ACCURACY OF CLOSE-RANGE ANALYTICAL RESTITUTIONS 347 
The quantity e is the residual random error of the measuring system S. As a rule the distri- 
bution of the random error e is normal: 
e e AN'(0, os). H c 
The precision of X, within the measuring system S, indicates the closeness of X to E, X and 
is characterized by the standard deviation ofe, o;, It is a well known experimental fact that, for 
the standard deviation o; of the arithmetic mean X computed from a large number of 
measurements in a stable measuring system, we have 
og; = 0. 
The quantity B is the bias (rather than the systematic error, the meaning of which is too 
restrictive). The principal reasons for the bias are systematic effects, lack of definition of the 
measured quantity, and resolving power of the measuring procedure. 
The accuracy of the estimation X indicates the closeness of X to X. It is characterized by 
RMS error VE, e?, with 
E,e;?2E(ec- BP? = EE + ßEe + ß? = g;? 4 2. 
If we consider all the measuring procedures which give approximately the same precision 
(for example all the comparators with the same trade-work), we have 
Ee;? = o,2+ E 2. (2) 
The quantity VEB2 may be called the RMS bias. 
From Equation 2 it can be seen that the RMS error is statistically superior to the RMS 
bias,* which characterized the average maximum accuracy of all measuring procedures with 
the same features, i.e., 
RMS error = RMS bias (3) 
It is important to notice that, in any case, even though the systematic effects are well cor- 
rected, there is in the RMS bias some irreducible part due to the lack of definition of the 
measured quantity and to the resolving power of the measuring procedure. It can be 
observed otherwise that for many precise equipments the RMS bias is often approximately 
equal to the standard deviation o of an elementary measurement of X (estimated from the 
repetition of measurements under the same conditions with the same operator. For example, 
for comparators and theodolites, for the arithmetic mean of n elementary measurements we 
obtain the RMS error i 
V1 + fe» a). 
Finally, we can say that if we consider a one-dimensioned physical quantity whose true 
value is X, the estimator X of X from a particular type of measuring procedure is a normal 
variable : 
X € N°(X, Vo? + E p?) (4) 
where c, is the precision of X and VE f? is the RMS bias. 
As a matter of course, Equation 4 is only a mathematical model. In particular, the dif- 
ficulty is to assign a meaning to the symbol X, the "true value" of the physical quantity 
(what are the “true values" of the coordinates of a geodetic signal, of a physical object point, 
of an image point?). The best way is probably to consider X as the mean of estimations 
coming from a great number of “optimal” measuring procedures. Then other problems 
occur: the true value depends on the chosen type of measuring procedure. For example, it 
can be thought that the center of an irregular, or even regular, spot is appreciated in different 
ways when using microscopic or long distance procedures, and in analytical aerotriangula- 
tion one cannot be sure that the true geodetic definition of an object point is identical to 
the photogrammetric one, particularly when there are no targets. 
Anyway, there is a gap between the symbol X and the reality owing to the lack of defini- 
tion of the physical quantity and the resolving power of the measuring procedure. It is of no 
physical interest, and perhaps impossible, perhaps a statistical nonsense, to want to attain X. 
What is of interest is to eliminate random error and systematic effects. In other words, we 
can say that the maximum accuracy (systematic effects and random errors eliminated) is 
characterized by the RMS bias, and that all estimations of maximum accuracy are equivalent 
to describing the true value of the quantity. 
* In French, the RMS bias characterizes what is called justesse (accuracy = exactitude, precision = 
precision).