UNIVERSITY OF 
 
 AGMCUJLTURr 
 
NQN CIRCULATING 
 
 CHECK FOR UNBOUND 
 CIRCULATING COPY 
 
UNIVERSITY OF ILLINOIS 
 
 Agricultural Experiment Station 
 
 BULLETIN No. 148 
 
 ON THE MEASUREMENT OF CORRELATION 
 
 WITH 
 
 SPECIAL REFERENCE TO SOME CHARACTERS 
 OF INDIAN CORN 
 
 BY HENRY L,. RIETZ AND LOUIE H. SMITH 
 
 URBANA, ILLINOIS, NOVEMBER, 1910 
 
CONTENTS OF BULLETIN NO. 148 
 
 PAGE. 
 
 Introduction 291 
 
 The Correlation Table 292 
 
 Nature of the correlation coefficient r 294 
 
 Details of the computation of the correlation coefficient 296 
 
 Modification of the method of computing r 301 
 
 Probable error 301 
 
 Use of the correlation coefficient 301 
 
 The regression coefficient 302 
 
 Use of the regression coefficient 302 
 
 Determination of the correlation coefficients for certain physical characters 
 
 in corn 303 
 
 Source of material 304 
 
 Discussion of results 306 
 
 Two year rotation corn 306 
 
 Illinois corn 307 
 
 Appendix on the Mathematical Theory of Correlation 303 
 
 Mathematical definition of correlation '. 309 
 
 Standard deviation of arrays 311 
 
 Correlation surfaces 312 
 
 Derivation of the shorter formula for numerical calculation of r 313 
 
OX THE MEASUREMENT OF CORRELATION WITH 
 SPECIAL REFERENCE TO SOME CHAR- 
 ACTERS OF INDIAN CORN 
 
 BY HENRY L. RIETZ, Statistican, and LOUIE H. SMITH, Assistant 
 Chief, Plant Breeding 
 
 INTRODUCTION 
 
 In Bulletin 119 of this station, there are presented methods of 
 dealing with problems involving variability of a single character 
 and these methods are there applied to the study of type and 
 variability of some characters in corn. 
 
 But the breeder deals with many characters in the same organ- 
 ism, each with its own variability. After treating separate char- 
 acters, what he needs next to know is whether, and to what extent, 
 any bond may exist between characters by virtue of which, if one 
 character varies, other characters of the same organism tend also 
 to move in the same or in opposite directions. 
 
 If such a bond exists the characters are said to be correlated 
 (co-related), and it is the purpose of this bulletin to describe 
 methods by which such a correlation may be detected if present 
 and the strength of its bond be measured. 
 
 A second purpose of the bulletin is to present data concerning 
 certain definite correlations for corn bred at the Illinois Station. 
 
 The great value to the breeder of definite knowledge of corre- 
 lations within a species is that it gives reliable information, enab- 
 ling him to predict from the presence of certain characters the 
 most probable values of associated characters. 
 
 More technically speaking, when we are dealing with two sys- 
 tems of variable characters in correspondence, we are, in general, 
 much concerned about whether fluctuations of variates in one sys- 
 tem are in sympathy with fluctuations of corresponding variates 
 in the other, and with establishing causal relations between the 
 two series of phenomena. For example, in breeding corn for com- 
 mercial purposes, we are much interested in knowing what charac- 
 ters of the seed ears should be modified or selected to increase 
 the yield ; and, if we should select directly one character, it is im- 
 portant to know to what extent other characters are being se- 
 lected indirectly, because of the tendency of the two to fluctuate in 
 the same or in opposite directions. 
 
 291 
 
292 BULLETIN No. 148. [November, 
 
 These examples illustrate the following technical definition of 
 correlation; Two characters say length and circumference of 
 ears of corn are said to be correlated when with any selected 
 values (.r) of the one character, we find that values of the other 
 character, a given amount above and below the mean of that char- 
 acter, are not equally likely to be associated. 
 
 As the first and simplest method, it may possibly occur that 
 correlation is so pronounced that it may be necessary merely to 
 look at two sets of figures to note that corresponding values have 
 a tendency to change simultaneously in the same or in opposite 
 directions. The existence of such decided correlation may be 
 known by inspection.. 
 
 As a second, and somewhat more effective method, one may plot 
 curves for each of two systems of variates, and if correlation is 
 very pronounced, it may sometimes be discovered by noting 
 whether the curves have a tendency to rise and fall together, or 
 if, when one rises, the other falls. 
 
 Not only do these methods prove inadequate to detect correla- 
 tion unless it is exceedingly pronounced, but they lack precision in 
 that they do not give a measure of correlation. It is not enough 
 to know whether correlation exists, its quantitative measure is 
 usually a matter of importance. 
 
 Our power to measure the correlations among associated phe- 
 nomena has been enormously increased during the past two dec- 
 ades by methods introduced by Galton and developed by Pearson 
 and those associated with him. An application of these methods 
 has not until very recently been made to problems in agriculture.* 
 
 It is the purpose of this Bulletin to present in a form useful to 
 agricultural students the methods of correlation measurement with- 
 out presuming more mathematics than is absolutely necessary, and 
 to give the results of our investigations into the correlation of 
 certain characters in corn bred at the Illinois Station. 
 
 I. The Correlation Table. The first step in the process of 
 measuring correlation is to construct a double, entry table (Fig. i) 
 called a "correlation table" out of the measurements of the 
 characters in a large number of individuals. One mark at the 
 intersection of the proper column and row in the table records a 
 pair of corresponding variates with reference to two characters. 
 
 Put in tabular form as it appears in the actual work, we have 
 the following (Fig. i) for the correlation table between the num- 
 
 *Davenport, Principles of Breeding, pp. 452-472, 703-711. 
 Pearl and Surface, Bulletin 166, Maine Agr. Exp. Sta. 
 Clark, Bulletin 2~<), Cornell University Agr. Exp. Sta. 
 
SOME CHARACTERS OF INDIAN CORN. 
 
 293 
 
 Correlation of Circumference and Rows. 
 Number of R ows. Crop 190 7- Plot 401 
 10 12 14- 16 Id 20 P_Z ^4. 
 
 $ 
 
 500 
 
 5.50 
 5.75 
 6.00 
 
 :6.50 
 675 
 
 125 
 
 7.50 
 7.75 
 6.00 
 025 
 
 
 
 it 
 
 1 
 
 
 
 
 
 i 
 
 
 i 
 
 
 1 
 
 1 
 
 
 
 
 in 
 
 
 II 
 
 M 
 
 1 
 
 
 
 
 
 MM 
 
 IN mini 
 
 mil 
 
 1 
 
 
 
 
 nan 
 
 N (,' ;Y'/ ft!! V/ 
 
 III 
 
 till IHI IN IHI 
 /WIW//II 
 
 IH! [Hi IHI IHI 
 
 Illl 
 
 III 
 
 
 
 
 i 
 
 IHllHJIHIHtl 
 
 II 
 
 MINI IHI IHI 
 1 
 
 Ml IN! IV! IHI 
 Illl 
 
 HI 
 
 
 
 
 i 
 
 IHUHIIHIN 
 Ml 
 
 IN IN IHI IN 
 IN/MM 
 NTH! IN IHI 
 IHI III 
 
 IWtHIIWIHJ 
 
 ai mi IHI mi 
 III 
 
 IHI IHI mil 
 
 in 
 
 
 
 i 
 
 IHI III 
 
 nu mi IHI mi 
 
 HI IN IHI INI 
 
 11 
 
 mi IHI mini 
 
 KNK'tl 
 
 IH1IH1 
 
 mi mi nan 
 
 mi 
 
 II 
 
 
 
 IHI 
 
 IN IHI IN M 
 
 mi 111 
 
 IHIIWfHlN 
 IHItHlltillN 
 
 IHIIWIHJIHI 
 IHI III 
 
 tftllHIIHl 
 
 11 
 
 
 i 
 
 
 IN Illl 
 
 IHSIHIMIN 
 
 mi 
 
 IHI mi nu IHI 
 'it 
 
 H 
 
 U 
 
 
 
 
 III 
 
 IN Mil . 
 
 its nu in 
 
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 it 
 
 
 
 
 
 Ill 
 
 in 
 
 
 
 
 
 
 
 II 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 no. /. 
 
294 BULLETIN No. 148. [November, 
 
 her of rows of kernels on ears and the circumference of the ears 
 for a certain plat of corn (Plot 401) grown in 1907 at the Illinois 
 Experiment Station. 
 
 It will be observed that this table consists of a double system 
 of arrays each of which is a frequency distribution as explained 
 in Bui. 119, and has its own mean and standard deviation as has 
 any other frequency distribution. 
 
 To show, in a concrete way, how such a table is made, suppose 
 an ear of corn has 18 rows of kernels and a circumference be- 
 tween 6.375 an d 6.625 in., a mark is made in the rectangle at the 
 intersection of the column headed 18 and the row of the table 
 marked 6.50. A second table (Fig. 2) exhibits the result of 
 counting the marks in each of the rectangles of Fig. i. 
 
 Any number in this table, say 43, in the column headed 18 
 and the row marked 6.50, indicates that 43 ears of the total of 
 769 ears had 18 rows of kernels and a circumference of class 
 mark 6.50. 
 
 By adding the numbers in horizontal arrays, we obtain the 
 frequency distribution of the population with respect to circum- 
 ference of ears, and by adding the numbers in columns, we ob- 
 tain the frequency distribution of the population with respect to 
 rows of kernels on ears (See Fig. 3). 
 
