Appendix 191 



The square root of this quantity is 1220. With <jy as the 

 symbol for the standard deviation of the Y series, the expres- 

 sion is <jy equals 1220. Similarly from column 7, <jx equals 249. 



3. It was noted in the text that another useful measure in deter- 

 mining whether or not high population increases were asso- 

 ciated with high farm increases in individual cases is the 

 product of the two deviations. These XY products are shown 

 in column 5, table 17 and their corrected total after dividing 

 by the number of counties is 246,210, i.e. 2xY=246,210. 



4. A coefficient expressing the sums of these deviation products 

 in terms of the two standard deviations constitutes a measure 

 of the real relationship between the two series which is duly 

 weighted for each case. Provided a straight line is the best 

 description of the distribution of the two series, the coeffi- 

 cient which should be developed is one which will describe 

 the best fitting straight line in terms of the deviation products 

 and the standard deviations. 



Derivation (Based on the "Mathematics of correlation, Moore, 

 Forecasting the Yield and Price of Cotton). — Two series of ob- 

 servations are taken on the same cases. Example, let the coun- 

 ties studied in Chapter II, Part II, be the cases and the first 

 series of observations be on the increases in farms operated by 

 Negroes, the second on increases in Negro population. Call one 

 set of variables (the observations made on the increases in 

 farms) x 1 , x 2 , x 3 , x„. Call the other set of variables, (the ob- 

 servations made on the increases in population) y v y 2 , y 3 y„. 



Compute the averages of the two series and by subtracting the 

 average from each individual observation, obtain the deviation of 

 each observation from its average. (See table 17, columns II 

 and III.) For the first series call these deviations X , X 2 , X 3 

 . ...X n . For the second series call these deviations Y, Y , Y 3 

 ....Y n . i.e. 



x x — Av. x series=X 1 

 x 2 — Av. x series=X 2 

 y t — Av. y series=Y 1 

 y 2 — Av. y series=Y 2 etc., etc. 



2. Plot the cases on a system of coordinates with the aver- 

 ages of the two series as the zero point: (See Diagram III op- 

 posite page.) That is, locate the intersection of the x and y 



