right 
an be 
lS 1S 
the 
r and 
1e Ho 
] this 
it 1s 
than 
, this 
oeing 
asing 
mum 
ed as 
s not 
| the 
id as 
If the 
Ho 
more 
> not 
e Qu 
This 
mber 
50%, 
ed as 
o the 
number of samples as in the second graph of Fig. 2.2. 
Increasing the number of samples also means increasing the 
cost of production, so an optimum level should be selected. 
  
  
  
  
  
  
Fig 2.2 Producer risk and consumer risk 
2.2 Determining number of samples using hypergeometric 
distribution 
Hypergeometric distribution is widely used for quality control 
purposes and is the probability model for a fixed amount of 
observations with two values. If number of maps satisfying the 
required accuracy is M and inaccurate maps are defined as D, 
the probability f(x) that x amount of accurate maps are selected 
from a sample of n is as follows. 
f(x) =nCr Chix IN+DCn 
where x2 0,1,2,..,nandx «nand. n«M 
Applving this to the consumer and producer risk, the number of 
maps satisfying the minimum required accuracy Qr, when n 
samples are selected from N number of maps is as follows. 
M = N.Qi, 
And the number of maps satistying the required accuracy is as 
follows. 
D=N-M 
Consumer risk, i.e. the probability from a 95% confidence test 
that the number of inaccurate maps ( ny) from sample maps (n), 
can be defined as follows. 
413 
CR= |Z (mCus.nCy)/(NEn)] < 0.05 (for y —0 to n,) 
If the consumer risk is 5%, the producer risk will depend on 
the whole number of maps N and the size of sample n and is 
defined as follows. 
PR 5. [X dun, i0Cy QU] (for y=ny + 1 to D) 
From this equation the optimum size of samples considering 
producer risk can be known when the consumer risk is fixed. 
3. EXPERIMENTS 
In some recent data conversion projects, standard sampling 
methods were suggested with superimposition tests for 10 - 
20 % of the whole number of maps. The following is an 
application of the aforementioned method of considering both 
the consumer risk and producer risk using the hypergeometric 
distribution. 
If the consumer risk is fixed at 5%, the minimum required 
accuracy can be known invariable to the size of the sample and 
from the producer position, an appropriate size of sample can 
be determined at an appropriate producer risk. 
From Fig. 3.1, it can be seen that generally, if 90% for 
minimum required accuracy, 95% for digitizing level and 30 % 
for the producer risk are selected, the appropriate size of 
samples are 83 for 100, 104 for 200 and 121 for 500. As can 
be seen the size of sample decreases relatively as the number 
of whole maps increase. For a size of 100 sheets, the sample 
size is about 80% but for 500 sheets the sample size is only 
25% to make the same test. 
From the consumer position, 90% is selected for minimum 
required accuracy and 97% or 99% is selected for digitizing 
level, and 1 map sheet is the criteria for not passing the test 
from a sample size of 47 from the total 100 sheets, as can be 
seen from Fig. 3.1, the producer risk is 21.8% which makes it 
difficult to pass the test is the overall quality is not high. It can 
be assumed that the both the consumer and the producer can 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B2. Vienna 1996