Appendix II 
II: 1. Tests of Assumed Normal Distribution 
II: 1.1 Abstract 
In the theory of errors of measurement, the normal distribution of error 
(normal law, error law, Gaussian law) is important. Indeed, reliable results 
from an adjustment procedure depend on the errors of the fundamental meas 
urements being normally distributed. The laws for error propagation from the 
observations through various functions are also usually founded upon the 
assumption of a normal distribution, as are various significance and confidence 
tests. In reports on photogrammetric research, the distribution of the errors, 
discrepancies, or residuals is seldom given, and statistical tests of the normal 
distribution are rarely found in photogrammetric literature. A practical ex 
ample of such a test is shown below. The procedure is borrowed from Cramer 
1946. The example is taken from actual investigations into fundamental photo 
grammetric operations. 
II: 1.2 The Principles of the Test 
The errors, discrepancies or residuals can be considered as a sample from a 
population, the distribution of which is expected to be normal. This hypothesis 
is to be tested. 
The sample cannot be expected to be exactly normally distributed. There 
will always be deviations from the theoretical normal distribution. These 
deviations must be used for the analysis, and certain limits for the deviations 
must be applied. The c///-square test is generally used for this purpose and is 
discussed below. The limits must be determined in relation to certain predeter 
mined levels of probability. If the upper limit is exceeded, the normal distri 
bution hypothesis is rejected. On the other hand, very small differences be 
tween the theoretical and practical distributions cannot be accepted either. 
For the test, the actual statistical material is first grouped — see the table 
below. The interval which contains all values is divided into a certain number 
of class intervals. It is desirable that about ten values fall within most of the 
class intervals and, if possible, the number of class intervals should not be less 
than ten. Sometimes, classes can be pooled in order to increase the number of 
values. The number of class intervals is denoted k. The class mean (class mid 
point) of the class i is denoted t t and the corresponding number of values (the 
class frequency) / ; . Consequently, the sum [/] is the number n of all values. 
The mean value of the sample is found as