Calibration and Detection Limits
Rüdiger Kaus
Thomas Nagel
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
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Contents
1
Introduction
2
Basics of Calibration
3
Excel-charts for calibration
4
Limits of detection, determination
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
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1 Introduction
Calibration is an important process in
 establishing traceability
 method validation to get the
performance characteristics
 the routine use of modern analytical
equipment
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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What is Calibration?
Calibration is the process of establishing
how the response of a measurement
process varies with respect to the parameter
being measured.
The usual way to perform calibration is to
subject known amounts of the parameter
(e.g. using a measurement standard or
reference material) to the measurement
process and monitor the measurement
response.
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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Calibration has Two Major Aims:
 Establishing a mathematical function
which describes the dependency of the
system’s parameter (e. g. concentration)
on the measured value
 Gaining statistical information of the
analytical system, e. g. sensitivity,
precision
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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Calibration Concepts
 External standard
 Internal standard
 Standard addition
 Definitive calibration methods
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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Goals of Calibration
“Ability to calculate
a (measurement) result
in a secure (safe) working range”
Funk, W., Dammann, V., and Donnevert, G.,
“Quality Assurance in Analytical Chemistry”,
VCH Weinheim 1995
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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First Steps to the Goal:
 Establishing the calibration function
 Choosing the working range
 Measuring several calibration standards
 Linear regression
 Test of non linear regression
 Test of variance homogeneity
 Calculate performance characteristics
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
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Next Steps to the Goal:
 Calculating the (measurement) results
 Conversion of the calibration function
 Reporting the measurement results
 Calculating the statistical limits
 Securing the lower working range
 critical value of detection limit
 calculation of the quantitation limit
 Securing the higher working range
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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2 Basics of Calibration
Mathematical Functions
 simple linear function with intercept in zero origin
y=mx
 linear function with intercept a and slope b
y = ax + b
 quadratic function
y = ax2 + bx + c
 cubic function
y = ax3 + bx2 + ax + d
 exponential function
y = a ebx
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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Principle of Linear Regression
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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Simple Example of Linear Regression
No.
x
y
x2
y2
yx
1
0,5
0,47
0,25
0,221
0,235
0,491
2
1,0
1,01
1,00
1,020
1,010
0,997
3
1,5
1,54
2,25
2,372
2,310
1,503
4
2,0
1,98
4,00
3,920
3,960
2,009
Xm=1,25
ym=1,25
Σ=7,5
Σ=7,533
Σ=7,515
a = 1,012
b = -0,015
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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Simple Example of Linear Regression
No.
x
mg/L
y
1
1,0
37
2
2,0
120
3
3,0
170
4
4,0
205
Xm=2,50
ym=133
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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International Standards
ISO 8466 “Water quality– Calibration and
evaluation of analytical methods and estimation of
performance characteristics
- Part 1: Statistical evaluation of the linear
calibration function”
- Part 2: Calibration strategy for non-linear second
order calibration functions”
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
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3 Excel-charts for Calibration
University of Applied Sciences
Quality Management Manual
Version:
2.0
Date :
proved:
13.04.2001
Part B
Page: 1 von 3 Pages
released:
Ka
Department of Physical Chemistry
Prof Dr. R. Kaus
QA-guideline: QA_04_N10
Calibration and process data
for the determination of
Test data
me thod
Ca libra tion da ta
value 1
value 2
value 3
value 4
value 5
value 6
value 7
value 8
value 9
value 10
number
10
measured at
x
y_1
mg/ L
Area / 1000
Proce ss da ta of a ca libra tion function ... of 1. orde r
0,050 3,060
0,100 3,522
0,150 3,707
0,200 4,280
0,250 5,058
0,300 5,510
0,350 5,703
0,400 6,205
0,450 6,950
0,500 7,178
mean
0,275
5,117
slope
0,342
b=
intercept
0,106
a=
residual standard deviation
s(y) =
process standard deviation
s(x0) =
process variation coefficient
V(x0) =
auxiliary value for the determination of x_P
y_P =
testing value to secure the low er range limit
x_P =
MDL = LC
XN =
detection limit (DIN 32645)
XE =
quantitation limit (DIN 32645)
XB =
k=
lowest
mean
highest
1
2
3
4
5
6
7
8
9
10
Data
factor of
Results
Area / 1000
dilution
mg/L
3,06 0,0581
5,12 0,275
7,18 0,4922
5,00 0,2626
0,995
c=
0,362
9,487
2,508
0,155
0,016
0,060
b=
9,288
a=
2,528
s(y) =
0,166
s(x0) =
0,017
E=
9,487
0,206
Calibration function 1.O: y = 2,508
Sa mple
no.
... 2. order
regressio n 2. Ordnung is significant no t better
Testing linearity:
correlation coefficient
1
1
1
1
1
1
1
1
1
1
1
1
1
0,058
0,275
0,492
0,263
+
9,487
±
±
±
±
mg/L
0,035
0,032
0,035
0,032
t(95%, single-sided) 1,86
t(95%, single-sided) 1,86
t(99%, single-sided) 2,90
t(99%, double-sided) 3,36
x
confidence
interval for
results
±
2,858
0,071
0,057
0,115
0,177
3
Standard uncertainty
rel. in %
0,0351 60,3%
0,0319 11,6%
0,0351 7,1%
0,0319 12,2%
e.g. double probe
M= 2
Standard
relative
expanded
uncertainty
S.U.
S.U.
0,0127
4,8%
0,025
±
±
±
±
±
±
±
±
±
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
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3 Excel-charts for Calibration
University of Applied Sciences
Version:
Quality Management Manual
Department of Physical Chemistry
Prof Dr. R. Kaus
Part B
QA-guideline: QA_04_N10
i_01
i_02
i_03
i_04
i_05
i_06
i_07
i_08
i_09
i_10
x
y
A rea / 1000
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
3,06
3,522
3,707
4,28
5,058
5,51
5,703
6,205
6,95
7,178
number
10
released:
Ka
residual analysis
0,2
residuals
0,15
0,1
0,05
0
-0,05
-0,1
-0,15
-0,2
mean