 The mere superficial inspection of a correlation table may sug- 
 gest that a certain amount of correlation exists. For example, 
 ears of corn of circumference 7.50 inches, from this population, 
 are much more likely to have 20 rows of kernels than are ears of 
 circumference 6 inches. It is pretty clear that there is a tendency, 
 in general, for the marks in the table of Fig. i to arrange them- 
 selves in a region along the diagonal from the upper left hand cor- 
 ner to the lower right hand corner of the table. This signifies that 
 a positive correlation exists; that is, in general, for this popula- 
 tion, ears that have a large number of rows of kernels are more 
 likely to be large in circumference than are ears with a smaller 
 number of rows of kernels. But it is not our purpose merely to 
 detect the existence of correlation. What we seek is a statistical 
 coefficient that will serve to measure correlation, and that will en- 
 able us to predict with as high a degree of probability as possible, 
 from an assigned character, the value of the associated character 
 in the related system of variates. The coefficient of correlation, de- 
 noted by r in this paper, is useful for this purpose. 
 
 2. Nature of the coefficient r. A discussion of the mathemat- 
 ical theory of correlation will be given in the Appendix to this 
 Bulletin, but the general character and common sense significance 
 of r may well be stated here. The value of the coefficient is within 
 the limits i and +i. If r=i, there is said to be perfect positive 
 
SOME CHARACTERS OF INDIAN CORN, 
 
 295 
 
 Correlation of Circumference and Rows. 
 Number of Rows. Crop IQ07-Pht40L 
 10 12 14- 16 16 20 22 
 
 CO 
 
 00 
 
 -C 
 o 
 C 
 
 c: 
 ^> 
 
 i. 
 
 6 
 
 5.00 
 5.25 
 5.50 
 5.75 
 6.00 
 6.25 
 6.50 
 6.75 
 7.00 
 7.25 
 7.50 
 7.75 
 6.00 
 
 
 
 2 
 
 1 
 
 
 
 
 
 1 
 
 
 / 
 
 
 1 
 
 i 
 
 
 
 
 3 
 
 6 
 
 2 
 
 5 
 
 i 
 
 
 
 
 a 
 
 15 
 
 13 
 
 6 
 
 / 
 
 
 
 
 7 
 
 23 
 
 34 
 
 24 
 
 3 
 
 
 
 
 / 
 
 22 
 
 4! 
 
 24 
 
 7 
 
 1 
 
 1 
 
 
 / 
 
 26 
 
 66 
 
 43 
 
 16 
 
 3 
 
 
 
 / 
 
 d 
 
 42 
 
 50 
 
 Id 
 
 5 
 
 a 
 
 
 
 5 
 
 2d 
 
 40 
 
 26 
 
 15 
 
 'a 
 
 
 / 
 
 
 9 
 
 25 
 
 23 
 
 5 
 
 a 
 
 
 
 
 3 
 
 12 
 
 13 
 
 6 
 
 a 
 
 
 
 
 
 3 
 
 3 
 
 4 
 
 i 
 
 
 
 1 
 
 
 a 
 
 1 
 
 
 
 
 
 
 
 
 1 
 
 
 
 FIG. a. 
 
2 96 BULLETIN No. 148. [November, 
 
 correlation ; that is, for any assigned value of the character in one 
 system, the value for each corresponding individual of the related 
 system is known, and the ratio of the deviations of any two vari- 
 ates of a pair from their mean values is a constant for all pairs. 
 In other words, perfect correlation (r=i) indicates complete cau- 
 sation in the sense that the two characters go together perfectly. 
 If r= i, there is said to be perfect negative correlation. In this 
 case, the ratio of the deviations from mean values are negative and 
 constant for all pairs. If no correlation exists, the two characters 
 appear indifferent to each other, and this fact is expressed by r=o. 
 In a general way, we may say that the correlation should be judged, 
 in any application, by the value that r takes between i and +i. 
 For our applications to characters in corn, there is usually a posi- 
 tive correlation, and the amount of correlation is measured by the 
 value of r between o and i. 
 
 The correlation coefficient may be defined as the mean product 
 of deviations of corresponding variates from their mean values in 
 units of the standard deiiations. 
 
 The meaning of the standard deviation of a frequency distri- 
 bution is shown in Bulletin 119 of this Station. If a variate is 
 below the mean, its deviation is negative; while if it is above the 
 mean, it is positive. Hence, if each individual of a pair of variates 
 is above the mean of the system to which it belongs, or if each of 
 the pair is below, the pair tends to contribute to positive correla- 
 tion. On the other hand, if one variate of a pair is below the mean 
 of its system and the other above, the product is negative, and 
 such a pair tends to contribute to negative correlation. While it 
 appears from this that the coefficient of correlation, as defined 
 above, has a common sense justification, we shall require the math- 
 ematical methods of the Appendix to see more fully how this co- 
 efficient with the standard deviations of the two systems of vari- 
 ates are descriptive of the correlated population exhibited on a 
 correlation table such as is shown in Fig. i. 
 
 3. Details of the computation of r. In algebraic form 
 
 r = 
 
 - - - - (1) 
 
 where Sxy means the sum of the products of the deviations of 
 corresponding variates from their mean values ; and <r x , o y are 
 standard deviations, while n is the number of pairs of variates. 
 
 As we shall use the correlation table of Fig. 2 to illustrate a 
 systematic arrangement of the work in the computation of r, the 
 formula (i) may appear more significant in the application by 
 writing it in the form 
 
/p/o.] SOME CHARACTERS OF INDIAN CORN. 297 
 
 D D 
 
 where the subscripts c and R refer to circumference and rows of 
 kernels respectively, while the D's represent deviations of charac- 
 ters indicated by subscripts. That is to say, D c is the deviation 
 of the circumference of an ear from the mean circumference, and 
 D R the deviation of the number of rows of kernels from. the mean 
 of the number of rows of kernels. 
 
 There is derived in the Appendix, pp. 313-314, a formula which 
 gives the same numerical value as 
 
 ^^ *"~"~^ 
 
 *^ XV *^ D D 
 
 _ or c R 
 
 and, while its algebraic expression is a little more complicated, it 
 is much better adapted to numerical computation than the above 
 formula, as it avoids the use of decimals until almost the end of 
 the work. In this respect, it is analogous to the shorter method 
 presented in Bulletin 119 for finding the mean and the standard 
 deviation. If applied to the case of the number of rows of kernels 
 and circumference of ears of corn, the formula is 
 
 _C R C C ) 
 
 ..--(3) 
 
 where D K ', D c ' are deviations from our guesses at the means instead 
 of deviations from the means themselves ; and C B , C c are the cor- 
 rections applied to the guesses at the mean number of rows of 
 kernels and circumference respectively in finding the means and 
 standard deviations. 
 
 In the actual work of calculating r from formula (3), it is 
 highly important to have a systematic form in which to arrange 
 the work, in order to avoid confusion in the somewhat compli- 
 cated details. It seems desirable, for this teason, to describe the 
 arrangement of the actual work as shown in Fig. 3. 
 
 Having given the correlation table, we first add the numbers 
 in the arrays with respect to both characters ; that is, add numbers 
 in rows and columns of the table. This gives two frequency dis- 
 tributions the one with respect to circumference exhibited in the 
 vertical column headed / c , and the other with respect to rows of 
 kernels shown in the horizontal column of figures marked / B . For 
 each of these frequency distributions, the means and standard devi- 
 ations are calculated by the shorter method explained and applied 
 in Bulletin 119. 
 
298 
 
 BULLETIN No. 148. 
 
 [November, 
 
 Correbtbn of Cirtumfertrxt and Rows 
 Number of Rowi of Kerne/j. 
 10 12 14 16 16 20 22 24 
 
 Crop /907-P/of 401. 
 
 5.00 
 525 
 5.50 
 575 
 ~ 6-00 
 
 ; V5 
 
 630 
 E 6.75 
 
 3 
 
 7-00 
 o 
 
 225 
 7.50 
 7.75 
 6.00 
 &25 
 
 
 
 2 
 
 / 
 
 
 
 
 
 3 
 
 4 
 17 
 57 
 
 91 
 97 
 
 157 
 
 lid 
 65 
 3d 
 II 
 4 
 1 
 
 1 
 
 
 1 
 
 
 / 
 
 I 
 
 
 
 
 3 
 
 6 
 
 2 
 
 5 
 
 1 
 
 
 
 
 2 
 
 15 
 
 15 
 
 6 
 
 1 
 
 
 
 
 7 
 
 23 
 
 54 
 
 24 
 
 3 
 
 
 
 
 / 
 
 22 
 
 41 
 
 24 
 
 7 
 
 / 
 
 1 
 
 
 / 
 
 26 
 
 66 
 
 43 
 
 16 
 
 3 
 
 
 
 / 
 
 6 
 
 42 
 
 50 
 
 16 
 
 5 
 
 2 
 
 
 
 5 
 
 26 
 
 40 
 
 26 
 
 15 
 
 2 
 
 
 / 
 
 
 9 
 
 25 
 
 23 
 
 5 
 
 2 
 
 
 
 
 3 
 
 12 
 
 13 
 
 a 
 
 2 
 
 
 
 
 
 3 
 
 3 
 
 4 
 
 1 
 
 
 
 / 
 
 
 2 
 
 / 
 
 
 
 
 
 
 
 
 / 
 
 
 
 r -2|5^S^S 
 
 769 
 
 ^y 
 
 f 
 
 DC 
 -150 
 
 -1.25 
 -1.00 
 -.75 
 -.50 
 -.25 
 
 25 
 .50 
 75 
 IM 
 1.25 
 1.50 
 1.75 
 
 *o 
 
 -500 
 -17.00 
 -27.75 
 -45.50 
 -2425 
 
 6.75 
 625 
 /7.00 
 20AI25 
 22.75 
 6.0625 
 
 7.675 
 29.50 
 36.5625 
 3&00 
 17.1675 
 9.00 
 
 3.0625 
 
 769J22Q.&I25 
 , 02&7I 
 
 (1 = 02777 
 C =Q527 
 
 06333 - 01 
 
 15.00 
 12.50 
 44-00 
 72.00 
 96.00 
 3&.00 
 
 -1350 
 
 <?6.00 
 40.50 
 64.00 
 3500 
 -3.00 
 3J50 
 
 3 
 
 4 
 
 17 
 
 37 
 
 91 
 
 97 
 
 157 
 
 -124.00 
 
 3150 
 5900 
 4475 
 3300 
 13.75 
 6.00 
 1.75 
 
 126 
 
 lid 
 
 65 
 
 36 
 
 II 
 
 4 
 
 1 
 
 769 
 
 "OJvO 
 lf)fO 
 
 ^qlT\ 
 
 1 1 
 
 196.75 
 769J74.75 
 
 t 
 
 -(450 44&J50 
 16.50 
 
 05616 
 C^ -0.07 15 
 0.6333 
 
 01 0.016. 
 
