FISCHER FISCHERSCOPE X-RAY 5000 Series, Руководство оператора, Страница: 107 / 188

Pre-run for determining the control limits Measurement device monitoring for the Fischerscope X-RAY
FISCHERSCOPE
®
X-RAY 107
The standard deviations s
i
of groups 1 to 10 are used to build the
arithmetic mean value s
I
:
(3)
s
I
= (1/m) ·

s
i
The index i runs across the number m of the groups.
The estimated value s
I
serves as the ”current” standard deviation of
the individual readings. The variations between the group mean
values due to drift, interference or changes in the position of the
reference samples during the pre-run are taken into account by
calculating the standard deviation of the mean values
s
II
from the
group mean values x.
i
of the m groups 1 to 10:
(4)
s
II
= {

(x.
i
– x..) 2 /(m-1)} ½
s
II
takes into account the ”extraordinary” influences (potentially
present during the pre-run) on the variations of the group mean
values that may come into existence due to sample re-positioning
for a new group or by changing the measurement location or the
operator – as will also occur during subsequent measurements in
the control chart.
Now, the standard deviations
s
I
and s
II
are used to form the
estimated value of the standard deviation s of the population:
(4)
= (s
I
2 + s
II
2 ) ½
The two limit values UCL(x) (upper control limit for the mean value
x. ) and LCL(x) (lower control limit for the mean value x. ) are the
product of the nominal value x_est and the standard deviation s
with:
(5a) UCL (x) = x_est + 3 · s /

n
(5b)
and
LCL (x) = x_est – 3 · s /

n
n is, as has already been mentioned above, the group size. It has
been taken into account that for every control limit an acceptance
probability of 99.5% applies, i.e., an upper and lower violation of the
control limits occurs
randomly only at a total of 1% of all cases.
An upper control limit UCL(s) applies also for the standard
deviation that is calculated from the estimated standard deviation s
as follows:
(6) UCL (s) = B´
a0
· s = 2.09 · s
The limiting factor B´
a0
is a value that corresponds to the size of the
random sample n, in this case for n = 5, B´
a0
= 2.09.
B´
a0
is arithmetically determined as follows:
(7a)
B´
a0
= 1 + 3 ((1 -C(n)²) ½ / C(n))
with
(7b)
C(n) = 1 - (0.25057315 / n) - (0.20580811 / n²) - (0.202729377 / n
3
)

ˆ