0,275
13.04.01
proved:
Page: 1 von 3 Pages
data sheet with the complete data
mg/ L
2.0
Date :
-0,25
5,117
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
m g/ L
Outlier testing
F(f1=1,f2=N-2)= 5,317645
F-Test
PW =
1
t-Test VB(yA) = 4,245976
3,616836
data sheet with the reduced data
i_01
i_02
i_03
i_04
i_05
i_06
i_07
i_08
i_09
i_10
x
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
number
10
y
3,06
3,522
3,707
4,28
5,058
5,51
5,703
6,205
6,95
7,178
mean
0,28
data pair with suspected outlier i_03
no outlie r
above
no outlie r
for testing: line
33
0,15
3,707
above above
should be eliminated
Process data of the new linear calibration function (no outlier)
slope
intercept
residual standard deviation
process standard deviation
process variation coefficient
b=
a =
s(y)_a =
s(x0) _a =
V (x0)_a =
9,487
2,508
0,155
0,016
6,0 %
residual analysis
0,2
0,2
0,1
0,1
0,0
-0,1
-0,1
-0,2
-0,2
-0,3
0,05
5,117
0,1
0,15
0,2
0,25mg / L0,3
test datas:
y_P = 2,858
Relevance: Comparing the slopes
0,35
0,4
0,45
0,5
x_P =
0,07138
0,00% deviation
8
7
Area / 1000
6
5
x_P
4
3
2
1
0
0,000
0,100
0,200
0,300
m g/L
0,400
0,500
0,600
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
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in Analytical
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3
Excel-charts for Calibration
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
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3.1
University of Applied Sciences
Excel-charts for calibration
Insert Data
Quality Management Manual
Version:
2.0
Date :
proved:
released:
13.04.2001
Part B
Department of Physical Chemistry
Prof Dr. R. Kaus
QA-guideline: QA_04_N10
Page: 1 von 3 Pages
Calibration and process data
Ka
for the determination of
Test data
part from sheet: Insert data & presenting result
method
Calibration data
... 2. Order
Process data of a calibration function ... of 1 . order
10
0,050
3,060
0,100
3,522
0,150
3,707
0,200
4,280
0,250
5,058
0,300
5,510
0,350
5,703
0,400
6,205
0,450
6,950
0,500
7,178
mean
0 ,2 7 5
5 ,1 1 7
Testing linearity:
slope
0,342
0,106
process variation coefficient
0,995
0,362
9,487
2,508
0,155
0,016
0,060
9,288
2,528
0,166
2,858
0,071
0,057
0,115
0,177
3
auxiliary value for the determination of x_P
testing value to secure the lower range limit
Calibrat ion f unct ion 1 .O: y 2
=,5 0 8
+
3,06 0,0581
5,12
0,275
7,18 0,4922
5,00 0,2626
1
1
1
1
1
1
1
1
1
1
1
1
1
9 ,4 8 7
mg/l
±
0,058
0,275
0,492
0,263
±
±
±
±
t(95%, single-sided) 1,86
rel. in %
0,0351
0,0319
0,0351
0,0319
60,3%
11,6%
7,1%
12,2%
e.g. double probe
M= 2
mg/ L
Standard-
relative
expanded
uncertainty
S.U.
S.U.
0,0127
4,8%
0,025
value 1
±
value 2
±
±
±
±
±
value 3
±
±
±
8,000
7,000
x
t(99%, single-sided) 2,90
t(99%, double-sided) 3,36
Standard uncertainty
... mg/l
0,035
0,032
0,035
0,032
t(95%, single-sided) 1,86
x
confidence
interval for
results
Results
Sample
Nr.
XB
6,000
Calibration data
9,487
0,206
value 4
Calibration
x_P
Area / 1000
5,000
value 5
4,000
----- regression curve
----- confidence interval
3,000
2,000
value 6
1,000
0,000
0,000
0,100
0,200
0,300
0,400
0,500
0,600
mg/l
value 7
value 8
value 9
value 10
number
10
0,050
0,100
0,150
0,200
0,250
0,300
0,350
0,400
0,450
0,500
y_1
sums
Area / 1000
3,060
3,522
3,707
4,280
5,058
5,510
5,703
6,205
6,950
7,178
mean
0,275
5,117
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
in Analytical
in Analytical
Laboratories
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3
3.2
University of Applied Sciences
Excel-charts for calibration
Linear regression and performance characteristics
Quality Management Manual
Version:
2.0
Date :
proved:
released:
13.04.2001
Part B
Department of Physical Chemistry
Prof Dr. R. Kaus
QA-guideline: QA_04_N10
Page: 1 von 3 Pages
Calibration and process data
Ka
for the determination of
Test data
method
Calibration data
... 2. Order
Process data of a calibration function ... of 1 . order
10
0,050
3,060
0,100
3,522
0,150
3,707
0,200
4,280
0,250
5,058
0,300
5,510
0,350
5,703
0,400
6,205
0,450
6,950
0,500
7,178
mean
0 ,2 7 5
5 ,1 1 7
Testing linearity:
slope
0,342
0,106
process variation coefficient
0,995
0,362
9,487
2,508
0,155
0,016
0,060
9,288
2,528
0,166
9,487
0,206
2,858
0,071
0,057
0,115
0,177
3
auxiliary value for the determination of x_P
testing value to secure the lower range limit
Calibrat ion f unct ion 1 .O: y 2
=,5 0 8
+
3,06 0,0581
5,12
0,275
7,18 0,4922
5,00 0,2626
1
1
1
1
1
1
1
1
1
1
1
1
1
9 ,4 8 7
mg/l
±
0,058
0,275
0,492
0,263
±
±
±
±
t(99%, single-sided) 2,90
sums
t(99%, double-sided) 3,36
Standard uncertainty
... mg/l
0,035
0,032
0,035
0,032
t(95%, single-sided) 1,86
x
confidence
interval for
results
Results
Sample
Nr.
t(95%, single-sided) 1,86
rel. in %
0,0351
0,0319
0,0351
0,0319
60,3%
11,6%
7,1%
12,2%
e.g. double probe
M= 2
Standard-
relative
expanded
uncertainty
S.U.
S.U.
0,0127
4,8%
0,025
Process data of a calibration function of 1. order
±
±
±
±
±
±
±
±
±
correlation coefficient
8,000
7,000
XB
6,000
0,995
Calibration
x_P
Area / 1000
5,000
4,000
----- regression curve
----- confidence interval
3,000
slope
b=
intercept
a=
2,000
residual standard deviation
1,000
0,000
0,000
0,100
0,200
0,300
0,400
0,500
s(y) =
0,600
mg/l
process standard deviation
s(x0) =
process variation coefficient
V(x0) =
9,487
2,508
0,155
0,016
0,060
t (95%, single-sided) 1,86
0,206
auxiliary value for the determination of x_P
y_P =
testing value to secure the lower range limit
x_P =
MD L = LC
XN =
de te ction limit (D IN 32645)
XE =
qua ntita tion limit (D IN 32645)
XB =
k=
2,858
0,071
0,057
0,115
0,177
t (95%, single-sided) 1,86
t (99%, single-sided) 2,90
t (99%, double-sided) 3,36
3
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
in Analytical
in Analytical
Laboratories
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Excel-charts for Calibration
3.2 Linear Regression and Performance Characteristics
Performance characteristics of the calibration function (DIN 38402 Teil 51)
slope
b = =SLOPE(C4:C13;B4:B13)
intercept
a = =INTERCEPT(C4:C13;B4:B13)
residual standard deviation
s(y) = =STEYX(C4:C13;B4:B13)
process standard deviation
s(x0) = =s_y/b
process variation coefficient
V(x0) = =s_x0/x_m
Q= =SUMQUADABW(B4:B12)
auxillary value for the determination of x_P
y_P = =a+t*s_y*SQRT(1+1/N+x_m^2/Q)
testing value to secure the lower range limit
x_P = =2*s_x0*t*SQRT(1/N+1+(y_P-y_m)^2/b^2/Q)
XN ==s_x0*t99e*SQRT(1+1/N+x_m^2/Q)
detection limit (DIN 32645)
XB ==k*s_x0*t99z*SQRT(1+1/N+(k*NG-x_m)^2/Q)
quantiation limit (DIN 32645)
k= 3
y P  a  sY  t 
x P  2  s xo  t 
XN  s xo  t 
12
x2
1? ?
x
1  yP a sY ?t ? 1 ? ?
2
N
 ( xiN -x? )(x2i ?x )
2
2?
(
)
y
y
1
1 ? ? ?(
y
y
)
P
2 sxo 2t ? 1 ?P ? 2
1  xP 
bi ?? x(x)i ?2x )2
N
b  N( x
1
XB  k  s xo  t 
1

