161 
3. PRACTICAL EXAMPLE AND CONCLUSIONS 
E[sl] = o * 2 B 
var[s 2 b ]=- 
(9) 
The first one simply confirms what has been observed 
about (6) and (7); it also shows that our formal develop 
ment is correct; the second highlights the dependence of 
the variance of in respect of . In other words: 
is a correct estimator of the variance of B because its 
mean value coincides with , but the goodness of the 
estimation depends inversely on 
Another well known way of understanding the properties 
of an estimator is to study its confidence interval at a 
certain confidence level . From (8) it is easily obtain 
able 
2 Si U 2 
V 2 < â <y 2 
Xn-a 2 — 2 . _ 2 ~ Xn\ 1- 
G , +G„ 
P = 1 
where xl a / and xl \-a/ are two real numbers with the 
following properties 
P [°>X fl 2 ;«/ 2 ]IX n 2 =Y 
P [^Xn^lXn =1 — 
The confidence intervals for the estimated variance and 
standard deviation of B are easily calculated 
Y 2 ■ 
A./»; a/2 
n 
—g 2 <5]<x„ 2 i^/ 2 
_ 2 , _ 2 
°A + °B „2 
A 
_ 2 2 
2 °A +Cy B _ 2 
A»w;a/2 
n 
v 2 ÎLl 
A.n, l-a/2 
The role played by the size of the variance of the instru 
ment A is also shown in this case: the bigger it is, the 
larger the confidence interval becomes. So it confirms the 
rule that high values of mean low quality estimation 
of the variance of B. In the next section a practical exam 
ple will clarify the orders of magnitude of the phenome 
non. 
For a research job devoted to the determination of the 
precision of a DPW at different resolutions, the position 
of 25 control points has been measured with an analytical 
stereoplotter. The operator has completed not only one 
cycle of orientation and point measurements, but twelve 
cycles, so to determine the precision of the analytical 
measurements. For the X component the standard devia 
tion was <3 x =6. cm; having twelve independent meas 
urements, mean values have been calculated for each 
point 
i 12 
=-y 
12 jS 
Ai 
Their estimated standard deviation of the analytically 
measured coordinates is g . = 
VÎ2 
= 1.9 cm. The 
DPW measurements of the X component, at the resolu 
tion of 300 dpi, have shown an estimated standard devia 
tion s B = 21. cm; this will be assumed as the true value 
for the instrument B ( g b = 21. ). 
The following picture shows the confidence interval 
width of the estimated standard deviation as a func 
tion of ; the value of and n are kept fixed; the con 
fidence level is a = 0.0 . 
Length of the confidence interval of S 0 as a function of CTA 
The interval width obviously increases with ; there is 
a lower limit, corresponding to the value g a , that can 
be improved only by increasing the number of the points, 
n\ this limit is unfortunately high in our case, because the 
standard deviation estimation improves very slowly, in 
respect to the number of the measurements.