196 REPORTS ON THE STATE OF SCIENCE. 



It remains to compute the error in the total from the errors in 

 the details. Let n stand for the number of non-taxpayers in each 

 class, c for its modulus — given in thousands — s for the average income 

 in £, and 1c for its modulus, c and k are to have the meaning dis- 

 cussed above and to be regarded as measuring that particular distance 

 from the average known as the modulus in mathematical statistics. 



Let Aj be the estimate and A the true measurement, and let A,= 

 A(l + c). 



Let A 1 be the measurement if A, was in defect equal to the modulus 



in both its factors, so that A' = A,, (l + -) (l + -} = A, (l + - + -) 



approximately. Then by the theory of error e 2 = ( - ) + ( -) 



approximately, 



and A, = A + \/ [cV + & 2 « 2 ] approximately = A + K\ 

 Then K' is the modulus for the product. 



K' 



In the table K is taken as — — , to correspond with the units used, 



and is worked out for each class and sub-class separately. 



To get the modulus for the aggregate we have by a known principle 

 to add the squares of the moduli of the items and take the square root 

 of the sum. If C is this final modulus, 



1 C 2 = 2(K 2 ) = S(c 2 s 2 + F« 2 )= S(c 2 s 2 )+ S(Z; 2 « 2 )= 6,300 + 80,000=86,300 

 and C = 294, viz. 29,4O0,000Z. 

 The first term is due to the margin (c) in numbers, the second to the 

 margin (fc) in income. The second term of itself would give C = 283 ; in 

 fact, the errors in numbers are not important, and we need not there- 

 fore discuss the effect that our approximate knowledge of the total 

 number in the group has on the accuracy of the result. 



We thus arrive at the conclusion that the aggregate earned income 

 in the intermediate group is 285,000,000*., with modulus 30,000,0001., 

 using round numbers. This is subject to the definitions and limitations 

 of the income dealt with as stated throughout the Eeport, and is con- 

 ditioned by the accuracy of the occupational returns in the census. 



If Classes 20, 25, 26, 27, 28, and 29 are excluded, the margin in 

 the aggregate of the remaining classes is small; the result for Classes 

 1 to 19, 21 to 24, 30, 31 being- 

 No. of persons 1,263,000, modulus 33,000; aggregate income, 

 108,700,000*., modulus 5,700,000*. 



Unearned Income. 



Estimates for this can only be satisfactorily made by the authorities 

 of Somerset House, but we have endeavoured to interpret the tables 

 given in the Eeports of the Income Tax Commissioners. 1 It appears 

 that 45,000,000*. under Schedules A, C, and D is ' unearned ' income 

 belonging to persons with less than 160*. a year from all sources. Of 

 this the tax on 18,000,000*. is collected from and repaid to about 



1 We have had access to some details not published in the reports, which have 

 enabled us to split up the tables ; but the estimates based on them are not authoritative. 