N
1
( x - x )2
 ( xi -x ) 2
1
( k  XN - x ) 2

N
b 2   ( xi -x ) 2
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
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Excel-charts for Calibration
3.2 Linear Regression and Performance Characteristics
part from sheet: Insert data & presenting result
t (95%, single-sided) 1,86
t (95%, single-sided) 1,86
t (99%, single-sided) 2,90
t (99%, double-sided) 3,36
Do we need tables of statistics?
No, in EXCEL are a lot of functions integrated
Example:
=TINV(0,1;N-2)
= 1,86
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
in Analytical
in Analytical
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3
3.3
Excel-charts for Calibration
Variance Homogeneity (Heteroscedasticity)
from sheet: Calibration (10 values) with test for homogeneity
of the variances
Calibration and process data
Values
unit:
mg/L
extinction
no.
x
y_1
0,05
2
0,1
3
0,15
4
0,2
5
0,25
6
0,3
7
0,35
8
0,4
9
0,45
10
0,5
10 0,275
1
0,05
0,14
y_4
0,140 0,143 0,140 0,146
0,281
0,405
0,535
0,662
0,789
0,916
1,058
1,173
1,303 1,302 1,303 1,304
0,7262
y_5
y_8
y_9
y_10
0,144 0,145 0,144
y_6
y_7
0,146
0,145
0,148
0,1441
1,300 1,296 1,295
1,301
1,296
1,306
1,3006
test for homogeneity of the variances
o.k.
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
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in Analytical
in Analytical
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3
3.4
Excel-charts for calibration
Comparing function of 1. order with function of 2. order
part from sheet: Insert data & presenting result
University of Applied Sciences
Quality Management Manual
Version:
2.0
Date :
proved:
released:
13.04.2001
Part B
Department of Physical Chemistry
Prof Dr. R. Kaus
QA-guideline: QA_04_N10
Page: 1 von 3 Pages
Calibration and process data
Ka
for the determination of
Test data
method
Calibration data
... 2. Order
Process data of a calibration function ... of 1 . order
10
0,050
3,060
0,100
3,522
0,150
3,707
0,200
4,280
0,250
5,058
0,300
5,510
0,350
5,703
0,400
6,205
0,450
6,950
0,500
7,178
mean
0 ,2 7 5
5 ,1 1 7
Testing linearity:
slope
0,342
0,106
process variation coefficient
0,995
0,362
9,487
2,508
0,155
0,016
0,060
9,288
2,528
0,166
9,487
0,206
2,858
0,071
0,057
0,115
0,177
3
auxiliary value for the determination of x_P
testing value to secure the lower range limit
Calibrat ion f unct ion 1 .O: y 2
=,5 0 8
+
3,06 0,0581
5,12
0,275
7,18 0,4922
5,00 0,2626
1
1
1
1
1
1
1
1
1
1
1
1
1
9 ,4 8 7
mg/l
±
0,058
0,275
0,492
0,263
±
±
±
±
sums
t(99%, single-sided) 2,90
t(99%, double-sided) 3,36
Standard uncertainty
... mg/l
0,035
0,032
0,035
0,032
t(95%, single-sided) 1,86
x
confidence
interval for
results
Results
Sample
Nr.
t(95%, single-sided) 1,86
e.g. double probe
M= 2
rel. in %
0,0351
0,0319
0,0351
0,0319
60,3%
11,6%
7,1%
12,2%
Standard-
relative
expanded
uncertainty
S.U.
S.U.
0,0127
4,8%
0,025
±
±
±
±
±
±
±
... 2. order
Process data of a calibration function ... of 1. order
±
±
8,000
7,000
XB
6,000
Calibration
Area / 1000
regression of 2. order is not significantly better
Testing linearity:
x_P
5,000
4,000
correlation coefficient
3,000
----- regression curve
----- confidence interval
2,000
0,995
c=
0,362
9,487
2,508
0,155
0,016
0,060
b=
9,288
a=
2,528
s(y) =
0,166
s(x0) =
0,017
E=
9,487
1,000
slope
0,000
0,000
0,100
0,200
0,300
0,400
mg/l
intercept
0,500
0,600
0,342
b=
0,106
a=
residual standard deviation
s(y) =
process standard deviation
s(x0) =
process variation coefficient
V(x0) =
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
in Analytical
in Analytical
Laboratories
Chemistry
– Teaching
– Training
Material
and Teaching
3
3.5
Excel-charts for calibration
Calculating the function
part from sheet: Insert data & presenting result
Calibration function: y = a+ b *x (+ c* x2)
University of Applied Sciences
Quality Management Manual
Version:
2.0
Date :
proved:
released:
13.04.2001
Part B
Department of Physical Chemistry
Prof Dr. R. Kaus
QA-guideline: QA_04_N10
Page: 1 von 3 Pages
Calibration and process data
