je
ne
VO
1e
gi
STANDARDIZATION OF EXPRESSIONS FOR ACCURACY, HALLERT 169
From the [vo] the accuracy of the measurements is defined as the standard error
(of a measurement) of unit weight according to the expression (ref. 6, p. 138, 2, p. 68).
/ [vv]
fs neat (10)
se
The expression n—2 denotes in this case the number of redundant measurements (de-
grees of freedom). For the corresponding treatment of a problem with 4 unknowns the
expression for the standard error of unit weight would become
(11)
9 n—u
Je) ;
live]
Sy = -
5. Expressions for the precision of different measures of accuracy.
The distribution of the errors is assumed to be at least approximately normal. The
precision is particularly dependent upon the number of redundant measurements.
5.1 Standard error of the estimates. (Ref. 2, p. 298 and 6, p. 158).
The standard error of the various measures of accuracy (expressions 3, 4, 5 and 11
above) can in general be determined as follows for the various expression
S
(3a) and (3b) Fonds
V 2n
Ss
(4) Sg = -
V 2(n—1)
: Su
(5) Say =
u VOn(n-—1)
S
(11) s, =
he V 2(n—u)
The use of standard error of the various measures of accuracy is mainly to demon-
strate the infuence of the number of redundant measurements upon the reliability of the
determination of the accuracy.
5.2 Confidence intervals. (Ref. 3, p. 507-524).
The expressions for accuracy under point 4 can be regarded as estimates of an
unknown parameter. The precision of an estimate can be expressed as a confidence inter-
val around the estimate, which will include the unknown parameter with a certain pro-
bability, lenoted the confidence coefficient. Usually the confidence interval is defined by
the confidence limits which are determined for the actual number of redundant measure-
ments (degrees of freedom) and a certain confidence level. Special tables for the y?-
distribution have to be used for this purpose. The method with confidence intervals is
particularly recommended in photogrammetry, also for tests of “maximum errors".
6. Error propagation.
For a determination of the accuracy in functions of basic measurements the special
or the general law of error propagation can be used for uncorrelated and correlated basic
data respectively.
6.1 The special law of error propagation. (Ref. 6, p. 54 and 9, p. 47).
Assume the equation (6) or the normal equations (8) to be solved so that x and y are
expressed as direct linear functions of the measurements I. We obtain for the normal
equations
ys gd cad + +. - + ah, (13)
vu =Ah+Ahlat-+-+hh