 {o5~c 
 
 >70)(<?.39t) 
 
 M c =6597 
 c =0-5^7 
 
 M= 17264 
 
 tfvi 
 
 = 6.597, 
 
 C = 0.527, 
 
 The results are 
 
 <T R = 2.396, 
 
 where M c , M B represent mean circumference and mean number of 
 rows of kernels respectively, while <r c , o- R are corresponding stand- 
 ard deviations. 
 
 The columns of figures marked f c , D c ', / C D C ', / C D C ' 2 , f^, D B ' r 
 / R D K ' , / B D B ' 2 are all self explanatory to one familiar with the 
 meaning of algebraic symbols, and who knows how to find the 
 mean and standard deviation. 
 
SOME CHARACTERS OF INDIAN CORN. 
 
 299 
 
 There remains the column of figures headed ^ D' B D c ', which 
 we shall endeavor to explain in detail, as this is the only part of 
 the computation that is actually new to one who knows how to 
 calculate the variability of a population. In finding the means, \ve 
 get the deviations of class marks from our guesses at mean cir- 
 cumference and mean number of rows of kernels on ears of corn. 
 These deviations are marked D c ', D R '. For example, in row i, 
 we find 3 ears of circumference 5 inches. These three ears devi- 
 ate 1.50 from our guesses at the mean circumference. 
 
 We next form the product of each number of the correlation 
 table and of the two corresponding deviations. For example, 
 where the column is headed 16 and the row is labelled 6.00 inches 
 intersect, occurs the number 34. These 34 ears have deviations 
 from our guesses of 0.50 and 2 as is indicated by the sym- 
 bols D c ' and D '. Hence, for this number 34, we form the product 
 34( o.5o)( 2) =+34. Without regard to labor, AVC should find 
 such a product for each compartment of the correlation table. The 
 sum of these products, with due regard to signs, is the ^ D R 'D C ' of 
 formula (3). The systematic way to carry out this work is to 
 record the results of this operation for each horizontal array in 
 line with the array under the heading ^D B 'D C ', and then to add the 
 results for separate arrays to obtain 432.00 \vhich is symbolically 
 ndicated by ^ D^D C '. 
 
 To illustrate the method of calculation, let us take the array 
 of circumference 6.00 inches as an example. We have, for this 
 array, 
 
 -6X7 
 
 -4X23 
 
 2X34 X( 0.50) 
 
 4 0X24 
 
 2X3 
 This gives 98.00, 
 
 For the array of mark 6.75 
 
 6X1 
 -4X8 
 > 
 
 X(0.2 5 ) 
 
 2X42 
 0X50 
 
 2X18 
 
 4X5 
 6X2 
 
 This gives I3-5O- 
 
300 
 
 BULLETIN No. 148. 
 
 [November, 
 
 Treat all arrays in this manner, and divide the sum of the 
 products thus obtained (that is, 432.00) by the number of variates 
 
 769. This gives 
 
 
 of formula (6) and equals 0.5618. 
 
 Xext, we subtract from this the product of our two corrections 
 in finding means. That is, C B C C = 0.071 5. 
 To subtract this negative number, we must add 0.0715. This 
 
 gives 0.6333 f r tne numerical value of c R - C C C R 
 
 r = 
 
 Cerre/of'on of Circumference and_Rows. 
 Number of Rowa of Kermis'. 
 
 O 
 
 = 0.501. 
 
 Crop 1907- Plot 401. 
 
 
 3 
 4 
 /7 
 37 
 9/ 
 97 
 157 
 126 
 IIQ 
 65 
 38 
 
 4 
 
 -6 
 -5 
 -4 
 -3 
 -2 
 -1 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 
 1 
 
 -20 
 -66 
 
 -in 
 
 97 
 
 (06 
 100 
 272 
 333 
 
 - 564 
 97 
 
 126 
 472 
 565 
 608 
 275 
 144 
 49 
 
 30 
 5 
 
 (44 
 
 196 
 76 
 
 -27 
 52 
 61 
 I2& 
 70 
 -6 
 7 
 
 
 
 
 
 / 
 
 
 -496 
 126 
 
 152 
 
 55 
 
 7 
 795 
 769299 . 
 
 2 
 
 2 
 
 2 
 
 2 
 
 i 
 
 
 
 o 
 
 = 
 
 *|i 
 
 76 & 
 o * 
 
 \j oo 
 Vj Oj 
 
 ?* 
 
 7693535 
 , 4.594 
 
 C=ai5i 
 
 a' ? IDA 
 
 -33 697 
 33 
 
 1.1235 
 -0.1450 
 
 
 C, /.<?665 
 
 K r /.<?&65 
 
 0^0/ 0.0/6. 
 
 Fig. 4 
 
SOME CHARACTERS OF INDIAN CORN. 301 
 
 4. Modification of the method of computing r (Fig. 4). The 
 
 computation of r may, in general, be further simplified by tak- 
 ing the difference between two successive class marks as the unit 
 of measurement. That is, the units of grouping are the units 
 throughout. Then, as shown in Fig. 4, the deviations from the 
 guesses are consecutive integers. This method gives precisely the 
 same value for the correlation coefficient as that explained under 
 Fig. 3. But the standard deviations and the corrections to guesses 
 at the means are expressed in terms of differences between consecu- 
 tive classes as units. When thus expressed, we note on Fig. 4, that 
 
 o- c '=2.io8, 
 
 0-^=1.198. 
 
 Where <r c ' is the standard deviation in circumference (expressed in 
 units equal to the difference between classes) and o- R ' is the stand- 
 ard deviation in rows of kernels (similarly expressed). To ex- 
 press <r c ' in inches, we must multiply by 0.25. This gives <r c =0.527 
 as before. 
 
 Similarly, to make <r R consistent with o- R of figure 3, we must 
 multiply by 2. This gives 
 
 5. Probable error. It still remains to find the probable error 
 in our computed value. The general meaning of the probable er- 
 ror, and its use in indicating the degree of confidence to be placed 
 in a result obtained from a random sample of a population has 
 been given in Bulletin 119. It seems sufficient here to give merely 
 the formula for the probable error in the coefficient of correla- 
 tion r. This formula is 
 
 0.6745 0-r') 
 
 0.6745 [ 1 - (0.501)' 1 
 
 Applied to our example, k r = 
 
 0.018 
 
 Hence, we write 
 
 r=o.5Oio.oi8 
 as the measure of the correlation in question. 
 
 6. Use of the correlation coefficient The general use of such 
 a precise measure of correlation as is given by r, has perhaps been 
 sufficiently discussed in the introduction. However, it seems well 
 to emphasize here that in the selection of one character, we, in 
 general, indirectly select correlated characters, and change their 
 means and standard deviations accordingly. Again, if in 
 
302 BULLETIN No. 148. [November, 
 
 the selection of what the breeder conceives to be the most 
 desirable type for parents, he artificially increases or de- 
 creases correlations between characters, this process will, in gen- 
 eral, change the correlation between these characters in pa- 
 rents and in offspring. For example, it appears that the corre- 
 lation between length of ears and circumference for different 
 types of corn which we have examined varies between the limits 
 0.128 and 0.623. Now, we have examined cases in connection with 
 this work where 48 parents are selected so as to exhibit a negative 
 correlation of 0.21 between length and circumference. If we 
 use the offspring of such a set of 48 ears and ask to what extent 
 length and circumference are inherited, or if we ask to what ex- 
 tent these characters in the parent are correlated with the yield, it 
 is pretty clear that we have much complicated our problem by the 
 selection and imposition of the correlation 0.21. Hence, it is 
 highly desirable to know what correlations actually exist among 
 different characters, before we should expect to obtain in even an 
 approximately precise way the correlation between parent and off- 
 spring (inheritance) or the correlation between characters in the 
 parent and yield. 
 
 7. The regression coefficient. From the correlation coefficient 
 and the standard deviations of each of the two characters, it is easy 
 to obtain what is known as the regression coefficient. For ex- 
 ample, to obtain the regression coefficient of circumference rela- 
 tive to the number of rows of kernels, multiply the coefficient of 
 correlation by the standard deviation in circumference and divide 
 the product by the standard deviation in the number of rows of 
 kernels. This gives, for the particular example above, 
 
 r =0.110. 
 
 Similarly, the regression of the number of rows of kernels rel- 
 ative to the circumference of ears: is 
 
 r = 1.200. 
 
 c 
 
 8. Use of the regression coefficient. In many systems of cor- 
 related variates, the regression is of a kind described in the ap- 
 pendix as linear regression. In such cases, the regression coeffi- 
 cient gives us a useful method of predicting, from a given value 
 of one character, the most probable value of the corresponding cor- 
 related character. That is to say, from the selected value of one 
 character, we calculate the mean value of the corresponding array. 
 