Ka
for the determination of
Test data
method
Calibration data
... 2. Order
Process data of a calibration function ... of 1 . order
10
0,050
3,060
0,100
3,522
0,150
3,707
0,200
4,280
0,250
5,058
0,300
5,510
0,350
5,703
0,400
6,205
0,450
6,950
0,500
7,178
mean
0 ,2 7 5
5 ,1 1 7
Testing linearity:
slope
0,342
0,106
process variation coefficient
0,995
0,362
9,487
2,508
0,155
0,016
0,060
9,288
2,528
0,166
9,487
0,206
2,858
0,071
0,057
0,115
0,177
3
auxiliary value for the determination of x_P
testing value to secure the lower range limit
Calibrat ion f unct ion 1 .O: y 2
=,5 0 8
+
Sample
Nr.
3,06 0,0581
5,12
0,275
7,18 0,4922
5,00 0,2626
1
1
1
1
1
1
1
1
1
1
1
1
1
9 ,4 8 7
mg/l
±
0,058
0,275
0,492
0,263
±
±
±
±
sums
t(99%, single-sided) 2,90
t(99%, double-sided) 3,36
Standard uncertainty
... mg/l
0,035
0,032
0,035
0,032
t(95%, single-sided) 1,86
x
confidence
interval for
results
Results
t(95%, single-sided) 1,86
rel. in %
0,0351
0,0319
0,0351
0,0319
60,3%
11,6%
7,1%
12,2%
e.g. double probe
M= 2
Standard-
relative
expanded
uncertainty
S.U.
S.U.
0,0127
4,8%
0,025
±
±
±
±
±
±
±
±
±
8,000
7,000
XB
6,000
Calibration
x_P
Area / 1000
5,000
4,000
----- regression curve
----- confidence interval
3,000
2,000
1,000
0,000
0,000
0,100
0,200
0,300
0,400
0,500
0,600
mg/l
Calibration function 1.O: y = 2,508
+
9,487
x
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
in Analytical
in Analytical
Laboratories
Chemistry
– Teaching
– Training
Material
and Teaching
3
3.6
Excel-charts for calibration
Graphically representation the data
part from sheet: Insert data & presenting result
University of Applied Sciences
Quality Management Manual
Version:
2.0
Date :
proved:
released:
13.04.2001
Part B
Department of Physical Chemistry
Prof Dr. R. Kaus
QA-guideline: QA_04_N10
Page: 1 von 3 Pages
Calibration and process data
Ka
for the determination of
Test data
method
Calibration data
... 2. Order
Process data of a calibration function ... of 1 . order
10
0,050
3,060
0,100
3,522
0,150
3,707
0,200
4,280
0,250
5,058
0,300
5,510
0,350
5,703
0,400
6,205
0,450
6,950
0,500
7,178
mean
0 ,2 7 5
5 ,1 1 7
Testing linearity:
slope
0,342
0,106
process variation coefficient
0,995
0,362
9,487
2,508
0,155
0,016
0,060
9,288
2,528
0,166
9,487
0,206
2,858
0,071
0,057
0,115
0,177
3
auxiliary value for the determination of x_P
testing value to secure the lower range limit
Calibrat ion f unct ion 1 .O: y 2
=,5 0 8
+
3,06 0,0581
5,12
0,275
7,18 0,4922
5,00 0,2626
1
1
1
1
1
1
1
1
1
1
1
1
1
9 ,4 8 7
mg/l
±
0,058
0,275
0,492
0,263
±
±
±
±
t(99%, single-sided) 2,90
sums
t(99%, double-sided) 3,36
Standard uncertainty
... mg/l
0,035
0,032
0,035
0,032
t(95%, single-sided) 1,86
x
confidence
interval for
results
Results
Sample
Nr.
t(95%, single-sided) 1,86
rel. in %
0,0351
0,0319
0,0351
0,0319
60,3%
11,6%
7,1%
12,2%
e.g. double probe
M= 2
Standard-
relative
expanded
uncertainty
S.U.
S.U.
0,0127
4,8%
0,025
±
±
±
±
±
±
±
±
±
8,000
7,000
XB
6,000
Calibration
x_P
Area / 1000
5,000
4,000
----- regression curve
----- confidence interval
3,000
2,000
1,000
0,000
0,000
0,100
0,200
0,300
0,400
0,500
0,600
mg/l
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
in Analytical
in Analytical
Laboratories
Chemistry
– Teaching
– Training
Material
and Teaching
3
3.7
3.8
Excel-charts for calibration
Calculating outliers
Graphically representation the outliers
part from sheet: Outlier
residual analysis
University of Applied Sciences
Version:
Quality Management Manual
Department of Physical Chemistry
Date :
Prof Dr. R. Kaus
Part B
QA-guideline: QA_04_N10
x
y
mg/ L
A rea / 1000
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
3,06
3,522
3,707
4,28
5,058
5,51
5,703
6,205
6,95
7,178
number
10
released:
Ka
residual analysis
0,2
residuals
0,15
0,1
0,05
0
-0,1
-0,15
-0,2
-0,25
5,117
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
m g/ L
Outlier testing
F(f1=1,f2=N-2)= 5,317645
F-Test
PW =
1
t-Test VB(yA) = 4,245976
3,616836
data sheet with the reduced data
i_01
i_02
i_03
i_04
i_05
i_06
i_07
i_08
i_09
