected. Therefore, it is neither necessary nor, in fact, expedient that the technic of the simple feeding experiment be carried to the same degree of refinement as that of the nutrition experiment proper. Any great refinement of the former, is objectionable from the standpoint of the practical availability of the results of feeding trials. Thus, the individual feeding of animals in ordinary feeding trials seems unnecessary, if not inadvisable, because we are here imposing an experimental condition entirely out of harmony with ordinary practical live-stock raising, and while the experimental error may very probably be reduced by seeing to it that each animal obtains the same amount of feed per 100 lbs. live weight for instance, SS6 Bulletin No. 165 [July, the practical availability of the experimental data obtained would undoubtedly be greatly reduced. The individual feeding of ani- mals may yield valuable data for some purposes. However, the variability in the consumption of feed and consequently in the gains produced cannot be presumed to be the same in individual feeding as in lot feeding. (i6) Publication of Results.' — The results of feeding experi- ments should be published, not only with the idea of describing a particular investigation, but also with the idea of determining, in so far as such a determination is possible, whether a reasonable probability exists that the practical live-stock farmer will actually benefit himself by applying the results of the investigation to his own live stock. If no such probability exists, the farmer should be specifically warned. The elaborate analysis necessary for an- swering such a question will very probably not be appreciated by the majority of the readers of experiment station bulletins, but this is no excuse for not using such analytical methods at the expense of accuracy in the formulation of conclusions and recommenda- tions. As a matter of fact, the analysis undertaken need consti- tute no part of the bulletin published, the purpose of such analysis being primarily simply to check or rectify conclusions. (17) Formulating Conclusions. — In formulating the conclu- sions of feeding experiments, the necessity of keeping in mind the possibility that several of the specific experimental conditions may seriously limit the applicability of the results of the investigation should not be lost sight of. Thus, Ration A may be superior to Ration B under some, but not all, conditions. The possibility, if not the probability, exists that if the constituents of Ration A are not up to a certain standard, the reverse relation may hold ; hence the necessity of a chemical analysis of the rations used in order that one may know the actual conditions under which the experi- mental conclusions may reasonably be applied. It is not sufficient simply to enumerate the individual feeds of which the rations are constituted and the proportions in which they enter into the ra- tions. An exhaustive and repeated chemical anlysis of rations is neither necessary nor especially advantageous. In fact, if a fairly complete analysis of feeds be made at the beginning of the experi- ment and substantially the same feeds be used thruout the subse- quent feeding period, it may be necessary to run only moisture determinations on the feeds from time to time during the experi- ment. If variation in the moisture content of feeds is not appre- ciable during storage, even the repetition of moisture determina- tions will be