SOME CHARACTERS OF INDIAN CORN. 303 
 
 For the particular case in hand, suppose we select ears with twenty 
 rows of kernels, such ears deviate 2.736 above the mean number 
 of kernels on ears, the regression coefficient (o. no) of circumfer- 
 ence on number of rows of kernels indicates that we should expect 
 the mean circumference of ears of 20 rows of kernels to be 
 (o. no) (2. 736) =0.301 inches above the mean circumference for 
 the entire population. That is, ears with 20 rows should have a 
 mean circumference 
 
 6-597+0.301=6.898. 
 
 By actual computation, the mean of the array of class mark 
 20 rows is 6.927 0.12 1, so that the regression coefficient gives 
 the mean of the array to within deviations due to random sampling. 
 
 To be more general, if we select an ear of any deviation x in 
 the number of rows of kernels from the mean, we should expect 
 the deviations in circumference of corresponding ears to center 
 about o.i i ox. 
 
 Similarly, if we select an ear of any deviation y in circumfer- 
 ence from the mean circumference, we should expect the deviation 
 in number of rows of kernels to center about 
 
 i. 20 y. 
 
 In beginning the discussion of the use of the regression coeffi- 
 cient, we limited our remarks to linear regression. This means 
 that if the correlation table is constructed to scale, and the mean 
 values of arrays be plotted, these mean values will lie along a 
 straight line to within deviations to be attributed to random samp- 
 ling. Fortunately, this condition is, in general, w;ell satisfied in 
 our applications. 
 
 It is important in every case to examine the correlation table 
 to ascertain whether the means of systems of parallel arrays lie 
 reasonably near a straight line. 
 
 9. Determination of the correlation coefficients for cer- 
 tain physical characters in corn. Corn grown on experi- 
 mental plots of the Illinois Station in 1907, 1908, 1909, furnishes 
 the material for the present study of the correlation between phys- 
 ical characters in corn, and for the problem of quantitative laws of 
 inheritance of these characters. It should be understood that the 
 problem of inheritance is a problem of the correlations between 
 ancestry and offspring. The characters with which we propose to 
 deal here are : length, weight, circumference of ears, and the num- 
 ber of rows of kernels on ears. 
 
 While we have done considerable work on the inheritance of 
 these characters and hope to publish these results, it appears better 
 to present in the present bulletin the correlations between charac- 
 ters as we find them in large populations, with very little reference 
 to heredity. We do this because, with our material, as is very 
 
304 BULLETIN No. 148. [November, 
 
 general in the quantitative study of inheritance in plants, the ques- 
 tion of the precision of the results is complicated by the fewness 
 of parents relative to the number of offspring, as well as by the 
 rather stringent selection of parents. We think it expedient to 
 defer to a later bulletin the treatment of these difficulties. Further, 
 it is important to know these correlations before attempting a study 
 of inheritance or of the correlation between characters in parent 
 ears and yield. 
 
 In the comparison of two statistical results, the difference be- 
 tween the two results compared to its probable error is of great 
 value. In general, we may take the probable error in a difference 
 to be the square root of the sum of the squares of the probable 
 errors of the two results. If the difference does not exceed two or 
 three times the probable error thus obtained, the difference may 
 reasonably be attributed to random sampling. If the difference be- 
 tween the two results is as much as 5 to 10 times the probable error, 
 the probabilty of such differences in random sampling is so small 
 that we are justified in saying that the difference is significant. In 
 fact, a difference of ten times its probable error is certainly signifi- 
 cant in so far as there is certainty in human affairs. 
 
 Such significant differences in our applications may perhaps be 
 well divided into three classes : 
 
 1 i ) Those due to differences in variety of corn. 
 
 (2) Those due to seasonal influences. 
 
 (3) Those due to difference in soil treatment. 
 
 (4) Those to be attributed to selection of parents. 
 
 10. Source of material. The material for this study is fur- 
 nished by the crops obtained from a number of the regular experi- 
 ment plots which are being conducted for different purposes. These 
 plots may be considered as belonging to two different groups, one 
 of which is devoted primarily to soil investigation and the other 
 to experiments in corn breeding. 
 
 The soil plots comprise what are designated as the 400 and 500 
 series. They are devoted to a two-year rotation consisting of corn 
 alternating with oats, that is to say, corn occupies the 400 series 
 one year and the 500 series the next year. 
 
 Each series is divided into ten plots of one-tenth acre number- 
 ed from 401 to 410 and from 501 to 510. To these plots various 
 soil treatments have been applied as follows : 
 
 401 and 501 None (check plot). 
 
 402 " 502 Legume catch-crop and crop residues.* 
 
 *Corn stalks and oat straw plowed under, removing only the grain from 
 the land. 
 
SOME CHARACTERS OF INDIAN CORN. 305 
 
 403 and 503 Farm manure. 
 
 404 504 Legume and crop residues and lime. 
 45 55 Manure and lime. 
 
 406 506 Legume and crop residues, lime and phosphorus. 
 
 47 507 Manure, lime and phosphorus. 
 
 408 508 Legume and crop residues, lime, phosphorus and 
 
 potassium. 
 
 409 509 Manure, lime, phosphorus and potassium. 
 
 410 510 Legume and crop residues with extra heavy ma- 
 
 nure and phosphorus.* 
 
 The yields from these variously treated plots are given in the 
 following tables in connection with the other data. 
 
 This brief description will serve to explain in a general way 
 the significance of the various plots and the following data per- 
 taining to them. The reader who may be interested in a more de- 
 tailed account regarding the arrangement, description and his- 
 tory of these soil plots is referred to Bulletin 125 of this Station, 
 "Thirty Years of Crop Rotations on the Common Prairie Soils of 
 Illinois," where are given the complete records. 
 
 The particular variety of corn grown upon these plots has been 
 two strains of Learning which have been under selection for a 
 number of years for high-protein and low-protein content respect- 
 ively, the work being controlled by the method of "mechanical se- 
 lection" described in Bulletins 55-87-100. 
 
 To be more definite, in 1907, 1909, seed corn low in protein 
 content was planted, while in 1908, seed corn high in protein was 
 planted on the 400 and 500 series concerned in this investigation. 
 
 Although a material difference in composition between the two 
 strains has been effected through this method of selection, this 
 difference does not seem to have significantly affected the correla- 
 tion values tinder consideration, as will appear from the results of 
 this bulletin. 
 
 The corn breeding plots from which samples have been taken 
 for these correlation studies represent four lines of selection which 
 have been under way since 1896, the object in view being to change 
 the normal composition of the grain of a variety of corn by pro- 
 ducing strains of special chemical characteristics. 
 
 In this manner four strains of markedly different chemical 
 composition have arisen from a single variety by selecting contin- 
 uously for 
 
 *Five times the ordinary application of manure and of phosphorus. 
 
306 BULLETIN No. 148. [November, 
 
 I High-protein content 
 2 Low-protein content 
 3 High-oil content 
 4 Low-oil content. 
 
 For the history and the results of the first ten generations of 
 this work the reader is referred to Bulletin 128, "Ten Generations 
 of Corn Breeding." The variety under experiment contained orig- 
 inally in 1896 an average of 10.92 percent of protein and 4.70 
 percent of oil. 
 
 The composition of the different strains for the years herein 
 concerned is as follows : 
 
 High Protein. Low Protein. Hjigh Oil. Low Oil. 
 
 1907 J 3-89 7-32 7-43 2.59 
 
 1908 13.94 8.96 7.19 2.39 
 
 1909 I3-4I 7-65 . 6.96 2.35 
 
 II. Discussion of results. With a set of the four characters 
 
 under consideration, there are possible six pairs of variates between 
 each pair of w r hich we can determine the correlation. From the 
 results of the tables, it will be observed that we have carried out 
 the determination of the correlation coefficient for some plots for 
 each of these six pairs of characters. As the correlations coeffi- 
 cients we have determined and given in this paper seem sufficient 
 to give a good general notion as to the value of correlations be- 
 tween these characters, it has appeared as well to defer further cal- 
 culations of correlation coefficients until we ascertain whether fur- 
 ther special determinations will be of service in problems of in- 
 heritance of these characters and their correlations with yield, 
 the problems to which we regard the present investigation as pre- 
 liminary. The accompanying tables include the results of 141 de- 
 terminations of correlation. 
 
 Two year rotation corn. In length and circumference, the 
 correlation centers about 0.33 for the year 1907, 0.47 for 1908, and 
 0.49 for 1909. For the three years together, the values center 
 about 0.43. The smallest correlation is given by data from plot 
 409 of 1907. This correlation is 0.203. The greatest correlation 
 is 0.623 furnished by data of plot 402 in 1909. These extreme 
 values are very different as seen by comparison with their prob- 
 able errors. They belong to different years, and the value 0.623 
 corresponds to a decidedly low yield while 0.203 corresponds to a 
 high yield. In fact, there seems to be a somewhat general ten- 
 dency towards high correlation of length and circumference when 
 the yield is low and vice versa. There are three small values of 
 correlation between length and circumference. These belong to 
 
jp/o.] SOME CHARACTERS OF INDIAN CORN. 307 
 
 three consecutive plots of 1907. Inspection of the correlation table 
 shows that, in these cases, there is considerable deviation from 
 linear regression the ears of extremely large circumference 
 tended to be shorter than ears of less extreme circumference. 
 
 In length and number of rows, the correlations are insignifi- 
 cant except possibly in one case where r is more than four times 
 its probable error. 
 
 In circumference and rows, the correlation centers about 0.486 
 in the year 1907, 0.499 m 1908, 0.467 in 1909. The extremes pre- 
 sented are 0.425 and 0.608, which do not differ so much as the 
 extremes for length and circumference. While the deviations are 
 too great to be assigned to random sampling, it appears that there 
 is both a difference in season, soil, and parentage. 
 