i_10
x
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
number
10
y
3,06
3,522
3,707
4,28
5,058
5,51
5,703
6,205
6,95
7,178
mean
0,28
data pair with suspected outlier i_03
no outlie r
above
no outlie r
for testing: line
33
0,15
3,707
above above
should be eliminated
slope
intercept
residual standard deviation
process standard deviation
process variation coefficient
b=
a =
s(y)_a =
s(x0) _a =
V (x0)_a =
9,487
2,508
0,155
0,016
6,0 %
-0,2
-0,25
residual analysis
0,2
0,2
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
0,1
0,1
0,0
m g /L
-0,1
-0,1
-0,2
-0,2
-0,3
0,05
5,117
0,1
0,15
0,2
0,25mg / L0,3
0,35
0,4
0,45
0,5
x_P =
0,07138
0,00% deviation
8
data pair with suspected outlier
i_03
no outlier
above
no outlier
for testing: line
33
7
6
Area / 1000
0
-0,05
-0,1
-0,15
Process data of the new linear calibration function (no outlier)
test datas:
y_P = 2,858
Relevance: Comparing the slopes
5
0,2
0,15
0,1
0,05
-0,05
mean
0,275
13.04.01
proved:
Page: 1 von 3 Pages
data sheet with the complete data
i_01
i_02
i_03
i_04
i_05
i_06
i_07
i_08
i_09
i_10
2.0
x_P
4
3
2
0,15
3,707
above
above
should be eliminated
1
0
0,000
0,100
0,200
0,300
m g/L
0,400
0,500
0,600
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
in Analytical
in Analytical
Laboratories
Chemistry
– Teaching
– Training
Material
and Teaching
3
3.7
3.8
Excel-charts for calibration
Calculating outliers
Graphically representation the outliers
part from sheet: Outlier
University of Applied Sciences
Version:
Quality Management Manual
Department of Physical Chemistry
Part B
QA-guideline: QA_04_N10
i_01
i_02
i_03
i_04
i_05
i_06
i_07
i_08
i_09
i_10
y
mg/ L
A rea / 1000
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
3,06
3,522
3,707
4,28
5,058
5,51
5,703
6,205
6,95
7,178
number
10
Page: 1 von 3 Pages
released:
0,2
Ka
residual analysis
0,2
0,1
residuals
0,15
0,1
0,1
0,05
0
0,0
-0,05
-0,1
-0,1
-0,15
-0,2
mean
0,275
0,2
13.04.01
proved:
data sheet with the complete data
x
2.0
Date :
Prof Dr. R. Kaus
-0,1
-0,25
5,117
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
m g/ L
F(f1=1,f2=N-2)= 5,317645
F-Test
PW =
1
t-Test VB(yA) = 4,245976
3,616836
data sheet with the reduced data
i_01
i_02
i_03
i_04
i_05
i_06
i_07
i_08
i_09
i_10
x
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
number
10
y
3,06
3,522
3,707
4,28
5,058
5,51
5,703
6,205
6,95
7,178
mean
0,28
data pair with suspected outlier i_03
no outlie r
above
no outlie r
for testing: line
33
-0,2
0,15
3,707
above above
should be eliminated
-0,2
Process data of the new linear calibration function (no outlier)
slope
intercept
residual standard deviation
process standard deviation
process variation coefficient
b=
a =
s(y)_a =
s(x0) _a =
V (x0)_a =
9,487
2,508
0,155
0,016
6,0 %
-0,3
-0,2
-0,3
0,05
0,1
0,15
8
0,2
0,35
0,4
0,45
Area / 1000
x_P
4
Area / 1000
6
6
x_P
5
3
2
1
0,200
0,25
0,3
0,35
0,4
0,45
0,5
0,300
m g/L
4
3
2
x_P =
0,07138
0,00% deviation
0,5
x_P =
0,07138
0,00% deviation
7
0,100
0,2
8
0,25mg / L0,3
7
8
0
0,000
0,15
test datas:
y_P = 2,86
Relevance: Comparing the slopes
0,1
0,1
0,0
-0,1
-0,1
-0,2
test datas:
y_P = 2,858
Relevance: Comparing the slopes
5
0,1
residual analysis
0,2
0,2
5,117
mg/ L
0,05
Area / 1000
Outlier testing
0,400
0,500
0,600
7
6
5
x_P
4
3
2
1
0
0,000
0,100
0,200
0,300
0,400
0,500
0,600
mg/L
1
0
Kaus, R.,Titel
Author:
Nagel, T.: Calibration
and Detection
Limits
Berlin
Heidelberg 2003
0,000
0,100
0,200
0,300
0,400 © Springer-Verlag
0,500
0,600
mg/L
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
in Analytical
in Analytical
Laboratories
Chemistry
– Teaching
– Training
Material
and Teaching
3
3.9
Excel-charts for calibration
Calculating analytical results
part from sheet: Insert data & presenting result
The analytical results will be calculated by the inverse
of the calibration function: x = (y-a)/b
University of Applied Sciences
Quality Management Manual
Version:
2.0
Date :
proved:
released:
13.04.2001
Part B
Department of Physical Chemistry
Prof Dr. R. Kaus
QA-guideline: QA_04_N10
