Metrohm 746 VA Trace Analyzer - Manual (Page 486)

6.9 Content calculation
746 VA Trace Analyzer / 747 VA Stand
6-23
As a consequence of the weighting of the least squares, the contribution of the
measured points x
ii
, y
ii
for the determination of the curve parameters differs in ac-
cordance with the position of y
ii
. With y
ii
> 15 nA, the influence on the calibration
curve becomes smaller, the greater y
ii
.
The calculated calibration curve is used in subsequent measurements to determine
the associated result x
M M
from the mean value
–
y
M M
of the m measured quantities y
M M
.
Mean value
–
y
M M
and scatter σ
y,M y,M
of the individual values are defined as follows:
The estimation of the total error σ
xx
of the result x
M M
is performed by the 746 VA Trace
Analyzer with a linear error calculation which takes into account both the error
contribution from the measurement and that from the calibration. As the two contri-
butions are statistically independent, their variances σ 2 and not the individual errors
σ are added:
( σ
x x
) 22 = ( σ
x,M x,M
) 22 + ( σ
x,C x,C
) 22
The error contribution from the actual measurement is calculated from the x, y de-
rivative of the calibration function, the measured scatter σ
y,M y,M
and the Student factor
t
M M
as follows:
For the calculation of the error contribution from the calibration, the errors of the in-
dividual parameters of the calibration function used are determining. As these pa-
rameters z
rr
(a, b, c) are statistically dependent on one another, here all covariances
cov (z
rr
, z
ss
) must be taken into account (t
C C
is again the Student factor):
In measurements with the 746 VA Trace Analyzer, from the statistical point of view
only small samples (<10) are determined from a population with gaussian distribu-
tion. These samples have a Student distribution. Both the error contribution from
the measurement and that from the calibration are thus multiplied by the Student
factor t 2 . This factor depends on the number of measurements n and the number of
degrees of freedom f and is defined for a probability of 68.3% as follows:
n – f t n – f t n – f t f
1
2
3
4
5 1.837
1.321
1.197
1.142
1.111 6
7
8
9
10
1.091
1.077
1.067
1.059
1.053 15
20
30
50
100
1.035
1.026
1.017
1.010
1.005 t
M M
t
C C
for y = bx
t
C C
for y = a + bx
t
C C
for y = bx + cx 4
t
C C
for y = a + bx + cx 4
1
1
2
2
3
The total error σ
xx
of the result x
M M
consequently gives the range x
M M
± σ
xx
in which the
result x
M M
may be expected with a probability of 68.3%.
( y
M,i M,i
– ^ ^ y
M M
) 2
m m
Σ
i = 1 i = 1
m – 1
σ
y,M y,M
=
–
y
M M
=
) ( 22
( σ
x,M x,M
) 22 = (t
M M
) 22 ( σ
y,M y,M
) 22
m
1
y
M,i M,i
m m
Σ
i = 1 i = 1
∂ y
∂ x
cov (z
rr
, z
ss
) ( σ
x,C x,C
) 22 = (t
C C
) 22
Σ
r,s r,s
∂ z
rr
∂ x
∂ z
ss
∂ x