 In length and weight of ears, the mean value of the correla- 
 tion is 0.810 in 1909, and these correlations did not show very 
 great differences for different plots. We may regard 0.8 as sort 
 of rough value for this correlation. 
 
 In weight and rows of kernels, we have values from 0.178 to 
 0.345. In weight and circumference, we have values from 0.648 
 to 0.840. 
 
 The Illinois Corn. In length and circumference, the correla- 
 tion are very different for selected strains. The conditions are 
 complicated by differences of soil, and season. The low oil plot of 
 1907 gives the lowest correlation, whereas the low oil plot of 1908 
 gives the highest correlation of the entire series. 
 
 In correlation between circumferences and rows of kernels, 
 there are much smaller differences between plots than for length 
 and circumference. 
 
 Between length and rows of kernels, there appears to be no 
 significant correlation, except possibly in one case. 
 
 Between length and weight, the correlations are not very dif- 
 ferent, and fall roughly between, 0.65 and 0.85. 
 
 Arranging the pairs of systems of variates in descending order 
 as to correlation, we have the following order : 
 
 (1) Length and weight. 
 
 (2) Circumference and weight. 
 
 (3) Circumference and rows of kernels. 
 
 (4) Length and circumference. 
 
 (5) Weight and rows of kernels. 
 
 (6) Length and rows of kernels. 
 
 For this arrangement the odds are pretty large except in the 
 case of (3) and (4), and possibly (i) and (2). As a sort of gen- 
 eral conclusion, we may say that the correlations between length- 
 weight and circumference-weight are high. The correlations of 
 
308 
 
 BULLETIN No. 148. 
 
 [November, 
 
 circumference-rows of kernels, and length-circumference are con- 
 siderable. The correlation of weight-rows of kernels is low, while 
 that of length-rows is probably insignificant. 
 
 These correlations mean that the tendency towards a relative 
 proportioning of length and circumference; and circumference 
 and rows of kernels is considerably greater than that towards a 
 relative proportioning of weight and rows of kernels or length 
 and rows of kernels. 
 
 It seems somewhat disappointing that the correlation coeffi- 
 cients differ so widely, as this fact complicates the problem of as- 
 sessing the influence of the selection of parents in a precise meas- 
 ure of heredity. The wide difference for different pairs of char- 
 acters may be compared with the correlations between different 
 pairs of characters of the human body, where it has been found 
 that between measurement of the long bones of the arms and legs 
 a high correlation exists, while between different measurement of 
 the skull a much lower correlation exists.* 
 
 TABLE 1. CORRELATION AMONG CERTAIN CHARACTERS OF EARS OF CORN 
 400 SERIES. CROP 1907. SEED: Low PROTEIN BY MECHANICAL SELECTION 
 
 
 Yield 
 
 t 
 
 Value of r for 
 length and circum- 
 ference 
 
 Value of r for 
 length and rows 
 
 Value of r for 
 circumference and 
 rows 
 
 401 
 
 68.4 
 
 423+0.019 
 
 0.044+0.024 
 
 0.501 0.019 
 
 402 
 
 69.9 
 
 0.438+0.019 
 
 +0.007+0.024 
 
 0.446+0.020 
 
 403 
 
 69.4 
 
 0.312+0.020 
 
 
 0.484+0.018 
 
 404 
 
 75.9 
 
 0.462+0.018 
 
 
 0.548+0.017 
 
 405 
 
 66.6 
 
 0.452+0.018 
 
 
 0.5020.019 
 
 406 
 
 84.6 
 
 0.403+0.019 
 
 
 0.470+0.018 
 
 407 
 
 68.6 
 
 0.282+0.021 
 
 
 0.440+0.020 
 
 408 
 
 84.1 
 
 0.278+0.021 
 
 
 0.554+0.016 
 
 409 
 
 71.4 
 
 0.203+0.021 
 
 
 0.487+0.019 
 
 410 
 
 95.6 
 
 0.411+0.017 
 
 
 0.432+0.018 
 
 
 
 Value of r for 
 length and weight 
 
 Value of r for 
 rows and weight 
 
 Value of r for 
 weight and circum- 
 ference 
 
 401 
 
 
 0.781+0.008 
 
 0.275+0.023 
 
 0.768+0.009 
 
 405 
 
 
 0.786+0.009 
 
 0.223+0.024 
 
 0.721+0.011 
 
 *Biometrika, Vol. i, pp. 408-467. 
 fComputed at 80 Ib. per bushel of ear corn. 
 
SOME CHARACTERS OF INDIAN CORN. 
 
 309 
 
 TABLE 2. CORRELATION AMONG CERTAIN CHARACTERS OF EARS OF CORN 
 500 SERIES. CROP 1908. SEED: HIGH PROTEIN BY MECHANICAL SELECTION 
 
 
 Yield 
 
 * 
 
 Value of r for 
 length and circum- 
 ference 
 
 Value of r for 
 length and rows 
 
 Value of r for 
 circumference and 
 rows 
 
 501 
 
 37.5 
 
 0.590-1-0.014 
 
 0.090+0.025 
 
 0.514+0.019 
 
 502 
 
 45.0 
 
 0.528-1-0.015 
 
 0.0610.026 
 
 0.506+0.019 
 
 503 
 
 33.1 
 
 0.562+0.015 
 
 0.120+0.027 
 
 0.432+0.022 
 
 504 
 
 39.3 
 
 0.526-1-0.016 
 
 
 0.486+0.021 
 
 505 
 
 39.6 
 
 0.519-t-0.016 
 
 
 0.6080.017 
 
 506 
 
 76.1 
 
 0.422+0.015 
 
 
 0.480+0.016 
 
 507 
 
 46.1 
 
 0.385 -(-0.018 
 
 
 0.444+0.020 
 
 508 
 
 74.1 
 
 0.360+0.016 
 
 
 0.517+0.016 
 
 509 
 
 39.8 
 
 0.4440.017 
 
 
 0.5080.019 
 
 510 
 
 75.0 
 
 0.344+0.017 
 
 
 0.497+0.015 
 
 
 
 Value of r for 
 length and weight 
 
 Value of r for 
 rows and weights 
 
 Value of r for 
 weight and circum- 
 ference 
 
 501 
 
 
 0.855+0.006 
 
 0.345+0.021 
 
 0.771+0.009 
 
 505 
 
 
 0.871+0.005 
 
 0.348+0.021 
 
 0.763+0.009 
 
 Computed at 80 Ib. per bushel of ear corn. 
 
 TABLE 3. CORRELATION AMONG CERTAIN CHARACTERS OF EARS OF CORN 
 40 SERIES. CROP 1909. SEED: Low PROTEIN BY MECHANICAL SELECTION 
 
 
 Yield 
 
 * 
 
 Value of r for 
 length and circum- 
 ference 
 
 Value of r for 
 length and rows 
 
 Value of r for 
 circumferences and 
 rows 
 
 401 
 
 46.2 
 
 0.548+0.016 
 
 0.004+0.027 
 
 0.452+0.021 
 
 402 
 
 38.6 
 
 0.623+0.016 
 
 0.027+0.034 
 
 0.466+0.026 
 
 403 
 
 48.4 
 
 0.453+0.118 
 
 -0.044+0.026 
 
 0.514+0.022 
 
 404 
 
 43.6 
 
 0.534+0.016 
 
 
 0.463+0.022 
 
 405 
 
 47.2 
 
 0.461+0.018 
 
 
 0.487+0.022 
 
 406 
 
 57.6 
 
 0.506+0.017 
 
 
 0.482+0.019 
 
 407 
 
 46.0 
 
 0.443+0.019 
 
 
 0.524+0.020 
 
 408 
 
 58.8 
 
 0.432+0.018 
 
 
 0.428+0.021 
 
 409 
 
 51.6 
 
 0.409+0.019 
 
 
 0.458+0.022 
 
 410 
 
 72.6 
 
 0.539+0.012 
 
 
 0.425+0.018 
 
31Q 
 
 BULLETIN No. 148. 
 
 [November, 
 
 TABLE 3. Concluded. 
 
 
 
 Value of r for 
 length and weight 
 
 Value of r for 
 weight and rows 
 
 Value of r for 
 weight and circum- 
 ference 
 
 401 
 
 
 0.818-1-0.008 
 
 0.216+0.025 
 
 0.840+0.007 
 
 402 
 
 
 0.844-1-0.008 
 
 0.225+0.032 
 
 0.746+0.012 
 
 403 
 
 
 0.815+0.008 
 
 0.178+0.027 
 
 0.648+0.013 
 
 404 
 
 
 0.801-1-0.010 
 
 0.212+0.029 
 
 0.757+0.012 
 
 405 
 
 
 0.810+0.008 
 
 0.229+0.027 
 
 0.728+0.011 
 
 406 
 
 
 0.791+0.008 
 
 
 
 407 
 
 
 0.800+0.008 
 
 
 
 408 
 
 
 0.798+0 008 
 
 
 
 409 
 
 ' , 
 
 0.785+0.008 
 
 
 
 410 
 
 
 0.843+0.005 
 
 
 
 * Computed at 50 Ib. per bushel. Shelled corn (dry substance). 
 