Page: 1 von 3 Pages
Calibration and process data
Ka
for the determination of
Test data
method
Data
Calibration data
... 2. Order
Process data of a calibration function ... of 1 . order
10
0,050
3,060
0,100
3,522
0,150
3,707
0,200
4,280
0,250
5,058
0,300
5,510
0,350
5,703
0,400
6,205
0,450
6,950
0,500
7,178
mean
0 ,2 7 5
5 ,1 1 7
Testing linearity:
slope
0,342
0,106
process variation coefficient
0,995
0,362
9,487
2,508
0,155
0,016
0,060
9,288
2,528
0,166
9,487
lowest
0,206
2,858
0,071
0,057
0,115
0,177
3
auxiliary value for the determination of x_P
testing value to secure the lower range limit
Calibrat ion f unct ion 1 .O: y 2
=,5 0 8
+
3,06 0,0581
5,12
0,275
7,18 0,4922
5,00 0,2626
1
1
1
1
1
1
1
1
1
1
1
1
1
9 ,4 8 7
mg/l
±
0,058
0,275
0,492
0,263
±
±
±
±
... mg/l
0,035
0,032
0,035
0,032
t(95%, single-sided) 1,86
t(95%, single-sided) 1,86
mean
t(99%, single-sided) 2,90
t(99%, double-sided) 3,36
highest
x
confidence
interval for
results
Results
Sample
Nr.
rel. in %
0,0351
0,0319
0,0351
0,0319
60,3%
11,6%
7,1%
12,2%
Standard uncertainty
1
e.g. double probe
M= 2
2
Standard-
relative
expanded
uncertainty
S.U.
S.U.
0,0127
4,8%
0,025
±
3
±
4
±
±
±
±
5
±
±
±
6
8,000
7,000
XB
6,000
Calibration
7
x_P
5,000
Area / 1000
8
4,000
----- regression curve
----- confidence interval
3,000
9
2,000
1,000
0,000
0,000
Sample
no.
0,100
0,200
0,300
0,400
0,500
0,600
10
mg/L
Area / 1000
3,06 0,0581
5,12 0,275
7,18 0,4922
5,00 0,2626
confidence
interval for
results
Results
1
1
1
1
1
1
1
1
1
1
1
1
1
sums
0,058
0,275
0,492
0,263
±
±
±
±
±
mg/L
0,035 0,035
0,032 0,032
0,035 0,035
0,032 0,032
±
±
±
±
±
±
±
±
±
mg/l
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
Quality Assurance
Quality Assurance
in Analytical
in Analytical
Laboratories
Chemistry
– Teaching
– Training
Material
and Teaching
3
3.10
Excel-charts for calibration
Estimating the uncertainty
part from sheet: Insert data & presenting result
University of Applied Sciences
Quality Management Manual
Version:
2.0
Date :
proved:
released:
13.04.2001
Part B
Department of Physical Chemistry
Prof Dr. R. Kaus
QA-guideline: QA_04_N10
Page: 1 von 3 Pages
Calibration and process data
Ka
for the determination of
Test data
method
Calibration data
... 2. Order
Process data of a calibration function ... of 1 . order
10
0,050
3,060
0,100
3,522
0,150
3,707
0,200
4,280
0,250
5,058
0,300
5,510
0,350
5,703
0,400
6,205
0,450
6,950
0,500
7,178
mean
0 ,2 7 5
5 ,1 1 7
Testing linearity:
slope
0,342
0,106
process variation coefficient
0,995
0,362
9,487
2,508
0,155
0,016
0,060
9,288
2,528
0,166
9,487
0,206
2,858
0,071
0,057
0,115
0,177
3
auxiliary value for the determination of x_P
testing value to secure the lower range limit
Calibrat ion f unct ion 1 .O: y 2
=,5 0 8
+
3,06 0,0581
5,12
0,275
7,18 0,4922
5,00 0,2626
1
1
1
1
1
1
1
1
1
1
1
1
1
9 ,4 8 7
mg/l
±
0,058
0,275
0,492
0,263
±
±
±
±
rel. in %
0,0351
0,0319
0,0351
0,0319
60,3%
11,6%
7,1%
12,2%
e.g. double probe
M= 2
Standard-
relative
expanded
uncertainty
S.U.
S.U.
0,0127
4,8%
0,025
±
±
±
±
±
±
Standard-
relative
expanded
uncertainty
S.U.
S.U.
0,0127
4,8%
0,025
±
±
XB
6,000
e.g. double measurement
M=
2
±
8,000
7,000
sums
t(99%, single-sided) 2,90
t(99%, double-sided) 3,36
Standard uncertainty
... mg/l
0,035
0,032
0,035
0,032
t(95%, single-sided) 1,86
x
confidence
interval for
results
Results
Sample
Nr.
Standard uncertainty
t(95%, single-sided) 1,86
Calibration
x_P
Area / 1000
5,000
4,000
----- regression curve
----- confidence interval
3,000
2,000
1,000
0,000
0,000
0,100
0,200
0,300
0,400
0,500
0,600
mg/l
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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4
Limits of Detection, Determination
4.1
Limit of Detection
4.2
Limit of Determination (Quantitation)
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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4. Limits of Detection, Determination
4.1 Limit of Detection
Detection limit
DIN 32645
from blanks
from calibration data
Funk
dynamic model
xD 
sL
1 1
 t f;α 

