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2010)
k
ni
k
ni
1 XX
s =
( yi j − ȳ)2
n − 1 i=1 j=1
2
1 XX
=
( yi j − ȳ + ȳi − ȳi )2
n − 1 i=1 j=1
=
k ni
2
1 XX
( ȳi − ȳ) + ( yi j − ȳi )
n − 1 i=1 j=1
=
k ni
1 XX
( ȳi − ȳ)2 + ( yi j − ȳi )2 + 2( ȳi − ȳ)( yi j − ȳi )
n − 1 i=1 j=1
s2b
2
sw
}|
{ z
}|
{
k
k
X
X
1
1
ni ( ȳi − ȳ)2 +
(ni − 1)si2 +
=
n − 1 i=1
n − 1 i=1
z
(B.2)
k ni
2 XX
( ȳi − ȳ)( yi j − ȳi )
+
n − 1 i=1 j=1
ni
ni
k X
Š
€X
2 X
2
2
ȳi
yi j −
= s b + sw +
( ȳi − ȳ)
n − 1 i=1
j=1
j=1
k
2 X
+
( ȳi − ȳ)(ni ȳi − ni ȳi ) ⇒
n − 1 i=1
=
s2b
2
+ sw
s=
Ç
2
s2b + sw
Therefore, the uncertainty in repeatability (precision) is computed from equation
v
u
k
u 1 X
uran = s b = t
ni ( ȳi − ȳ)2
n − 1 i=1
(B.3)
The standard deviation within samples is the within sample sigma (noise), sw , computed from
equation
v
u
k
u 1 X
t
sw =
(ni − 1)si2
n − 1 i=1
(B.4)
and represents the noise in COP measurements. It is used later to obtain the error variance for the
smoothing process.
The overall mean ( ȳ) and its standard deviation (s) are computed using the following equations
ni
1 X
ȳi =
yi j
ni j=1
and
v
u
u
si = t
ni
1 X
( yi j − ȳi )2 ,
ni − 1 j=1
153
with i = 1, . . . , k.