 TABLE 4. CORRELATION AMON^ CERTAIN CHARACTERS OF EARS OF CORN 
 ILLINOIS COK>T. CROPS 1907, 1908, 1909 
 
 
 Value of r for 
 length and 
 circumference 
 
 Value of r fo 
 circumference 
 and rows 
 
 Value of r for 
 leiio*^ 1 an d 
 rows 
 
 Value of r for 
 length and 
 weight 
 
 Crop 1907 
 
 High protein . 
 Low proein . . 
 High oil 
 
 0.202 0.031 
 0.368+0.028 
 0.317+0.029 
 128+0.035 
 
 0.4900.026 
 0.520+0.025 
 0.431+0.027 
 0.5620.024 
 
 -0.051+0.032 
 0.0170.034 
 
 0.822+0.013 
 0.725+0.016 
 0.838+0.009 
 0.7270 017 
 
 Crop 1908 
 
 High protein . 
 Low protein . . 
 High oil 
 
 0.310+0.027 
 0.293+0 027 
 132+0.027 
 
 0.435+0.025 
 0.488+0.024 
 0.464+0 022 
 
 0.106+0.030 
 0.017+0.031 
 
 0.776+0.012 
 0.809+0.012 
 673+0 015 
 
 Low oil 
 
 Crop 1909 
 
 High protein . 
 Low protein . . 
 High oil 
 
 0.569+0.020 
 
 0.183+0.030 
 0.437+0.022 
 299+0.025 
 
 0.459+0.025 
 
 0.592+0.021 
 0.305+0.029 
 0.335+0 026 
 
 0.081+0.035 
 
 0.868+0.007 
 
 0.681+0.016 
 0.770+0.011 
 769+0 013 
 
 Low oil . 
 
 0.255+0.024 
 
 0.480+0.020 
 
 
 0.675+0.014 
 
APPENDIX ON THE MATHEMATICAL THEORY OF 
 CORRELATION. 
 
 1. Mathematical function. A variable y is said to be a 
 mathematical function of a variable x if they are so related that 
 to assigned values of x there correspond definite values of y. 
 Thus, if y=2x+4, y is a function of x; since, for any assigned 
 value of x, we can compute y. Those who are familiar w r ith ana- 
 lytic geometry know that a curve is useful for representing and 
 following the variations of a mathematical function. 
 
 \Ye shall assume, in the present treatment of correlation, a 
 knowledge* of the use of a system of co-ordinate axes to represent 
 numbers and functions. 
 
 In order to place the notion of correlation on a precise basis, 
 we lay down the following special 
 
 2. Definition. t TWo measurable characters of an individual 
 or of related individuals are said to be correlated if to a selected 
 scries of sizes of the one there correspond sizes of the other zvhose 
 mean values are functions of the selected values. The word "sizes" 
 is used in the sense of numerical measure, and the function is to 
 be different from zero for some of the selected values. 
 
 To be concrete, we may think, for example, of measuring the 
 correlation between length and circumference of ears of corn, or 
 the correlation of fathers and sons with respect to stature. 
 
 To render the above definition in symbolic language and to 
 develop the methods of determining the function mentioned in the 
 definition are the first points in the application of mathematics to 
 the theory of correlation. For this purpose, let x and y be variables 
 such that y=f(x) gives the mean value of a system of variates 
 which correspond to a selected x. Suppose the following system 
 of corresponding values results from measurement: (x', y'), 
 (x", y"), . . . , (x (n) , y (n) ), where n is a large number indicat- 
 ing the total number of pairs observed. These observations are 
 said to form a total population or universe of observations. As it 
 is more convenient to deal with the deviations of the observations 
 
 *See Davenport's Principles of Breeding, pp. 687, 689. 
 
 ^Philosophical Transactions of the Royal Society, Vol. i8~A, pp. 256-257. 
 
 .'11 
 
312 
 
 BULLETIN No. 148. 
 
 [November, 
 
 from their mean values than with the observations themselves, let 
 ( x i> yi)> ( X 2> V 2)> > ( x n> y n ) represent the deviations of the 
 observations from their mean values. These deviations may be 
 conveniently represented with respect to co-ordinate axes (Fig. 5). 
 The origin then represents the mean of the two characters. In 
 fact, we may think of the co-ordinate axes as passing through the 
 mean of the table and drawn parallel to arrays. The vertical par- 
 allel lines of the figure may then be looked upon as separating the 
 observations into arrays. The values of the y's which correspond 
 to a given class mark x are said to form a y-array. Suppose there 
 are s such arrays. 
 
 x-*- 
 
 Y 
 
 FIG. 5. 
 
 Let the crosses ( X ) in Fig. 5 represent the means of the y's in 
 each of the ^-arrays. If correlation exists, these means do not lie 
 at random over the field, but arrange themselves more or less in 
 the form of a smooth curve called the "curve of regression." This 
 curve is a crude picture of the function which defines the correla- 
 tion of the ^-character relative to the .^-character. Experience has 
 shown that, in many sets of measurements, this line is approxi- 
 mately a straight line. For this reason, and for simplicity, the line 
 subjected to the condition that the sum of the squares of the devia- 
 tions (measured parallel to the ^-axis and weighted with number 
 of points in array) of the means from it shall be a minimum, is 
 called the "line of regression." When the means lie exactly on 
 the line, the regression is said to oe "truly linear." 
 
/p/o.] SOME CHARACTERS OF INDIAN CORN. 313 
 
 Let y=mx+b be the function which represents the line of re- 
 g>ression, then the problem of determining the line is that of de- 
 termining m and b by means of the above minimal condition. The 
 algebraic details of subjecting a line to this minimal condition are 
 well known to those familiar with the method of least squares or 
 the method of moments. The equation of the resulting line is 
 
 y = r^Lx, (1) 
 
 ff x 
 
 where o- x is the standard deviation of the population with respect 
 to the x-character, cr y is the standard deviation with respect to 
 the y-character, and r is the correlation coefficient given by 
 
 S xy 
 r 
 
 where the summation is extended to every pair of corresponding 
 variates of the population. Similarly, the regression of the x char- 
 acter on the y character is given by 
 
 x = r_!l y (2) 
 
 y 
 
 It should be noted that (2) cannot be obtained by solving (i) 
 for x, for the reason that the correspondence is one between se- 
 lected values and means. 
 
 3. Standard deviation of arrays. Suppose that regression 
 is truly linear, so that the means of the y-arrays fall on the line 
 
 a ; 
 
 y = r_il x j ; and, for the present, assume that the standard devi- 
 ations of arrays are equal. Then the standard deviation of an 
 array is given by 
 
 n 
 where the summation extends to the entire population. 
 
 = r + r 
 
 T * 2 r z a 
 y y 
 
314 BULLETIN No. 148. [November, 
 
 Hence, the standard deviation of a y-array is obtained from the 
 standard deviation o- y by multiplying v y by i/i-r 2 
 
 If the standard deviations of parallel arrays are unequal, then 
 <Ty i/ i_ r * is simply sort of an average value for the standard 
 deviation of an array. 
 
 Since the first member of (3) is a sum of squares divided by 
 n, the second member must be positive. Hence i<r^i. 
 
 This proves that the correlation coefficient takes values not 
 greater than +i nor less than i. 
 
 Equation (3) shows further that if r=H-i all the individual 
 points plotted from observations must lie on the line of regression, 
 and we can in this case, when one character is given, tell exactly 
 the magnitude of the associated character. Further, the ratio 
 
 x t x, x 
 
 = __= - - - = _ = a positive constant. 
 
 yi y 2 y a 
 
 Similarly, if r= i, the individual points plotted all lie on 
 the line of regression, but 
 
 X l X 2 x n 
 
 = = ... = = a negative constant. 
 
 y\ ?2 y a 
 
 4. Correlations among three or more'characters. The theory 
 of correlation can be extended to apply to any number of 
 variables. However, the complexity of the algebraic expression 
 for any number, say n-variables becomes so great that it does 
 not seem well to present a more extended discussion here, except 
 to say that the final result is expressed in standard deviations and 
 correlation between systems of variates in sets of two, so that the 
 problem is capable of reduction to the one which we have solved. 
 
 For the general case, the reader with considerable mathematical 
 training is refered to the treatment by Karl Pearson in the Philo- 
 sophical Transactions of the Royal Society, A, 187, 1896, and A, 
 200, 1903. 
 
 5. Correlation surfaces. If our frequency distributions 
 follow normal probability curves (Bulletin 119, pp. 30-31) there 
 can be derived a surface 
 
 z = /(x, y) 
 
 such that / (x, y,) h. k gives, to within deviations due to random 
 sampling, the number of the population with corresponding meas- 
 urements in the region bounded by x=x, y=y, x=x+h, y=y+k, 
 where the x and y are deviations from mean values and h and k 
 are any small numbers. For a considerable range of statistical 
 data, this surface takes the form 
 
SOME CHARACTERS OF INDIAN CORN. 315 
 
 1 .x 2 y' x y 
 
 Z= 2(l-r 3 
 
 2 / /-^ x 
 
 *" x y I/ l_r 2 
 
 ~1 
 
 where e is the base of natural logarithms and the other symbols 
 have been defined above. The symbol TT equals 3.1416, the ratio of 
 circumference to a diameter in the circle. 
 
 As the only parameters in this surface (aside from the total 
 number n) are the standard deviations and the correlation coeffi- 
 cient, we have in the standard deviations and its correlation coeffi- 
 cient a perfect description of a normally distributed population. 
 This fact adds much to the significance of r as a measure of cor- 
 relation. 
 
 6. Formula for the correlation coefficient r which are better 
 adapted to numerical calculation. In the first place, the calcu- 
 lation of the means and standard deviation of both systems of 
 variates should be done by the shorter method presented on pp. 
 9-1 1, Bulletin 119. 
 
 It may be well to give that method here in a more symbolic 
 form to prepare the way for the modified formula for r adapted 
 to calculation. 
 
 Let G represent a guess at the mean M given by 
 
 /.V.+A'- 2 +- + /. T. , _ 
 
 /, + u + --+/, 
 
 where the class marks and /'s are corresponding frequencies. Also 
 let c be the correction to the guess G which gives M. That is 
 
 M = G + c , 
 
 -G. 
 