b
m n
x D  s x 0  t f;α 
1 1 x2
 
m n Qx
1
yc  b  s y  t 
1
n
0 - x 
n
x
i 1
- x
2
1
xD  2 

1
a
n
n
2
a 2    xi - x 
2
i 1
 δαβ,v σ0  K 2t1-α,v σ0 K
xD  

 
A
I
 A  I
2
σ 
σ 
σA 

K  1  r(B,A)  B   t1-α,v A 
I  1 -  t1-α,v

σ0 
A
A

IUPAC
Coleman
i
 yc - y 
sy  t
2
Sc  t1-α,vs 0
recursive formula
1
1

2 2 
2 2


 1 x 

s 
1  xD - x 
x D    t n -2,1-α 1  
 
  t n - 2,1-β 1  
a 
Sxx
 n
 
 n Sxx 


1
x D  DL H - V 
explicit formula
A
a
s  t n -2,1-β
G  -2AB  Sxx

-J  J 2 - 4HK

2
2H
1
t n - 2,1-α  1 x 2  2
B
1  

t n - 2,1-β  n Sxx 
H  A 2  S-xx1
J  G  2x

 1 
F   B2 - 1     Sxx
 n 

K  F - x2
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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4. Limits of Detection, Determination
4.1 Limit of Detection
4.2 Limit of Determination (Quantitation)
A) DIN 32645
Detection limit
xD =
yk - a
1 1 x2
= s x0  t f;a 
+ +
b
m n Qx
by fast estimation:
x D = 1, 2  Φn;α  sx0
Capability limit
Determination limit
x C = x NG + s x0  t f;β 
x DT
1 1 x2
+ +
m n Qx
1 1  xD - x 
= k  s x0  t f;α 
+ +
m n
Qx
2
by fast estimation
x DT  = 1, 2  k  Φn;α  s x0
Factor for fast estimation
Φn;α = t f;α  1+
1
n
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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4. Limits of Detection, Determination
4.1 Limit of Detection
4.2 Limit of Determination (Quantitation)
B) Funk
Detection limit
dynamic model
1
yc  b  s y  t 
1
n
0 - x 
n
  xi - x 
2
1
xD  2 

1
a
n
i 1
Determination limit
 yc - y 
sy  t
2
2
n
a 2    xi - x 
2
i 1
dynamic model
xc 
sy  t
a

1
1
n
x2
n
x
i 1
x DT
i
- x
y - b sy  t 1
 h


1
a
a
n
1
yh  b  2  s y  t 
1
n
2
 xc - x 
n
x
i 1
 yh - y 
n
i
2
- x
2
2
a 2    xi - x 
2
i 1
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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4. Limits of Detection, Determination
4.1 Limit of Detection
4.2 Limit of Determination (Quantitation)
C) IUPAC
Detection limit
Sc  t1-α,vs 0
 δαβ,v σ0  K 2t1-α,v σ0 K
xD  

 
A
I
 A  I
K  1  r(B,A) 
σB 
σ 
  t1-α,v A 
σ0 
A
σ 

I  1 -  t1-α,v A 
A

2
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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4. Limits of Detection, Determination
4.1 Limit of Detection
4.2 Limit of Determination (Quantitation)
D) Coleman/HUBAUX-VOS model
Detection limit
recursive formula
1
1

2 2 
2 2


 1 x 

s 
1  xD - x 
x D    t n -2,1-α 1  
 
  t n - 2,1-β 1  
a 
Sxx
 n
 
 n Sxx 


1
x D  DL H - V 

-J  J 2 - 4HK

2
2H
explicit formula
A
a
s  t n -2,1-β
G  -2AB  Sxx
1
t
 1 x2 2
B  n - 2,1-α 1  

t n - 2,1-β  n Sxx 
H  A 2  S-xx1
J  G  2x

 1 
F   B2 - 1     Sxx
 n 

K  F - x2
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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4.3
Limits of Detection, Determination
Critical Discussion
FUNK et. al.
# BE ZUG!
P rüfgröße

