Bauhaus-Universität Weimar

On mean values for direct measurements. 
23 
The numerical uncertainty of the average of the ten results would 
0 25ß 
he indicated by a mean error of -i-r==0.08ß for the unit-place 
V 10 
or 0.8/3' for the first decimal. The median is thus at a disadvantage 
numerically. 
As the results gather more around a middle value in more accurate 
work, more values will coincide with the median, mt will become 
larger and the numerical error for a given number of results will 
become less. The numerical error of the average will remain the 
same. 
As mentioned on pages 2 and IV the influence of (æ) on the 
numerical error is negligible. Thus in calculating the numerical 
error of the median, as long as the unit of number does not exceed 
the size of the mean error, we can safely suppose the values xt to 
have arisen by rounding-off equally frequent values throughout ß. 
Dependence of accuracy on the number of results. Gatjss has 
deduced an expression for the accuracy of the mean value of any 
power of a variation as depending on the number of variations.1 
With slight changes the results can be stated as follows on the sup¬ 
position of (11). Let xk be the mean as determined from the k 
powers of the n observations. Then with not too small numbers of 
results the probable uncertainty, in the same sense as the probable 
0.VS2 ; 
error, for æ0=Af determined from xf xf . . ., x„ is ± -• 
\tn—1 
for x,=A determined from x1, x1, . . ., xn' it is ±--:. 
\/n — 1 
Other things being equal, it is necessary to take 249 observations 
to gain the same accuracy for the median as is given by 114 obser¬ 
vations for the average. 
Dependence on characteristic variations. As noted on p. ? the 
median is that representative value which corresponds to a minimum 
for the sum of the absolute values of the first powers of the varia¬ 
tions. The mean variation from the median will bear to the median 
a relation similar to that which the mean-square-error bears to the 
arithmetic mean. The median will thus be stated as 
M±a, M±l or M±s, where 
1 Gauss, Bestimmung d. Genauigkeit d. Beobachtungen, Zt. f. Astr., 1816 I 185; 
Werke, IY 109. 
Lipschitz, Sur la combinaison des observations, C. R. Acad. Paris, 1890 CXI 163.
        

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