94 C. E. VAN ORSTRAND 
this reason, no further consideration will be given the tabulations 
in these two columns in attempting to deduce an average gradient 
from the data of observation. 
In group J, where the weights are inversely proportional to the 
squares of the probable errors, the resulting values are not in agree- 
ment with those obtained by means of formulas 7, 8, 9. From what 
has just been said in regard to the effect of the curvature of the depth- 
temperature curves on the weights of the gradients, it follows that 
the results summarized in the J-groups are dependent chiefly on 
those depth-temperature curves which approach a straight line. 
In the last column are shown the different results obtained by the 
use of formulas 7, 8, 9. The mean values are not very consistent and 
differ widely from those given in the J-group. The inconsistencies 
emphasize the importance of assigning theoretically correct weights 
to the individual values. Formula g is the correct formula to use when 
true rock temperatures are recorded at a single point in each well. 
In group J, the weights have been computed from Formula 12. 
Since the observations have generally been made at practically the 
same points, 100, 250, 500, . . . feet, the values of the weights deviate 
only slightly from constancy, except for the column of total depth. 
It is to be expected, therefore, that the mean values in the first four 
columns of the J-group will approximate closely to those found in the 
A-group. Comparison of the results shows a sufficiently close agree- 
ment with theory. In comparison with the J-group, it amounts to 
assuming that 7, the probable error of an observation of weight unity 
in the J-group, is equal to a constant. 
In the B-group, those values of the gradient for which 2, the 
deviation from the mean, equals or exceeds 5r have. been rejected. 
This procedure is based on the statistical deduction that there is only 
1 chance in about 1,000 that any individual residual in a statistical 
system should equal or exceed 5r. The rule is a very useful guide. 
Like other statistical deductions, it is not absolute. The summary 
shows that one value in the first column and four in the last two 
columns were rejected. The rejected values are indicated by an 
asterisk in the main part of the table. 
In group C all values of 6 for which the reciprocal gradient in 
the B-group exceeds 1°F. in about 40 feet have been arbitrarily re- 
jected. The object of this procedure is to eliminate the abnormally 
high values of the gradients which are frequently found in the oil 
fields. The table shows that the reciprocal gradient for the first 1,000 
feet is changed by this operation from 65.21 to 70.71 feet per degree 
Fahrenheit. 
754 