 /, ( V, _ G ) +/ 2 (Y, - G) H ---- + /. (v. - G ) 
 
 /,+/,+ - - - - "T7T 
 
 Formula (2) gives the practical method of finding the cor- 
 rection to be applied to the guess to get the mean. 
 Next, the standard deviation is given by 
 
 / t ( v t - M )* + / 2 ( v 2 - M )' + - - - + / a ( v s - M )' , 
 
 
 ( v _ G - c )' + /-, ( v 2 - G - c )' + - + /, < v - G - c )* , 
 
316 BULLETIN No. 148. [November, 
 
 S/ t (v t -G)'-2c2/ t ( v t ^G) + c'2/ t , 
 2 /t 
 
 S / t ( y t _ G )' - 2 c S/ t ( v t - M + c ) + c S/ t , 
 
 /t(Vt-Q) a t 
 
 ~~i7 ~ " * ' 
 
 Formula (3) gives the practical method of calculating the 
 standard deviation. 
 
 The value of r is given by 
 
 2 x y 
 
 r : > 
 
 n (T G 
 
 x y 
 
 where x and y represent deviations from the means, and the sum- 
 mation extends to every pair of corresponding variates. 
 
 Let G x and G y represent class marks near the means of the sys- 
 tems of variates indicated by subscripts, and C x , C y corrections to 
 these class marks which give the correct mean values so that 
 
 M y = G y +c y . 
 
 Let x', y' be deviations from G x and G y which correspond to 
 deviations x, y from the mean. Then 
 
 '- C x 2y'+ 2 C x C y , 
 
 C y 
 
 2x'y' 
 
 C C 
 
 /xy 
 
 = ( - - 
 V n 
 
 This is a formula whose computation is shown on pp. 297-301. 
 

 UNIVERSITY OF ILLINOIS 
 
 Agricultural Experiment Station 
 
 BULLETIN No. 148 ABSTRACT 
 
 ON THE MEASUREMENT OF CORRELATION 
 
 WITH SPECIAL REFERENCE TO 
 SOME CHARACTERS OF INDIAN CORN 
 
 BY HENRY L. RIETZ AND LOUIE H. SMITH 
 
 URBANA, ILLINOIS, NOVEMBER, 1910 
 
The bulletin is a technical presentation of the methods of de- 
 termining the correlation coefficient for associated characters, to- 
 gether with considerable tabular matter giving the correlations 
 for various strains of Indian corn. It is the purpose of this 
 abstract to present the leading thought of the bulletin devoid of 
 technical terms, and, omitting all reference to methods of cal- 
 culation, to discuss briefly the meaning of the correlation coeffi- 
 cient, present the data involved, and assist the non-mathematical 
 reader to an understanding of its significance. Anyone desir- 
 ing to pursue the subject farther, particularly as to methods of 
 calculation of the correlation coefficient, can secure the complete 
 text upon request to the Agricultural Experiment Station. 
 
 Any one, who has at all considered the matter, is conscious 
 that there is correlation of some characters, both in animals and 
 plants. That is to say, that the different characteristics that go 
 to make up the individual animal or plant do not ex'st indepen- 
 dently of one another, but on the contrary are more or less co- 
 related, or bound together by such physiological bonds as com- 
 pel them to move more or less with reference to each other. 
 
 Among those who have not studied the matter carefully, but 
 rely merely upon personal impressions derived from unsystematic 
 observation, it appears that a notion prevails pretty commonly 
 that characters are either perfectly correlated or entirely uncor- 
 related, the conception being that the characters are either ab- 
 solutely bound together or else they move with complete inde- 
 pendence. The truth is, however, that characters are seldom per- 
 fectly correlated, just as they are seldom independent. The math- 
 ematician follows accurate methods in determining precisely what 
 correlations exist between characters in large populations. He 
 has no method of determining the bond between two or more 
 characters from a single or even a few individuals. He deals 
 only w 7 ith large numbers, and by his methods, he is able to dis- 
 tinguish very clearly whether, in general, two characters tend to 
 move together or in opposition to each other, and approximately 
 to what extent. If they move together, correlation is said to be 
 positive. If they move in opposite directions, the one tending to 
 increase proportionately as the other decreases, the correlation is 
 said to be negative. 
 
 The present bulletin treats the precise methods of measuring 
 this correlation. It is measured by a single number called the 
 correlation coefficient, denoted by r, which may take values from 
 -i to i, depending on how fluctuations in the two characters take 
 place. If, in general, the characters fluctuate together, say either 
 above or below the type, the value of (r) lies between o and i 
 depending upon how closely the characters are correlated. If, in 
 
general, two corresponding characters fluctuate in opposite direc- 
 tions, the correlation is between o and -i. The values r=+i, 
 and -i, indicate respectively perfect positive correlation, and per- 
 fect negative correlation, while indifference of the characters to 
 each others fluctuations leads to the value r=o when very large 
 numbers are used. 
 
 The general reader is not concerned with the methods by 
 which these values are obtained. He is concerned only with the 
 results, which are significant and extremely valuable, and which 
 with a little practice become easily apprehended by the non-mathe- 
 matical reader. But a single further word of introduction is nec- 
 essary, and that has reference to the so-called probable error, a 
 decimal always following the correlation coefficient, and preceded 
 by the + or sign. This probable error has no reference to mis- 
 takes which might be made in computation. It has reference to 
 the fact that any value which may be determined would probably 
 have been different if a larger number of individuals had been 
 involved. For example, if it is desired to ascertain what is the 
 weight of mature draft horses of a given breed, it could be ob- 
 tained approximately by weighing 100 such horses. It could be 
 ascertained with greater accuracy by weighing 1000 such horses, 
 but there is no absolutely accurate w r ay of determining the actual 
 weight until every draft horse of that age in the world has been 
 weighed. The so called probable error of a result is a number 
 that enables us to set limits within which we may reasonably ex- 
 pect the result to be found if we should use larger numbers in 
 establishing a result. For a more complete statement of the mean- 
 ing and applicability of the probable error, see Bulletin 119 of 
 this station. 
 
 The bulletin treats the correlations among four characters of 
 ears of corn length, circumference, weight, and number of rows 
 of kernels. The practical bearing of such information, as is con- 
 tained in the results, lies in the facts, ( I ) that in the selection of 
 parents for one character, we should know how this tends to change 
 other characters; (2) that the problem of the correlation of char- 
 acters and yield requires, for its solution, in case of a selected 
 parentage, a knowledge of the correlation of the characters among 
 themselves in the general population from which parents are se- 
 lected; (3) that the problems of inheritance of these characters 
 requires a knowledge of these correlations. 
 
 The following tables give the correlation coefficients and prob- 
 able error for a large number of determinations that have been 
 made at the Agricultural Experiment Station, and if the reader 
 will take the pains to compare the different correlation coefficients, 
 
TABLE 1. CORRELATION AMONG CERTAIN CHARACTERS OK EARS OF CORN 
 400 SERIES. CROP 1907. SEED: Low PROTEIN BY MECHANICAL SELECTION 
 
 
 Yield 
 * 
 
 Value of r for 
 length and circum- 
 ference 
 
 Value of r for 
 length and rows 
 
 Value of r for 
 circumference and 
 rows 
 
 401 
 
 68.4 
 
 0.423+0.019 
 
 0.044+0.024 
 
 0.5010.019 
 
 402 
 
 69.9 
 
 0.438-1-0.019 
 
 -f 0.007+0. 024 
 
 0.446+0.020 
 
 403 
 
 69.4 
 
 0.312+0.020 
 
 
 0.484+0.018 
 
 404 
 
 75.9 
 
 0.4624-0.018 
 
 
 0.548+0.017 
 
 405 
 
 66.6 
 
 0.452 -HO. 018 
 
 
 0.5020.019 
 
 406 
 
 84.6 
 
 0.403+0.019 
 
 
 0.470+0.018 
 
 407 
 
 68.6 
 
 0.282+0.021 
 
 
 0.440+0.020 
 
 408 
 
 84.1 
 
 0.278+0.021 
 
 
 0.554+0.016 
 
 409 
 
 71.4 
 
 0.203+0.021 
 
 
 0.487+0.019 
 
 410 
 
 95.6 
 
 0.411+0.017 
 
 
 0.432+0.018 
 
 
 
 Value of r for 
 
 Value of r for 
 
 Value of r for 
 
 
 * 
 
 length and weight 
 
 rows and weight 
 
 weight and circum- 
 ference 
 
 401 
 
 
 0.781+0.008 
 
 0.275+0.023 
 
 0.768+0 009 
 
 405 
 
 
 0.786+0.009 
 
 0.223+0.024 
 
 0.721+0.011 
 
 * Computed at 80 Ib. per bushel of ear corn. 
 
 TABLE 2. CORRELATION AMONG CERTAIN CHARACTERS OF BARS OF CORN 
 5 SERIES. CROP 1908. SEED: HIGH PROTEIN BY MECHANICAL SELECTION 
 
 
 Yield 
 * 
 
 Valus of r for 
 length and circum- 
 ference 
 
 Value of r for 
 length and rows 
 
 Value of r for 
 circumference and 
 rows 
 
 501 
 
 37.5 
 
 0.590+0.014 
 
 0.090+0.025 
 
 0.514+0.019 
 
 502 
 
 45.0 
 
 0.528+0.015 
 
 0.061 0.026 
 
 0.506+0.019 
 
 503 
 
 33.1 
 
 0.562+0.015 
 
 0.120+0.027 
 
 0.432+0.022 
 
 504 
 
 39.3 
 
 0.526+0.016 
 
 
 0.486+0.021 
 
 505 
 
 39.6 
 
 0.519+0.016 
 
 
 0.6080.017 
 
 506 
 
 76.1 
 
 0.422+0.015 
 
 
 0.480+0.016 
 
 507 
 
 46.1 
 
 0.385+0.018 
 
 
 0.444+0.020 
 
 508 
 
 74.1 
 
 0.360+0.016 
 
 
 0.517+0.016 
 
 509 
 
 39.8 
 
 0.4440.017 
 
 
 0.5280.019 
 
 510 
 
 75 
 
 0.344+0.017 
 
 
 0.497+0 015 
 
 
 
 Value of r for 
 length and weight 
 
 Value of r for 
 rows and weights 
 
 Value of r for 
 weight and circum- 
 ference 
 
 501 
 
 
 0.855+0.006 
 
 0.345+0.021 
 
 0.771+0.009 
 
 505 
 
 
 0.871+0.005 
 
 0.348+0.021 
 
 0.763+0. 09 
 
 * Computed at 80 Ib. per bushel of ear corn. 
 