unnecessary. However, an ordinary' analysis should be made of each new supply of feed from a sample fairly repre- sentative of the entire supply. jp/j] Uncertainty in Interpretation of Feeding Experiments 557 Other conditions than the composition of rations may limit the appHcabihty of conchisions. The manner in which the feeds are given to the animals, e.g., whether they be given ad libitimi or in restricted quantities, may determine to some extent the relative merits of rations. The breed or type of animals experimented upon may be still another limiting factor. The age or condition of the animals may be still other limiting factors. Such consid- erations as these, which are associated with greater or less degrees of probability, should receive due attention in interpreting feeding experiments, and the assertion that a given experiment indicates a superiority of one ration over another should be made only in close connection with a brief statement of the more important ex- perimental conditions. In conclusion, we take pleasure in acknowledging the valuable assistance of Professor H. L- Rietz in aiding us to a proper com- prehension of the technic of the statistical methods and of their general applicability to agricultural problems. SS8 Bulletin No. 165 U»ty, K m z o < < Q < < H 1-1 < a O O c o Is c CO -a c 1§ C3 * — en O d Length of experi- ment, days H d . d CO -^ "-I »n J> r- O ^- « •«»< t- rH o r- C5 c» C5 X O (M t^ CO to 1-1 X T-l CO c\> CJ iH t> C> X CJ IM W CO I- l^ C5 O X ■* t- o t- O X t> 1-1 CO r-l O -^ t- ^ 1-H 00 M C^ "-I i-l tH r-1 t^ ira t- CO 1-1 rn r-< CJ ■* t- X (M W T-l O CO 1-^ CO rt X O O X o (M >n (N CO 1-1 CO to N rH CO CI X CI 1-1 e* T-l T-l T^l If) (M >n in »n t- OJ O tH 05 O o CO 1-1 O o O tH !> «r5 ■^ to -^ X Tf t- O C-. X t- X t- O CO in X ■ 1- CO O t- CO ti to >0 lO ■>* !0 to X >n ■* I> CO to CO CO CO CO Ci 1*' t* It -^ t^ o o M >n ■* Ti< 05 IN in CO CO ■* to CO Tj" Tf .0 C! O (M l^ CO X C! O C! o t- to X o N »n O CO ■* o o ■* O «D CO C-. CO X X C! C: to o ^ X o X t- ■* in ■* O ■* CI X X X 1-1 1-1 o c; 1-1 t- to t- CO C5 l- 1.0 O T}< LO to CO t- to J> CO t- C^ LO X 0! to t> to X 't l> m ■* Ti< T}< Tf o o «o iH »r5 »-l o O (M t- X to X 05 I> ■* to 1-1 (M t- O to CI in to o c X o E < CO ■* Tf CO CO CO O 50 to O » «D o o o o o o — ' 'f w w !M (M :^ ^ J^ 1— ( f— ^ *— « f^ K*^ *^fc 1— 1 1— 1 K* KH H-( 1— ( ^^ 1 — 1 1 — 1 XXX O CV C". CO HH 1—1 t— H l-H HH 1 — II — 1 »— t f— 1 »— ( 1— 1 HH HH I—I t— I S.. >-^ 1—1 1— 1 1— 1 l'^ 1^ 1-^ 0! Oi T'J W O! W rv C5 C-. o c~. OS o o o o t- l^ £- I- o 9 ■n in in o o o o o c r^ r-t 1-1 r-l 1— 1 1— 1 -^ (M ^ ira fO J9I3] Uncertainty in Interpretation of Feeding Experiments 559 « d c o i> -^ o rt O > n TS n j= S.2 be "S rt QJ C ■- ? C3 > H-P (U c m-v c rrt o c a C "ti o Q .Si rt — ' o , CO -a < OS t- "c < W 1 — 1 J < H t/i ri; H < H L/J M-l hH og- .J n 6^ < t« H o J d y5 ■ -■ >> M Q< C X . iJ -^ 1 c 1— 1 Ui IL> ° H e .' ° hl^; ^t X c w y £ ;0 (M O 00 O T-l ■^ n ^•r-roi.ooo«>THrH ■*C0Cllr-OOt-00 O TficOOOOOOrHcOOOOO oioowooinocMOO t- m 05 00 rH rH t- •rH CO CI 1— ! O >0 tH r-( IM rH CO tOOOCOIMOOOt--* rH rH rH C 00 o o to o •*^t»-CO«DOi«rH rHt-OOlMt-OOt-rH iMeOOOrHt-rHtO-^rHf^ COrHO!