5 
possible. Each setting should be replicated at least three times and the average 
of the readings should be used as the result of the observations. In this way 
the precision can be increased because the standard deviation of the average 
decreases with the square root of the number of replications. 1 ) The points 
should be measured in a well defined order and check repetitions should be 
made in a number of points. 
The measured data are then compared with the corresponding given data, 
if necessary after a preliminary transformation, such as by translations, rota 
tions and scale changes. 
The differences between the measured and (eventually) transformed co 
ordinates and the given coordinates are then computed for each point, and 
the discrepancies are tabulated. 
The discrepancies are then considered as caused by regular and irregular 
sources of error. The main task of the test procedure is now to distinguish as 
well as possible the various regular (systematic) errors which may be present 
and to express the residuals in a reliable statistical manner. 
The discrepancies are indirect measurements of regular and irregular sources 
of errors. The regular sources of error are, consequently, assumed to contri 
bute to the discrepancies according to certain functions expressing mathema 
tical relations. If the regular sources of error are sufficiently small, the relations 
can be replaced by their differential formulas, developed from well defined 
approximate data. If the approximations are too crude, the principles of 
iteration can be applied. 
The first step after the determination of the discrepancies is therefore 
to estimate possible regular sources of error and to derive the corresponding 
mathematical relations between the sources of error and their contributions to 
the discrepancies. Usually this can be done in terms of differential formulas. 
Evidently, the sources of regular error to be considered in a specific case 
vary with the circumstances. In testing a comparator for image coordinate 
measurements, for example, there are primarily six regular sources of error 
to be expected, viz. two translations and one rotation of the grid, lack of 
orthogonality between the coordinate axes of the comparator, and two scale 
errors, one in each coordinate direction. More regular sources of error can be 
assumed as, for example, periodical errors of the screws, etc. Frequently, 
such additional sources of error can be discovered and their effects calculated 
after determination of the first set of errors. 
Because, in principle, there must always be more observed discrepancies 
than the estimated number of regular sources of error, no unique determination 
of these sources is possible unless a specific condition is imposed on the residuals. 
J ) It is always suitable to determine the actual precision from series of replicated and 
repeated settings.