TABLE 3. CORRELATION AMONG CERTAIN CHARACTERS OF EARS OF CORN 
 400 SERIES. CROP 1909. SEED: L,ow PROTEIN BY MECHANICAL SELECTION 
 
 
 Yield 
 
 * 
 
 Value of r for 
 length and circum- 
 ference 
 
 Value of r for 
 length and rows 
 
 Value of r for 
 circumferences and 
 rows 
 
 401 
 
 46.2 
 
 0.548+0.016 
 
 0.004+0.027 
 
 0.452+0.021 
 
 402 
 
 38.6 
 
 0.623-1-0.016 
 
 0.027+0.034 
 
 0.466+0.026 
 
 403 
 
 48.4 
 
 0.453+0.118 
 
 0.044+0.026 
 
 0.514+0.022 
 
 404 
 
 43.6 
 
 0.534+0.016 
 
 
 0.463+0.022 
 
 405 
 
 47.2 
 
 0.461+0.018 
 
 
 0.487+0.022 
 
 406 
 
 57.6 
 
 0.506+0.017 
 
 
 0.482+0.019 
 
 407 
 
 46.0 
 
 0.443+0.019 
 
 
 0.524+0.020 
 
 408 
 
 58.8 
 
 0.432+0.018 
 
 
 0.428+0.021 
 
 409 
 
 51.6 
 
 0.409+0.019 
 
 
 0.458+0.022 
 
 414 
 
 72.6 
 
 0.539+0.012 
 
 
 0.425+0.018 
 
 
 
 Value of r for 
 length and weight 
 
 Value of r for 
 weight and rows 
 
 Value of r for 
 weight and circum- 
 ference 
 
 401 
 
 
 0.818+0.008 
 
 0.216+0.025 
 
 0.840+0.007 
 
 402 
 
 
 0.844+0.008 
 
 0.225+0.032 
 
 0.746+0.012 
 
 403 
 
 
 0.815+0.008 
 
 0.178+0.027 
 
 0.648+0.013 
 
 404 
 
 
 0.801+0.010 
 
 0.212+0.029 
 
 0.757+0.012 
 
 405 
 
 
 0.810+0.008 
 
 0.229+0.027 
 
 0.728+0.011 
 
 406 
 
 
 0.791+0.008 
 
 
 
 407 
 
 
 0.800+0.008 
 
 
 
 008 
 
 
 0.798+0 008 
 
 
 
 409 
 
 
 0.785+0.008 
 
 
 
 410 
 
 
 0.843+0.005 
 
 
 
 * Computed at 50 Ib. per bushel. Shelled corn (dry substance). 
 
 he will learn something of the way corn behaves under a variety 
 of conditions. 
 
 The material for this study is furnished by the crops obtained 
 from a number of the regular experiment plots that are being 
 conducted for different purposes. These plots belong to two dif- 
 ferent groups, one of which is devoted primarily to soil investi- 
 gation (Tables 1-3), and the other to experiments in corn breed- 
 ing (Table 4). 
 
 The soil plots are designated as the 400 and 500 series. They 
 are devoted to a two year rotation consisting of corn alternating 
 with oats, that is to say, corn occupies the 400 series one year 
 and the 500 series the next year. Each series is divided into 
 ten plots, and various soil treatments are applied. For a detailed 
 account of the arrangement and the soil treatment of these plots, 
 the reader is referred to the bulletin of which this is an abstract, 
 or to Bulletin 125 of this station. 
 
TABLE 4. CORRELATION AMONG CERTAIN CHARACTERS OF EARS OF CORN 
 
 CORN. CROPS 1907, 1908, 1909 
 
 
 Value of r for 
 length and 
 circumference 
 
 Value of r for 
 circumference 
 and rows 
 
 Value of r for 
 length and 
 rows 
 
 Value of r for 
 length and 
 weight 
 
 Crop 1907 
 
 High protein . 
 Low pro ein . . 
 High oil . . . . 
 Low oil .... 
 
 0.2020.031 
 0.368+0.028 
 0.3170.029 
 128-1-0.035 
 
 0.4900.026 
 0.5204-0.025 
 0.431 0.027 
 562 024 
 
 -0.051 4-0. 032 
 0.0170.034 
 
 0.8224-0.013 
 0.7254-0.016 
 0.8384-0.009 
 727 017 
 
 Crop 1908 
 
 High protein . 
 Low protein . . 
 High oil 
 
 0.3104-0.027 
 0.2930.027 
 132-f-0.027 
 
 0.4354-0.025 
 0.4884=0.024 
 0.4644-0 022 
 
 0.1064-0.030 
 0.0170.031 
 
 0.7764-0.012 
 0.8094-0.012 
 6734-0 015 
 
 Low oil 
 
 0.5694-0.020 
 
 0.459-(-0 025 
 
 
 0.8684-0.007 
 
 Crop 1909 
 
 High protein . 
 Low protein . . 
 High oil 
 
 0.1830.030 
 0.4374=0.022 
 0.2994-0.025 
 
 0.5924-0.021 
 0.3054-0.029 
 0.3354-0.026 
 
 0.081 035 
 
 0.6814=0.016 
 0.7704-0.011 
 0.769-t-0.013 
 
 Low oil . ... 
 
 0.2554=0.024 
 
 0.4804-0.020 
 
 
 0.6754-0.014 
 
 The variety of corn grown upon these plots has been two 
 strains of Learning which has been under selection for high pro- 
 tein and low protein content respectively. In 1907 and 1909 the 
 seed corn planted was low in protein content while in 1908 it was 
 high in protein. 
 
 The corn breeding plots from which the material for Table 
 4 was taken, represents four lines of selection that have been un- 
 der way since 1896 for high protein content, low protein content, 
 high oil content and low oil content. 
 
 With a set of four characters under consideration, there are 
 six pairs of characters, between each pair of which the correla- 
 tion can be determined. From the results of the tables, it will 
 be observed that the correlations for some plots are given for 
 each of these six pairs of characters. The tables include the re- 
 sults of 141 determinations of correlation, and should give a good 
 general notion of the values of these correlations for the corn un- 
 der consideration. 
 
 In Table i are given, correlations between length and circum- 
 ference, length and number of rows, circumference and number 
 of rows, length and weight, weight and rows of kernels, weight 
 and circumference, for some plots of low protein corn, differently 
 treated, crop of 1907. A careful study of this Table shows, first 
 of all, a considerable tendency for length and circumference to 
 move together, that is, for long ears to be large in circumference, 
 
but that this correlation varies greatly in the different plots, rang- 
 ing all the way from 0.203 to 0.462. Second, there is practically no 
 correlation between the length of ear and the number of rows 
 it contains, that is to say, one is no index whatever to the other. 
 Third, there is a fairly high positive correlation between circum- 
 ference and the number of rows, which means that the large ears 
 have in general more rows than the small ears. However, the 
 correlation in no case approaches very near to i.o, which means 
 that the large ears have not only more rows than the smaller ones, 
 but the kernels are larger. There is a high correlation for length- 
 weight, and weight-circumference, but a rather low correlation 
 between weight and rows of kernels. 
 
 In Table 2 are also shown ten plots of high protein corn, dif- 
 ferently fertilized, crop of 1908. The same general traits are 
 maintained as in the former table, excepting that the correlation 
 runs somew r hat higher between length and circumference ; and, 
 that there is perhaps a slight positive correlation between the 
 length of the ear and the number of rows that it contains. 
 
 Table 3 exhibits the correlation for the same series as shown 
 in Table i, but for the crop of 1909, and a more complete list 
 of correlations between length-weight, weight-number of rows, 
 and weight-rows of kernels. In these later determinations, we 
 find what we should now 7 expect, namely, a high correlation be- 
 tween length and weight and between weight and circumference, 
 and a rather low 7 correlation between weight and the number of 
 rows. 
 
 Table 4 exhibits certain correlations for the so called Illinois 
 corn, which, as stated above, consists of four strains bred for 
 chemical composition. A gross comparison of this table will 
 show that the correlation in any two characters varies in the same 
 strain of corn in different years as it does by different methods 
 of treatment. For example, the correlation between length and 
 circumference in high protein corn, crop of 1907, was 0.202. The 
 next year it was 0.310, and the next, 0.183. The reader will be 
 interested in making this same sort of comparison for other 
 strains of corn and for other characters. 
 
 Arranging the pairs of characters in descending order as to 
 correlation, we have the following order : 
 
 1 i ) Length and weight. 
 
 (2) Circumference and weight. 
 
 (3) Circumference and rows of kernels. 
 
 (4) Length and circumference. 
 
 (5) Weight and rows of kernels. 
 
 (6) Length and ro\vs of kernels. 
 
8 . 
 
 For this arrangement, the odds are pretty large except in the 
 case of (3) and (4), and possibly of (i) and (2). 
 
 As a sort of general conclusion, we may say that correlations 
 for length-weight and circumference-weight are high. The cor- 
 relation for circumference-rows of kernels and length-circumfer- 
 ence are fairly high. The correlation of weight-rows of kernels 
 is low, \vhile that of length-rows of kernels is probably, in general, 
 insignificant. 
 
UNIVERSITY OF ILLINOIS-URBANA