>OOC>it-CO W 00 rH C5 O N C» CO ■* o >.o o >-0 CO ■* CO CO (M (M CO in [-cotoinooTtiinoco 00 rtl rH O a CO 00 t^ OJ NCJONNOMO inintomoowmNOo ON-* i.O •* 5D — 1 CO W CO (M — lOO-*'OtS«3i-l COWOOrHCOrHCOW 00 «o ^rHJ>CO-*rHOt-COO COCONWCJOOCOlMOOCJ Ci O 00 CO -^ CO O C5 m 00 00 CO l> rH C2COroOt-OOt~ TfiSDCOCOOJlMOOO ooco(M-*oorHco«oinTf< ■^01!0 Ci CO o CO w c^ 00 CO o >n (M rH CO r-l coot-oooocoo 00 iNrHtCrHCOThfOIMOO in ■* r^ O -t> M o T- O t- ■* OOOOCO^-<1-* T^ r1 W rH C. 35 C5 to >n 00 o 0(MOrjOOUOl>00>n o mininincoi>rHin-*co 00 O I> 00 ot 00 00 rH r- rH 00 00 -O CO 5D rHCloOIMl^iMOO 4.0 00 Tfi i- «0 X 'O 00 «0 OOOOOOr-.Oi-HO O CO c CO J> oc "u E < W CJ t- r- t- !^ oo«oo;do5050 OOOOOoc-.Cocr. rH rH rH i-H rH r^ Ci <^ c-. c) JO ciJ j3 rt^ rS^ rt^ rt-o o 1 — II — 11 — '^^.^l — 11—^1 — iK^ix* '"' S 1— 1 r^ t^ |I^ t— 1 1— H V— 1 HH HH l-H HH 1 — \ r^ r^ .-J >n o o r-( T— 1 O o O O t> I- t- I- OOOOOOOO i> t^ t- £- t- t>- t^ !> o o o 00 05 " n c > c« u 60 O IS H 560 Bulletin No. 165 [July. Ph Ui u K to !z o < < Q < < H CD a O .5? .5 _c 'rt O c o o > 1§ co-o lU V) o '5 l-H c n.2 o m 2 rt > ^ ooo>oosco>nco (M in " X •* IN t- ■* CO •>*< Tf X 1 CO •* w t- CD I- r<5 CD CO CO CO 0» Oi IM T-H M rH W r-l -hcdoocothcocooooim (M i-H 1-1 CO (M W (M O rt 1?! CO CO •<1< CO LO CO W CI t- in m CO T-l OS CD CD O -"f I- N « CO W W t- 05 O n ro Oi tH M •^1-ICO-^COOO^OO GO 00 CI t> CO (^ O t^ O I- 1^ LO O t^ ^ N CD O! CO OS O O CO CO O l- T-1 T-t 1— t COiOtOGOCOCOCOCOCOCO CO CJ LO '-' 1-1 in Ti< Tt CO OS CO o in -* > < •JS CO CO N (M O Tj*r»O^OOCOOWCS cococol^Jco<^^cococow X t^ N CO •* N X CD CO W r-l ■* o ^ in CO CD '!0 ■* O -^ t- Oi ■* o CO CO t- CO cOTficOOOCONscWcocO oirawcooooooococD CO CO lO t- CD « X X T-H ■* w in "a CO >0 CJ t- O 00 C^'XCOOOOJOOCO-*-*© ^H 1— i T— ( T-l tH rH r^ co CO CO CO X rt N CO X OS r-t 05 r-< CO O CO I> CO W 05 O CO C» OCOOSi-IOOiOt1<^OOC5 O W Tj< © •<*■>*< r»< CO C u X W bo •-0 OJ en "l- X -^ C; CO CS CO X CO C-. — ' O l^ CO t- N CO' OS T-. N (M O C! O Oi Noxo-^ost-cocom i.O CO o o (M O -* X O O 1-1 >- •5 OS iH tH o CQ ^ !> 00 CO 00 00 C» >OrJ!?•*■>!< OS u 1-1 1) s < 00 05' OS OS 00 cr: oooooooeoo IC LO in in in in t- t> t- K-( l-H !.> K^ _ f-1 ^^ t-H 1 — 1 t— 1 t-H ^ i-H K^ t— ' O o o o o c> xxxxxxxxxx OSOSOSOOiOOOSCSO X X X X X X O^ O^ C^ OS XXX CO CO CO r-l T-l T-l t3 c« '—I ^c s CO ■* in s I9I3] Uncertainty in Interpretation of Feeding Experiments 561 o O o < < Q < < (-1 in n < a. «> o .S c c5 O 5.1 'ii o > o I'E c 1 ,-1 TT t- O M O ■* t~. ^ rt O T)< o 01 05 CJ en X X X Ci X " CO o -f O — ' CO X c: tr o e >o X CO LO X -*" o> -^ lO — (D t- ■* O ^ <-! rt -4 M ■^ X O t- Jvj rt CM -^ X O i-O 1-0 O CO X CO o — Oi CO Ci CO N CO N CM X C! 1— 1 T-l ^^ 0! 1.0 Tf e-i CM -^ O CO .~5 t^ ■* X Tp O ■<1< X — 1 C O X X Ci •O X Ci -^ ;£ i-o o tr ro W -^ N o ■* w o « O 'O O I- «c Ci Ct LO CO X ^ £- O t- O X O X o CI CO «.o CO c. o t- o t- O O O X o t- «n t- t- »-o N 00 o o X •<*• w >-': t- t- O X O — ' Tf — • 01 o Ci X rJ S-J ro o t~ o ■* ■* CO IM O OJ cc o CO CO CO CO o OJ CO O -^ IN ■* TJ. Tj. T). -^ -H C5 t- O CO CJ W CO en (M CO Tt> Tt< Tf X t- -- X t- i-< (M —1 >0 CO CO CO x: -o n ci rt O to r^ ?^ LO X o CO Tf b f^ o CO O CS O N N C O ^ X O Ci rH CO CI If, C5 t- t- l> CO Tf CO CO CO CO t- <£> 5D >0 ?D t~- t- ".O to X CD !0 !> CO -^ X •O Tf !> t^ J> t- »-o to >.o c c X 3C c! r? C3 05 O Tf d x' ci X X o rj — Tf o -^ r: ci X X l> ^ U ^ c t-* 1-' t- ci -»• X CO X CI C-. o c CJ CC t- -^ ro c-1 -+ LO t- O 1.0 «0 lO •a o T-l N CO I- SQ o c «r c. I- o •* 1-1 ct ■* O T-f Oi Tj- •* LO LO t:- l> t> t/) r- t- r~ t- — o --^ — (C O! Oi CO CO CO rt C> (M T! (M t^ t- Tf O CO M ^ CO Ci Ci c-. C C c: CO «D CO c c c o o o o N o c o CM CM Ol N T-H W r- ■* Tf in irt w to in in -^ n ■<*l •* -^t N M W 1— ( ^^-> 1— ( ^H 1 — ( 1 — 1 H-i HH HH 1 — r o o c XXX CM t^ ,— ( Tf X X O C5 c C 3 X X X X C5 O C-. C! r- I- t> t- X X X X CO CO CO Ci Ci Ci r^ X o w CO CM c > >. OJ u bo _o 3 \3 562 Bulletin No. 1G5 \J»iy, s s o U Id Cd W m 2 o < < Q < < C/5 03 < .M C c o «-x: o *^ o > tn u^ o O •o C ou "H rt lU C .2 is rt > ■>-' t) r CAl -o c 05 s ID >: c ti o c •- .li rt u •.