126 MISC. PUBLICATION 218, U. S. DEPT. OF AGRICULTURE 
erty, and dividing the sum of the products thus obtained by the aggregate true 
value. Dividing the result by the average assessment ratio, R, gives the coeffi- 
cient of dispersion, or 
_ (R=) (R-1)m+...+(R—Ta) dn 
R(vj+v2+..-+0n) 
in which each term of the form (#—r,) is considered as positive regardless of 
sign. Also, by definition 
(2) N=) OpoeOiairoosarea)c 
Multiplying each member of equation (1) by the corresponding member of 
equation (2) 
(1) D 
(Sao Cen RY 0- | 
(3) De Pa ar enh OE at Nh AE ral 
However, by definition, 
tai tat... tan) =U (itm+...+0n); 
Ge Gyarecssr Gp. ih! ane 
Dee jean aoe Les 
Substituting for R in (3) 
t” , , i , / Dt , uf 
7h —r,t’ jv; + zt —rot! Jvot...+ 7 —T,t’ }vn 
D:- T= PRS a near RRR San ne eee 
oi 
whence 
Simplifying 
(4) D-T= (t’ — ryt) Vi + (t’ =F Tat) V2 +, o6 -- CH aaa T nt) Une 
The misplaced tax is determined directly as follows: 
DOG) se OY = Gh) scot O35 SG») 
oe 2 
in which each term of the form (v,t’—a,t) is considered as positive regardless of 
sign. 
“However, Q1=VUj"1, Ag=Ver2, etc. 
Substituting these values, and multiplying by 2, 
2M = (vt! — vit) + (wet? — veret) +...+ nt! —Unrat), 
or 
(5) 2M = (t’—rit)vui + & —ret) vo... + (t’ — rat) dn. 
M 
From equations (4) and (5) 2M=D.-T, or mas D-T. 
Thus the amount of taxes misplaced because of dispersion in assessment ratios 
is equal to one-half of the coefficient of dispersion times the total tax. 
In practice it is impossible to determine a precise coefficient of 
dispersion, since the estimates of true value are subject to error and 
where a sample is used it would not be likely to give a perfect repre- 
sentation of the group. Where there is equalization of the county 
tax levy among districts, as in Wisconsin, a coefficient of dispersion 
for the county tends to exaggerate the actual misplacement of county 
taxes provided there is greater uniformity in assessment, by and large, 
within the separate districts than within the county as a whole. 
It would be possible, in measuring inequality in assessment among 
individual properties, to treat all properties alike, regardless of their 
respective values. In that case the average deviation, measured from 
