6 BULLETIN 12*77, U. S. DEPARTMENT OF AGRICULTURE 
tions are analyzed by multiple correlation, 3 all four factors taken 
together have a multiple correlation of R= +0.215 ±0.037 4 with the 
value of fertilizer used, while the quantity of seed used proves to 
have a net correlation of t , = +0.191±0.038 with the value of fer- 
tilizer. On the average, for every increase of 1 bushel in the quan- 
tity of seed planted, there was an increase of $0.45 in the value of 
manure or fertilizer used. 
The net correlations with number of sprayings and depth of plow- 
ing were both negative, indicating that to the slight extent to which 
they are correlated, the more fertilizer used, the shallower the plow- 
ing and the fewer the sprayings. 
From this it is apparent that the application of fertilizer was 
practically independent of the factors considered, except that it was 
slightly correlated with the quantity of seed used. 
Similar analyses should be carried through to determine the other 
relations of variations in the value of fertilizer used, and to deter- 
mine the relations of the variations in the other inputs. Thus, for 
example, the apparent relation between the quantity of seed and the 
other factors would be shown by such a table as Table 3. Apparently 
the input of seed was to some extent related to each of the other 
cultural factors. Table 4 shows these relations in exact form, meas- 
uring the net instead of the apparent correlations. 
Table 3. — Average input of other factors per acre, oy given inputs of seed 
Average input of other factors 
Input of seed per acre 
Depth 
of 
plowing 
Cultiva- 
tions 
Spray- 
ings 
Value of 
manure 
and ferti- 
lizer 
6 to 7 bushels . .. . .. . .. 
Inches 
6.85 
7.23 
7.42 
7.62 
7.72 
7.87 
8.57 
Number 
7.20 
7.19 
7.52 
8.12 
7.92 
8.83 
8.57 
Number 
1.40 
1.72 
1.55 
2.05 
2.28 
3.25 
3.71 
Dollars 
10.10 
8 to 9 bushels . . ... . . - 
8.75 
10 to 11 bushels 
9.75 
12 to 13 bushels 
10.67 
14 to 15 bushels .. .. 
11.47 
16 to 17 bushels 
12.42 
18 to 23 bushels 
12.71 
7.56 
7.88 
2.04 
11.47 
3 The multiple and net correlation analysis reveals to what extent each factor was cor- 
related with the other, holding all other factors constant and assuming that the relations 
were all " linear." Thus in Table 2 there seems to be some slight positive correlation be- 
tween fertilizer and depth of plowing. But the same table shows that the slight increase 
in the depth of plowing was accompanied by a marked increase in the quantity of seed 
used. Now the quantity of seed used was positively correlated with depth of plowing as 
well as with quantity of fertilizer, so that when the effect of the quantity of seed used 
was eliminated, the " net " relation between fertilizer and depth of plowing became nega- 
tive. (See the discussion of " Statistical method " for further explanation of the methods 
of analysis. I 
4 The "±0.037 " following the coefficient, of correlation is the probable error, and should 
be read "plus or minus 0.087." This figure means that if the survey had been repeated 
on other farms in the same: area for the same year, and the coefficient of correlation com- 
puted for the same variables, the chances are even that it would be + 0.215 ±0.037 ; that 
is. thai it would lie between I 0.252 and i 0.178. Unless a coefficient of correlation is at 
least three or four times the size of its probable error not much reliance can be placed on 
it; in ibis case it is about six times as Large as the error, Indicating that there is almost 
no possibility thai the observed correlation is due to mere chance in selecting the farms 
surveyed. The coefficient of multiple correlation, It, is interpreted the same as the ordi- 
nary coefficient Of correlation, r; indicating no correlation and 1.00 perfect correlation. 
However, the r of simple correlation measures only the relation between two factors, while 
R measures the joinl relation between one factor and two or more others; in this case the 
R measuring the combined correlation of four factors— depth of plowing, number of cul- 
tivations, number of sprayings, and seed planted — with value of fertilizer and manure per 
jlcvq. 