— i£ S3 D. CI o > o ^ o O •a c -3 be C- n X . o n *- ■ c •"*«*-. ti o E e; 1 c •cz v 0.*J X c t1 a> in 1-1 m t- M O CO CO t- 1 00 t- O rH Q o N i-l •* 'H in CO to rH CO CO ri M CO tH (M CO C* iH rH j-i j-f rH C-: (M to w (M : ^^ ro re CO in t"- <-l •>*' •>*< •^ cc c-. C! m b- CO oa « w o CO th in N O rH ■^ in o m o o m •* M 05 «0 CO M" '-' t- c Oi to m rH rH < M W r- r- o c O fO T-l 00 m o 00 (N t- to t- to m m m O CO ^ CO o CO t~ Tf in 05 N CO T}< -f C> N n o «J •^ "^ o • u * ■■ — ' ■* M <0 00 CS 00 (M C/3 . > . -*-' Ttl t> CO rt (M (M CI IM t- GO m C-. o ■* m !< t- IM N c. o o o : o o C5 Cl CO CO CO CO Ir- CO 00 -^ •* M< ■ CO -a c CO-O c en O o a d Length of experi- ment, days t Tjf l> t- T-H 0 CO CO «D t- f^ CO M CO 00 CO CO CO t> «o .* t- O w X C-. t- cc r~ 02 c; t- 00 «o ^ CO N O O ;o ra rH o^ CO w i.o w CJ C5 Cl Ci O C2 O CO O CO *^ ^ HH I— I 1-^ »c CO en LO 05 a-. 00 a. n o c« O e JZ 564 Bulletin No. 165 [July, S s o w CO o < < P < < CO w l-l < a; (A O be o si •u.2 11 rt > CO T3 1 m s u u O +-> 'P o G t/5 O tn d Length of experi- ment, days i L) 1 l^ CO 05 05 QC CO -f CO CO r- — 1 CO lO t- oc CI Wi-HOt-t-csint-c-ic CO >0 CO CO CO CO (M en J> CJ O CO 1.0 O C>1 t^ •*00>Oi«0>-OOCO'-lt- T-tC^'— T— IT-IT-I r-lWCJ IM 1-1 Tj> CO tH Ol rH 1-1 oo5oo>niMO'-iC5 OOrioOOOOOTfOCOCO CO 0 -"l^ ■* rH CO OS CO 00 00 (M 00 CO 1-1 N CO O 00 !M 1-1 O O CO >.o W IM J3 OO'-ICOON-^eD'OlO ooooot^t-rHcooeac. CO 00 ■* CO CO CO oo 1-1 •^ W (M CO Tj. CO (M C3 CO Tf (M OV CO (M Wn t- ci i> c» >o >n QC 0 CO o o 00 00 00 00 iH 1-1 m lo N iH tH IM (M (M (M CJ rH 7-1 r-l tH o o T^ r^ rH (M CO CO O O tH rH E < ^H 1 — ( IN M W N o»o>o>n>n>o>nooo lo o m »n lo in in m )— 1 ^- t— hH 1 — 1 1 — 1 I—* 1— 1 1— ' r^ 1-^ i_ 1— 1 I— 1 l-H H^ S. I-H l-H f^ >> o o o o CO «D "o lo lo ira >-"5 o i.o m 'o >o (M (M M (M 00 00 00 00 05 Ci C> (M o o in in CO 1 CO ■4' •* Tf IJl bfl rt O. ^ o c (U > b£ >. x; ^ u c 03 t/5 J +-» J= >-. n ILI ILI ^ > E W) 3 C c lU CTJ c QJ c« .4> O OJ ^-^ ^ •a . J3 E c« D C o bo w rt (« u lU w j: > H< ^9^3] Uncekt.mnty in Interpret.xhon of Feeding Experiments 565 « u (d H O < < P < C/2 tn u (U o c a .an Is lU > U •a c 13.2 rt > c c 15.2 c5 4; 2 «n a d Lengtli of experi- ment, days - > d /< c < •!:; >n >n >n l-H l> OD W X OS X 00 1-H 00 T-) r-l r; rc ?! « 05 (M -l to X M t~ C5 X I> CC (M M N N —1 fC oc TJ. .^ en C 0/ E 1- c CJ . c, X X X >.-:: ■«■ br c •a OJ CJ U» Ci Ci ■-^ C5 r: •V rt C rt U XXX ~' ^ — ' H^ln; 7i Si >--5 w w rt r; .-^ t^ X ■*> t' «D •^ M in i-i W X « 00 M © © © CO © CO t^ © .* © © © N CO © © CO CO 1-1 CO OS rH W OJ »-i t- r-l N t- CO in —1 a Ti tn ?J ri CO © OS t- « CO rH t- N Ci t~ © (M © © CO X 10 © CO CI © © to © r- (M co X © f^ X "f .-4 Tj< LO CO l-H r-l W (M CO ©' -«•' W Ci X © >n -^ t- fc C* W 11 ■* ffl LO OJ X «0 Tj" © >« t- C5 © rl rl rH (M © CO >n CO NO©© r- w »n rH X W © Tj< rH ^— r^ TJH OS © ^ © ^ T»< -J. © X ^ 01 © © '- r^ >n 0} © LO t- .* © X in LO t~ ■* X © . LO s a X if' r^ iH © M •~' X — © •r M .* «C 46.4 61.3 88.4 67.3 © c t- CO — 10 ■* -- © CO CO CO >n 0! >0 © X lO •0 -.«• rt © — -s< X rt X t-H CJ " !M "-•^ LO Tf © © © © © © © C CQ OJ © © CO X rH r- © CO N N CO CO CI CO X X © © "So c '«• .