9 
The confidence limits are then found from this standard error by multiply 
ing with a specific factor ¿ p , which can be determined from the ¿-distribution 
of the errors. The ¿ p -factor is conveniently found from a ¿-table for the actual 
degrees of freedom (redundant observations) in the determination of the stand 
ard error of unit weight and a specific level, usually 5 percent. 
For the determination of the standard error of linear functions of basic 
observations the general law of error propagation has to be applied if there is 
algebraic correlation introduced by the adjustment procedure, Helmert 1907. 
For example, assuming the specific standard error of a parameter to be 
determined in this way and the degrees of freedom to be 10, the ¿ p -factor on 
the 5 percent level is ±2.2. 
It is evidently desirable to use many redundant observations for the deter 
mination of the standard error of unit weight if the tolerances based on it are 
to be kept small. 
Examples of linear functions of basic measurements where the establishment 
of tolerances is desirable include such adjustment data in instruments as cannot 
be compensated by other parameters in the photogrammetric procedure, i.e. 
non-projective data, and residuals after the adjustment of test measurements. 
The individual residuals can always be written as linear functions of the basic 
observations and the standard error can therefore easily be determined from 
the laws of error propagation. Then the ¿-test can be applied in order to deter 
mine whether or not the residual deviates excessively from its ideal value 
(usually zero). More sophisticated principles and procedures can be used for 
the determination of tolerance limits. Reference is made to the literature on 
statistics, see for instance Hald 1957. 
1.23 Root mean square values of standard errors. 
In many cases root mean square values of residuals or discrepancies can be 
computed and it may be of interest to judge the significance. If the quality of 
previous operations, in particular the basic measurements, has been deter 
mined as standard errors of unit weight and the error propagation to the 
residuals can be studied according to the usual laws, a theoretical value of the 
standard errors of the residuals can be computed and frequently a root mean 
square value of the standard errors can be determined. For the judgement of 
the significance of the difference between the theoretical and the practical 
(“true”) values the confidence interval around the theoretically computed root 
mean square value can be determined with respect to the degrees of freedom 
of the basic standard error of unit weight and a chosen confidence level. If the 
“true” value is included in this confidence interval the difference between the 
two values is not significant on the chosen level. This can be used as an addi 
tional tolerance criterion or for testing the applied theory of error propagation.