82 
A NEW METHOD OF ESTIMATING STREAM-FLOW 
The probable error of a single observation as shown above is the probable 
error, computed rigorously from the normal equations and the residuals, of the 
rise of the mean surface of the whole lake from one day to the next. From this it 
appears that the elevation of the mean surface of the whole of Lake Michigan- 
Huron on any day may be determined with a probable error of ±0.010 foot. 
Table 26 shows the best values of Ei and E 2 determinable from the obser- 
vations on Lakes Michigan-Huron and Superior. The values of £\ and E 2 and 
their probable errors are the same as in Table 25. From these are computed 
the weights assigned to each unknown, the weighted-mean value and its probable 
error, and the residuals of each value from its own weighted mean. The assigned 
weights are inversely proportional to the squares of the respective probable errors, 
corresponding to the assumption that all errors in the computed values of these 
constants are of the accidental character. Unit weight corresponds to a probable 
error squared of 0.1. 
Table 26 — Best values of E\ and E 2 
Probable 
error 
Assigned 
weight 
Value 
of— 
Resi- 
dual 
Sum of 
weights 
Weighted 
mean 
Probable 
error of 
weighted 
mean 
Best value of E\: 
Solution Vi, Lake M-H 
Solution BBi, Lake S 
Best value of E 2 : 
Solution Vt, Lake M-H 
Solution BB it Lake S 
±0.031 
± .052 
±0.148 
± .234 
104 
37 
4.57 
1.82 
Ex 
+0.266 
- .154 
E2 
+ 1.272 
+ 1.163 
-0.110 
+ .311 
-0.03 
+ .08 
141 
+0.157 
±0.027 
6.39 
+ 1.24 
±0.125 
The residual of the value of Ei in Solution BB t , viz, +0.311, is over five 
times the probable error of the weighted mean Ei from both solutions (5 X ± 0.027 
= ±0.135), therefore that value of E,, —0.154 ±0.052, is rejected. 
With the rejection stated in the preceding paragraph, the following are adopted 
as the best values of E h E 2 and x obtainable in this investigation. 
(21) 
#,= +0.266 = 0.031i 
E 2 = +1.24 ±0.125 
z = 2.6 i 
Using the adopted values, (21), equation (1), expressing the rate of natural 
evaporation from the surface of Lakes Michigan-Huron or Superior in units of 
0.001 foot of depth per day, may be written. 
#„=+0.266e+1.24 
[•&-")] 
(22) 
Expressed in units of 0.01 inch of depth per day, the unit used in Part II of 
this publication. 
*.- + 0.319e + 1.49 L QjL _ 2 .e)] (23) 
( = 0.037) (±0.150) 