^ .^ .^ .^ T^ T}< T»< Tj- ^ ^ .*■*•* Tf f -^ n! ^ rt j: ca rt j2^ ^H 1— « ■— < >^ 1—1 1—1 H-( 1— 1 1— 1 H- 1 © T-H X Tj- © © © X ;: ;: X © © -T-« Tf Tf t- t^ © © © © in CO 1.0 bo c > >^ Ci u « 42 E ^ c 4J n1 bO V , U rt «*- 1) u Vl-l -a c« lU c« c • w^ '5 VI taO u >> E c u bo u rt H < 566 Bulletin No. 1G5 [July. cc z o < < P <: o *4H ■^ c E C o > o o Mean Standard deviation (/I u — I O "O CO (M (M tH rH Cv! in tH .o C5 O IM C> --H c>} >n ■* o o tH ^ ^ ^ ^ ^ rt C« _Q J3 C<1 C! iM CI il Jp/J] U^XERTA1^•TY IN LNTEKPl be o ^2 •3 c 'S 2 rt > U5 -4-» . to Length of experi- ment, days I 1 d 1^ y ^ >c 10 rH TJH 00 CO N rH Ifl C >n N to rH rH C-- b- 00 05 to to l-H X >ra t- rH rt 00 00 in 00 rH T-H CO W rH r-^ 00 CQ (M tH CO (N (M CM rH rH (M CO to to rH rH rH IN CO ■* W iH 0* rH W ^ CO 00 CI (M oc ■* 00 >0 to Tf N rH 00 M C N to rH t- t- CO 00 to CO ■* 1— CO •* to to 00 M r-i CD Tji rH rH C5 00 tO 00 00 to 00 r-^ T^ y-i T-l r^ ^ TjH rH to rH rH W rH rH t- to X r- rH rH tH T- C5 L-: CO rH to ■0 ^ c w -^ •* fO C! to CC N to CO W rH l> rH to t- rH rH CO C! r- N to 1.0 CO tA. — C-. !> C N rH s to t- t- CO rt t^ 1-^ rH rH rH rn t- CR 05 ■* 0> TH C3 N N 1- P 2 Ci N rH Tj< 1.0 to C5 N CO CO CO 00 Hf i-O to CO ^ t- •* T*< 05 to 00 00 ci Hf r- to »n to to CO CO ts to CO rn OJ T- -5 rH T— 1 lO W C5 00 ocooi>£-oj a a c. CO CC! rJ C-. c rt lU < 1-H d tH t-1 X 00 r- 1-H c 'if rH C to Th to to CO CO d CO ■* t- ■* ^ ro X 00' 00 « to t- ■* CO Hj- CO CO 1-1 rH rt r- C5 C-. » rH r^ rH rH t- t- 00 to X as 00 CO 05 C5 C-. C-. 1 — I HH 1 — 1 1 — 1 HH 1— 1 1— 1 -0 1— I 1— 1 f— 1 >--» K,. 1— 1 1— I I— 1 1— ( HH 1— 1 I— 1 HH 1— 1 1— < S. Irt ^H ►-^ -0 11 •0 :r CO 00 00 X X CO t- C-. C5 Oi C5 -^ •* Ti- Tf X X X X c s c 000c rH rH rt r-. CO =r. 3 CO to to 568 Bulletin No. 165 [July. s o CO % o ■< < P 1-1 < to C/1 be '5. o _n O II il o > o •o n rt > CO-C C/5 'a o re 'c c o o > o 13.2 -g.re J- (U LO-O a (L> c/) O O en . be d Length of experi- ment, days P d ,-1 .* O C5 O C5 00 tH o w O CO to 00 CO to Q0OC0r-M50t-00CC00 i-ii-lr-iWr-l COOlT-<.^ T-l O CO t-l 1-1 00 Tf lO CO 1—1 T-l 00 w 1-1 CI Tt>fOO5C!00>O N ■* N CO OS to O Tf •* C>} CO !M O f>! fO Tt< >.0 CO CTi O M O .-^ CO r- c: 00 t~ t- o 1-1 ^ to O 00 o o 1-^ 1—1 C5 O O Tf iH T-l QOOOCOt-CCOOt-COC 00 o> o ^ en 00 (M a 00 »n O CO CM i-O to 1^ O c: •13 C! 1--5 to 00 ■* O »0 CC O •* (MOJNi-IIMr-IWi-INOO o o i> to t— 1 1—1 ■* CO 1—1 1—1 t^ CO i-i (M CO c CO TJ. t^>.-:tDC0C501MT-(OC CO C-. O (M Tj< 00 •* CO aC' oc t- 1— IM I- c p n, ro >c •'; O •* th »n th M c ci C-5 GO to 0-. 00 ,-j CO CO '- CO I- W to •5 u u C500COJ>i-iCTicO-t-hOSX iH 'i- Tf 05 ■* (M Tt- N O LO CO c i-o CO CO »-0 CO >o t- 00 CO CO CO LO CO r- o E < ooocnoooooo r-ti— IrH tHt— lr.HrHT-^T-n Tt< ■* 05 05 o c t- 00 (M IM 1— t T-l irs o 1 1 1 1 1— ( H^ HH 1 1 1 — 1 1 — 1 t—t 1 — 1 CO 00 (M (M (M M (M M (M W N C>} oc 00 o o o o to to 00 00 o I- o o o o ■o bO a o. n o c > 'bii >> re u bO _o 13.2 nS > c d i) ."2 c: c.S ^ > S.2 1.2 — ' OJ
