20 
distribution and the actual distribution of the sample is tested by comparison 
with the differences between the two distributions as expressed by the last 
function above. If the differences are too large, the normal distribution hypo 
thesis cannot be accepted. If the agreement is too good, the reliability of the 
sample distribution can be doubted from other points of view. The levels for 
both cases are usually chosen 2.5 and 100—2.5=97.5 percent, respectively. 
The degrees of freedom are in this case determined as k-3, where k is the number 
of class intervals. From a table of the chi-square distribution, which also is to 
be found in most textbooks on statistics, the values for the percentages mention 
ed, with k-3 degrees of freedom, can be determined and compared with the 
computed sum as indicated above. Usually only the upper limit is determined 
on the 5 percent level. If the sum is located below this value from the chi- 
square distribution, the hypothesis that the sample distribution is normal can 
be accepted. 
Finally, a histogram should be made from the distribution of the sample. 
In the histogram, the corresponding normal distribution frequency curve can 
be demonstrated simply with approximation by plotting the computed values 
of np { . 
A complete practical example of the computations is shown below together 
with the histogram and normal distribution curve. 
The complete computation procedure following the principles described 
above is shown in the following Table 1. In Diagram 3, the actual histogram 
and normal distribution curve are also shown. 
Sometimes, the degree of skew and the excess of a sample must be consid 
ered in a normal distribution test. See, for example, Cramér 1954. 
Nurhber 
of residuals 
chi 5 J % = 16.92 
Normal distribution hypothesis accepted. 
Diagram 3. Example of histogram and normal distribution curve. Residuals after test meas 
urements in a stereocomparator and adjustment.