detection limit
y_P =
quantita tion limit
BG =
x_P =
FUNK et. al.
P rüfgröße

detection limit
Figure 1:
291 4
0,086 2
0,1250
t (95%, onesided) 1,8 6
t (95%, onesided) 1,8 6
mg/l
# BE ZUG!
y_P =
x_P =
291 4
0,086 2
t (95%, onesided) 1,8 6
t (95%, onesided) 1,8 6
The values are calculated with the formulas from Funk’ s book [6] in an
quantita tion limit
BG =
0,1250
mg/l
Excel-
sheet
Figure 1:
The values
calculated
with with
the formulas
from Funks
book
in an
EXCEL[6]
sheet
The
valuesare
are
calculated
the formulas
from
Funk’
s book
in an
Excel-
sheet
DIN 32645
Nachw eisgrenze

d etection limit
Bestimmung sgrenze

q uant itation limit
NG =
BG =
k =
3
0,070
0,212
mg/l
mg/l
t(99%,einseit ig)2, 90
t (99%,zweiseit ig)3, 36
DIN 32645
Nachw eisgrenze

d etection limit
Bestimmung sgrenze

q uant itation limit
NG =
BG =
k =
3
0,070
0,212
mg/l
mg/l
t(99%,einseit ig)2, 90
t (99%,zweiseit ig)3, 36
The values are calculated with the formulas from DIN 32645 in an EXCEL sheet
Figure 2:
Figure 2:
The values are calculated with the formulas from DIN 32645 [5] in an Excel-sheet
The values are calculated with the formulas from DIN 32645 [5] in an Excel-sheet
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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4.3
Limits of Detection, Determination
Critical Discussion
Which Values of Detection limits and Quantitation limits
are correct?
Choosing a confidence range for the quantitation limit:
recommendation of the DIN 32645: k=3
+/- 33% ;
recommendation of the IUPAC:
+/- 10%
1/k=0,1
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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4.3
Limits of Detection, Determination
Critical Discussion
Detection limits from blanks - problems with
the normal distribution
detection limits from blanks give very low values,
but
- blanks don’t belong to the same statistically
population as the calibration and measuring data
- often they are normally distributed
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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4.3
Limits of Detection, Determination
Critical Discussion
Determination of the detection limit from blanks
(with test of normal distribution)
Sample-Nr.
1
2
3
4
5
6
7
8
9
10
number
10
Bl ank
33 9
19 3
19 6
20 3
37 6
0
28 8
0
0
0
IS
51559 2
51842 1
51557 2
52251 4
52988 7
52781 9
52557 1
52730 9
56354 4
51763 0
Bla nk/IS
0,0 00657 5
0,00 03722 8
0,00 03801 6
0,00 03885 1
0,00 07095 9
0
0,00 05479 8
0
0
0
mea n
standard deviatio n
Blan k/IS
sorte d
0
0
0
0
0 ,00037 2
0 ,00038 0
0 ,00038 9
0 ,00054 8
0 ,00065 7
0 ,00071 0
0 ,00030 6
0 ,00028 6
for:
Naphthalene
x-x_m
-0,0003 1
-0,0003 1
-0,0003 1
-0,0003 1
0,0000 7
0,0000 7
0,0000 8
0,0002 4
0,0003 5
0,0004 0
u =(x-x_m)/s
-1,06 8
-1,06 8
-1,06 8
-1,06 8
0,23 3
0,26 1
0,29 0
0,84 7
1,23 0
1,41 2
uexpecte d
-1,53 8
-0,99 9
-0,65 9
-0,3 8
-0,12 1
0,12 1
0,3 8
0,65 9
0,99 9
1,53 8
2,000
1,500
u measured
1,000
0,500
0,000
-0,500
-1,000
-1,500
-2,000
-2
-1,5
-1
-0,5
0
u exp ected
standard dev iation
slo pe from the clal ibration curve
detection limit
referred to the sample weig ht
0,5
1
1,5
2
0 ,00028 6
0,76 2
0 ,00065 4
0,0024 5
mg /l
mg/kg TS
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
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4.3
Limits of Detection, Determination
Critical Discussion
Quantitation limits and working range
Quantitation limits are often higher than some of the
calibration data
(in the procedure suggested by Funk the quantitation limit is
always higher than the 1st calibration point).
Now there is the difficulty:
Which is the lowest concentration I'm allowed to record?
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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4.3
Limits of Detection, Determination
Critical Discussion
Fa chhochsc hul e Nie derr hein
Q ua lity man ag em ent
Version : V _1. 0
Pa ge : 1 v on 4 p ag es
30.10.97
Date:
p repared by :
Fac hbere ich Chem ie
P rof. Dr . R. Ka us
Calibr ation
K aus
Ca rbo n
(m eas ured data are ta ke n from DIN 32 6 45)
s lop e
inte rce pt
res idu al sta nda rd dev iatio n
proc es s sta nda rd dev iatio n
pro ce ss v aria tio n coef fic ie nt
966 2
248 1
19 2
0 ,019 9
7,2 4%
0, 2062 5
291 4
0 ,086 2
a ux ili ar y va lu e fo r th e d e te rmi n atio n o f x_ P
te sti ng va lu e to se cu re the lo we r r a ng e lim it
XN =
XB =
k =
3
t( 95 %, si n g le -s id e d )
1, 86
t( 95 %, si n g le -s id e d )
1, 86
t( 99 %, si n g le -s id e d )
2, 90
t(9 9 %, d o u b le -s id e d )
3, 36
8 000
C alibra tion
__ XB
(F U NK )
7 000
XB
(D IN )
x_ P
6 000
5 000
Area
----- r eg re ssio n cu rve
----- con fiden ce int er val
4 000
3 000
2 000
1 000
0
0
0,1
0,2
0,3
0,4
0,5
0,6
mg/ l
Kaus, R.,Titel
Author:
Nagel, T.: Calibration and Detection Limits
© Springer-Verlag Berlin Heidelberg 2003
In: Wenclawiak, Koch, Hadjicostas (eds.)
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