  • n to r- ul CO -f M O M X O >-i ^ cy; o "-I i-H c-3 O O o n 00 f C T»< IM C^ W O ^ " --I 00 O t- M M IM O 00 T-l (M ^ O C; Tf « lit r~ o » w; ot oi CO ts o 00 ir: o (M r^ i-H ,- in 00 CO O O W i-H 00 W N W O! C^i «0 »n C\! to to 50 to ;^> 000c ':i:^> tfl c tc en n >> <\> c 'u u ti tc C y. ^« V ^ (\> X! tn (U bo f ) c rt I-. tl .(-f CI u-i r •o ^ c bO 570 Bulletin No. 165 [July, v. H hJ P O Pn o o •J < W .J pa < >> o o ,s bo .E _c 'rt O Si ■|| o Standard deviation c Age at beginning of experi- ment, days O X CO 1^ 6 :2; Length of experi- ment, days o W el d c E 00 N t- CO i-t ^ C-l 00 O t- t- O 00 (M CO ■* CJ (^> >n o C5 o o 00 o (M C CO 00 1-1 CO CM o> .-> CS 8> CO i-H CO CO C3 O t- (M W lO t- O tH r-l rH ro i-H Oi C\i CO l—t CM « tH t- W O 00 in O >c CO CO O Th CO in ■>^< CO ■^ CO O oc (M o CO C5 lO »o Tf in TH CO CO M CO t- CO O 00 t- iH 1-1 ^ o O IH in W 05 CO 1-1 rH O in to o o o ac CD ■>*< CO 1-1 a> 00 1-1 m •* CO CO CO CO CO CO ^ (M ■<1< (M oa ■* CO -*i CO CO CO l-H en o O O t- CO Tf r:)< t^ t- !> t- Tf Tt> O O n «5 O O !> CD OC 00 J> l^ f^ i> 00 00 05 O t- u 0) o< X br -n CJ 11 (U i_i r1 t— 1 1— 1 t— 1 > '3) >. '3, ctS u bo o 3 4= S _) n < < (U-T3 JiJ 1 U dj . - P <" 1 MOOOtO-^OSCO'OCQCJOfOMlOOMe'! (MOOOOOO>«OSOOC^l(M-*<(M(Mit>t-OCC: tfi rH T-iT-(--iT-it-4r-lr-t-«ctjici>M- •S (r: (M ^ T-^ T^ rt U5 Ph CO o o 55 .S o '*a>'HOto-*os^-*CJooroc-!>rtiroi-iT-i C.5 OOCOTfOrf 1- t-tCO t- coo C-. -*»C!-i o 11 5 rHr-IiHrHr-iNWNCMCOrOTfTfCOrfOil" rt rt •O h— i rt c o o 03 l_l CO N oi CO 05 t- N oj ■-' C n w (M 0-. r~- c^j ira c/l t~ C3 CO t- T? O t-- C-J O C OC t- t- X. t; t- O NTto«ct-c-x C^ o-;:; ^ O u (U ?^ !^ p 1^ u 2 5 ^ i- CV ^ "O O >ra 00-*0».0>-0;CC3C^ r^ THr-i i-HT-lT-li-lT-lTMrtT-li-iT-C-!?! a o J^ en C^ to j_, c s |m ^.2 C-1 (O X LO C Cs X CO W LO CO ■* I> uo m o ^ 'P •5.2 u rot-l-lT-^cxO(^!^T^I-tKO■*05l-^»clco ? Is 6^ O^XCOCO^i-IXrtO'Hi-iOiXCCC-. t/2 1—1 1 o o rHCv!r-lT-1r-ir-lr-) t— liHTHr-i r-li-^ n to 3 p •^ :: CO 1- O rt rt ■- U~t oto«5!0«OXi-HTHoeor- c r3 > O xt-uot-rooxoot-cocowwt-cowt- Or-iCJCiM-*-*COi.O;Ct-Xl>t-OOC~. t, , -^ CJ t-XOC5C: r-1 'jr. "a to .(TiVitntrttntfiir. fsi^ntritritj\rfiin^j\w. . ^^^^^J^^^^^^^^^^^^ rt ijoiLJa;oo U4>CJQJQJC^l>Q>QJQJl>0QJl>0Q-'0 _o tH C! CO ■* i.O O I- X 05 O "-1 C^ CO "^ to «D t- EEr:CECE::cc=:cc:cccc 3 c tao 574 Bulletin No. 105 [July, Ul a X -v. ^' 'n CO t- C5 C5 T-1 rt I— C\> (M !trt cccccccccccccc t™ C- f^ t^ IM M O cc c^ t^ CO «5 »o »n c- «o t- oooooicot-oooJc-jTfininoo cccccccccccccc jp-fi] Uncertainty in Interpretation of Feeding Experiments 575 a o H D i/i O U Q Is) Id tn -) < H O < c c o O U 9 bo in < < < O c >< < > O < 6 (d I-) (!) < •3 U5 "J ; , CsdOOOOCOOC -^ c c ot-^-^-t-^•^-t-t-^- o m 11 1 r^ ^ ooi-Tccxeoccic^ is u o o ^"^ Tl> — tc X C-. O C-. v- 1- T- -«rTrr}'Tr->r-T S tA «4-. - n bo t) ?- rt o U « o ■* C-. 1-"; -^ X X i-; — ' Ci .-t T- t- J- T- Ci «C C-. C! o 5'C — i> c :>! X o >.': o ■* rt 'rt u-S « bo C-. ■ -^t-xci cc — r-:-*-* • ^^ -w lU o 13 co-o o tr zz c3 OS X X ?■> ^ X C C: C"! O CO O fci;— -* « o ~ 12 o ~ 5^ X Er S "" " ' ' • ^ '- -^— - m o ^ c O 000©C©^-t-I> ^"■£. IT. -^ E o c 1> — > t/". — c u C U ,-^ .-Il.'^SSXOOO^XO 1-- t- X C Cv> W Ol c o t- c r:rCM-*^Tj<-^Tt CO *4-* ^ c O NOt-ClC. OOX— X '5 to t75 — iOOOXl-t-OOO CJTj'o-^'^oxiraoi--; "rt TfTjirOWNNt-lr-ini- O .r? p" data in wcig S.2 o 1—1 •*in«cj05Mxt-— 'C 11 M O O M ■* O CJ O O O H- ( r-:T}< < < > w u < < ^ O C-. ■* l-T 0-, Ci r-f-;-*OCCC-. 1— C!M-* -T3 en .2 I- OOMCOMMMNfOfOC*: O ooo«o«. "^ r7> 1> c CJ 5 > t" — o c (MO^Ot-LTot-t-r- 1-1 OOClOTtf^OOOOtO rt u o .''■^ CO ■* LO o »-'' o in o in «rt O U C H o U — CO •♦^ s £°-2 "o r-rcNt>oo,-iOostcc .« Cs> ir; N ,-1 C! N CQ O t> O 6.1 rt ^ oo^>cco'^J|^}oo■*>ncQ , , ,-|^lMrt nccoo»noi- "rt i/3 -a _tj tn n t)5 c5 Tj.oooooc^o>nO>-i(M 00 cc 'n o oo ■* c oc fo t- ,-iroM'!o«Occoowc<: T-i r-l r-t 1— 1 '5 bo .tnenentncncntne/)t/3 ^^-^^-^j«:js:^^^ OOI'IUS-'IUOOCJ'D 1 5iOlUo4'-< u c o 'V ^ H ^ 9 .=!