278 STATISTICS: PEARL AND REED Proc. N. A. S. 
The results are set forth in table 2, where Pritchett's figures are given 
for comparison. 
TABLE 2 
Showing {a) the Actual Population^ on Census Dates, {h) Estimated Population 
FROM Pritchett's Third-Order Parabola, (c) Estimated Population from 
Logarithmic Parabola, and {d) (e) Root-Mean Square Errors 
OF Both Methods 
CENSUS 
YEAR 
(a) 
OBSERVED 
POPULATION 
(b) 
PRITCHETT 
ESTIMATE 
(c) 
LOGARITHMIC 
PARABOLA ES- 
TIMATE 
(d) 
ERROR OP 
ib) 
(e) 
ERROR OF 
(c) 
1790 
3,929,000 
4,012,000 
3,693,000 
+ 
83,000 
236,000 
1800 
5,308,000 
5,267,000 
5,865,000 
41,000 
+ 
557,000 
1810 
7,240,000 
7,059,000 
7,293,000 
181,000 
+ 
53,000 
1820 
9,638,000 
9,571,000 
9,404,000 
67,000 
234,000 
1830 
12,866,000 
12,985,000 
12,577,000 
+ 
119,000 
289,000 
1840 
17,069,000 
17,484,000 
17,132,000 
+ 
415,000 
+ 
63,000 
1850 
23,192,000 
23,250,000 
23,129,000 
58,000 
63,000 
1860 
31,443,000 
30,465,000 
30,633,000 
978,000 
810,000 
1870 
38,558,000 
39,313,000 
39,687,000 
755,000 
+ 1,129,000 
1880 
50,156,000 
49,975,000 
50,318,000 
181,000 
+ 
162,000 
1890 
62,948,000 
62,634,000 
62,547,000 
314,000 
401,000 
1900 
75,995,000 
77,472,000 
76,389,000 
-f 1,477,000 
+ 
394,000 
1910 
91,972,000 
94,673,000 
91,647,000 
-f 2,701,000 
325,000 
1920 
114,416,000 
108,214,000 
935,000^ 
472,0002 
^ To the nearest thousand. 
2 Root-mean square error. 
It is obvious from the data of table 2 that, with the same number of 
constants, the logarithmic parabola gives a distinctly better graduation 
than a third-order parabola. 
The extreme precision of the present graduation is shown graphically 
in figure 1. 
It is evident that as a purely empirical representation of population 
growth in the United States equation (iii) gives results of a very high 
degree of accuracy. Indeed, interpolation on this curve for inter-censal 
years may obviously be relied upon with a greater probability that the 
estimated figures approximate the unknown true facts than is afforded 
by any other estimating expedient hitherto applied to the known data. 
An indication of the general exactness of this curve (iii) for estimating 
future population by extrapolation may be got in the following way. 
Suppose a mathematician of the Civil War period had desired to estimate 
the population of the United States in 1910, and had fitted a curve of the 
type of (ii), by the method of least squares to the known data available 