■ 1 1-4 '53 I 5 O ■J < H O H O >< H OS > bi o ^; <: X u bl < oooooooocorHooooooooooor COOOCOOOOOi— 'OOOOOOOOCCCO E V 50C050?00?0(DO(D*000 OCD "^ — ii-lrHi-lr-liHr-lr-INWWC-J rtTHT-lr-li-IT-IWNWWeMC'J c/1 -^'i i~> o u (U tu ^♦H ^^ re f-4 u >-':lC>^•00OMOO— IMMM XTflO^OOt-OWO^t- — O o c>iNoina3tDi>ooowcoTj< WC0«05D0000OT-cl-*»fit^ H u c o -Hr-lrHHrHr-lr-lrtWWNW "at rtrlT-lrtrHr-iWNWevlCQW ^1 _'5) '33 U V OOOOOC5«Ot-OCSOOMrJ< u s ct--j-!j'^os>omOooO'-' o oo-5tn>r5 c Ol-';CO-!»»0 tiM«>-*T»n ooooi>c5«n"*i>«05a«stc T!r-iT-lrtrtr-(rti-(i-(r-ir-l^ ■^C^Jr-lrHr-tTHT-tr-li-ii-HT-iT-i c o "^^ bo be OS bo "d c u o £ CL- rt-_n C'J C5r-5t-O'- 50-*rHTjHrJ>0000'-l'-l-*0 a y rt " CC0000030 0i--^^i-i«0 C^i— t-Ot-f^COOmCiOOO o r-OONM>-'^ (MCOCOOOt-OO^lM'^CCl^ V-H tH 1— 1 ^H T-l 1— 1 , 1 M-t t/i-v t— 1 o o H-1 o J -o ^^ c nowuoooi^ooWNiNr-c OOMOOCOMMMt'Iint-OOO (A ooi/j-*!>0'-^«-';-*^c:tc-* oxooctocxt-t-ccow !OMrtO>'-l;OtC■'-^^'-^00 iOi-HOCi^t-xtcectocvx T-lWW-^^Ottlt-OOOC 1-1 C^ IM ■* ^ O «C t- X C-. O c/} ( M c « t 10 c 3 C D .t/)trtc/)tr)c/)c/)t/5c/5crtt/:c/; • rntfitfivitr. tnViVirr. ifi^n ^^ji:^^^^j«i^^^^ ^^^^^^^^^.^J^^^ C>D'Ua-'OOCJ(U4;fU4> OWOOCJOQJQJCJQJOU 4;CJ4>l>4>CQ,J4>OOCJflJ ^&?^^?^&^^^^ ^^^iS^^iS^^^^^ rH(MM-*mOt-00050'-W rH(MM-*>.'5!Ot>X05C'-OT 1— T— T— f-~ T— CCCECCCCCC = = CECCCCCCCCCC 578 Bulletin No. 165 [July, Bibliography 1. Craig, S. J. and Melvin, L. R. III. Agr. Exp. Sta. Bachelor Thesis. 1906. 2. Craig, J. A. Wis. Agr. Exp. Sta. Bui. 32. 1892. 3. Coffey, W. C. 111. Agr. Exp. Sta. Bachelor Thesis. 1906. 4. Wilson, J. W. So. Dak. Agr. Exp. Bui. 119. 1910. 5. Wilson, J. W. Ibid. 6. Coffey, W. C. 111. Agr. Exp. Sta. Master Thesis. 1909. 7. Chilcott, E. C. and Thornber, W T. So. Dak. Agr. Exp. Sta. Bui. 71. 1901. 8. Carlyle, W. L. Wis. Agr. Exp. Sta. 16th Annual Report. 1899. 9. Carlyle, W. L. Wis. Agr. Exp. St^. 15th Annual Report. 1898. 10. Craig, J. A. W'is. Agr. Exp. Sta. Bui. 32. 1892. 11. Wilson, J. W. and Skinner, H. G. So. Dak. Agr. Exp. Sta. Bui. 86. 1904. 12. Wilson, J. W. and Skinner, H. G. So. Dak. Agr. Exp. Sta. Bui. 80. 1903. 13. Mumford, H. W. Mich. Agr. Exp. Sta. Bui. 136. 1896. 14. Arkell, T. R. N. H. Agr, Exp. Sta. Bui. 152. 1911. 15. Arkell, T. R. Ibid. 16. Grindley, H. S., Coffey, W. C, and Emmett, A. D. 111. Agr. Exp. Sta. Unpublished manuscript. 1912. 17. Voelcker, J. A. Journ. Roy. Agr. Soc. of Engl. Vol. 63. 1902. From Woburn Station. 18. Voelcker, J. A. Journ. Roy. Agr. Soc. of Engl. Vol. 62. 1901. From Woburn Station. 19. Voelcker, J. A. Journ. Roy, Agr, Soc. of Engl, Vol. 60. 1899. From Woburn Station. 20. Voelcker, J. A, Ibid. 21. Voelcker, J. A. Journ. Roy. Agr. Soc. of Engl. Vol. 59. 1898. From Woburn Station. 22. Voelcker, J. A. Journ. Roy. Agr. Soc. of Engl. Vol. 57. 1896. From Woburn Station. 23. Voelcker, J. A. Journ. Roy. Agr. Soc. of Engl. Vol. 53. 1892. From Woburn Station. 24. Voelcker, J. A. Journ. Roy, Agr. Soc. of Engl. Vol. 56. 1895. Woburn Station. 25. Lawes, J. B. Journ. Roy. Agr. Soc. of Engl. Vol. 10. Rothamsted Station. 26. Lawes, J. B. Ibid. 27. Lawes, J. B. Journ. Roy. Agr. Soc. of Engl. Vol. 12. Rothamsted Station. 28. Lawes, J. B. Journ. Roy. Agr. Soc. of Engl. Vol. 13. 1852, From Rothamsted Station. 29. Lawes, J. B. Journ. Roy. Agr. Soc. of Engl. Vol. 16. 1855. From Rothamsted Station. 30. Shaw, T. Minn. Agr, Exp. Sta, Bui, 76. 1902. 31. Shaw, T. Ibid. 32. Watson, G. C. and Risser, A. K. Penn. Agr. Exp. Sta. Bui. 57. 1901. 33. Carmichael, B. E. Ohio Agr. Exp. Sta. Bui. 193. 1908. 34. Sheppard, J. H. and Richards, W. B. No. Dak. Agr. Exp. Sta. Bui. 73. 1906. 35. Mairs, T. I. and Risser, A. K. Penn. Agr. Exp. Sta. Bui. 64. 36. Mairs, T. I. and Risser, A. K. Penn. Agr. Exp. Sta. Bui. 68. 37. Mairs, T. I. and Miller, N. G. Penn. Agr. Exp. Sta. Bui. 74. 1905. 38. Mairs, T. I. Penn. Agr, Exp. Sta. Bui 83. 1907. 39. Mairs, T. I. Ibid. 40. Mairs, T. I. Ibid. 41. Mairs, T. I. and Tomhave, W. H. Penn. Agr. Exp. Sta. Bui. 88. 1908. 42. Mairs, T. I. and Tomhave, W. H. Ibid. 43. Mumford, H. W. 111. Agr. Exp. Sta. Bui. 103. 1905. 44. Georgeson, C. C, Burtis, F. C, and Shelton, Wm, Kan. Agr. Exp. Sta. Bui. 34, 1895, From 1849. From 1851. From 1903. 1904. jp/j] Uncertainty in Interpretation of Feeding Experiments 579 45. Georgeson, C. C, Burtis, F. C, and Otis, D. H. Kan. Agr. Exp. Sta. Bui. 39. 1893. 46. Georgeson, C. C, Burtis, F. C, and Otis, D. H. Kan. Agr. Exp. Sta, Bui. 47. 1894. 47. Georgeson, C. C, Burtis, F. C, and Otis, D. H. Kan. Agr. Exp. Sta. Bui. 51. 1895. 48. Robertson, R. Canadian Experimental Farms. Report for 1901. 360-363. 49. Voelcker, J, A. Journ. Roy. Agr. Soc. of Engl. Vol, 63. 1902. From Woburn Station, 50. Voelcker, J. A, Journ. Roy. Agr, Soc. of Engl. Vol, 62, 1901. From Woburn Station, 51. Voelcker, J. A. Journ. Roy, Agr, Soc. of Engl, Vol, 60, 1899. From Woburn Station, '52. Voelcker, J. A, Journ. Roy. Agr. Soc. of Engl. Vol. 59. 1898. From Woburn Station, 53. Voelcker, J, A, Ibid. 54. Voelcker, J. A, Journ. Roy. Agr. Soc. of Engl. Vol. 56. 1895. From Woburn Station. 55. Voelcker, J, A. Journ. Roy. Agr. Soc, of Engl, Vol. 57. 1896, From Woburn Station. 56. Voelcker, J. A. Journ. Roy, Agr, Soc. of Engl, Vol. 53. 1892, From Woburn Station. 57. Voelcker, J. A. Ibid, 58. Carlyle, W. L. Wis. Agr. Exp. Sta. 15th Annual Report. 1898. 59. Henry, W. A. Wis. Agr. Exp. Sta. 16th Annual Report, 1899. 60. Carlyle, W, L, Ibid. 61. Michael, L. G. and Kennedy, W, J, Iowa Agr. Exp. Sta. Bui. 113. 1910. 62. Wing, H, H, N. Y. (Cornell) Agr. Exp. Sta. Bui. 220. 1904. 63. Henry, W, A. Wis. Agr. Exp. Sta. 15th Annual Report. 1898. 64. Kennedy, W. J. and Robbins, E. T. Iowa Agr. Exp. Sta. Bui. 91. 1907. 65. Kennedy, W, J. and Robbins, E. T, Ibid. 66. Henry, W, A, Wis. Agr. Exp. Sta. 17th Annual Report. 1900. 67. Henry, W, A, Wis. Agr. Exp. Sta. 15th Annual Report. 1898. 68. Henry, W, A, Ibid. 69. Carlyle, W. L, and Hopkins, A. G. Wis, Agr, Exp. Sta. 17th Annual Report. 1900. 70. Henry, W, A, and Otis D. H, Wis. Agr. Exp. Sta. Bui. 145. 1907. 71. Henry, W. A. and Otis, D. H. Ibid. 72. Henry, W. A. and Otis, D. H. Ibid. 73. Sheppard, J. H. and Richards, W. B. N. Dak. Agr, Exp, Sta. Bui. 84. 1909. 74. Georgeson, C, C, Burtis, F. C, and Otis, D, H, Kan. Agr. Exp. Sta. Bui, 47. 1894. 75. Grisdale, J. H, Canadian Experimental Farms. Report for 1899: 61. 76. Robertson, R. Canadian Experimental Farms. Report for 1900: 309-310, 77. Robertson, R, Canadian Experimental Farms. Report for 1899: 255-256. 78. Shutt, F. T. Canadian Experimental Farms. Report for 1902: 219-222. 79. Shutt, F. T. Canadian Experimental Farms. Report for 1902: 222-223. 80. Shutt, F. T. Canadian Experimental Farms. Report for 1902: 226-227. 81. Shutt, F. T. Canadian Experimental Farms. Report for 1902: 226-227. 82. Shutt, F. T, Canadian Experimental Farms. Report for 1902: 228-230, 83. Gilbert, A. G, Canadian Experimental Farms. Report for 1905: 256. 84. Shutt, F. T. Canadian Experimental Farms. Report for 1898: 214, i J I UNIVERSITY OF ILLINOIS-URBANA Q 630.7IL6B BULLETIN UBBANA 1651913 C008 iiiiiiiniini 3 0112 019529764