U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Basic Course Experiments to Demonstrate
Intercomparisons
(1) Description of underlying experiments
(2) Statistical evaluation of data
(3) Teaching material for subsequent tutorials
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Experimental
• volumetric method: titration of aliquots (25 mL) of an
aqueous solution of NaOH (c  0.01 mol L-1) with a
standard solution of H2SO4 (c1 = 0.1 mol L-1,c2 = 0.0100
mol L-1) employing a manual 50 mL-burette
• each student analyzes aliquots of the same solution
• each student repeats the titration three times with each
standard solution
 six titrations per student
• the results are handed out to supervisors via tables
containing the consumed volume of standard solution
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
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Statistical Data Evaluation: Formation of Classes
set of 54 experiments
N
Gaussian
normal distribution?
30
25
20
15
10
5
0
-2
25 26 27 28 29 30 31 32 33 34 35 35 37 38 39 V/(10 mL)
histogram
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Statistical Data Evaluation: the Probability
Function
cumulative frequency
60
N
50
40
30
20
10
0
25 26 27 28 29 30 31 32 33 34 35 35 37 38 39 V/10-2 mL
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Statistical Data Evaluation: the Probability
Function
normalized cumulative frequency
1
P(x)
0
25 26 27 28 29 30 31 32 33 34 35 35 37 38 39 x
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
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Basic Course Experiments to Demonstrate Intercomparisons
Normal Probability Plot
• special graph paper:
normal probability paper
• straight line
normal distribution

99,999
P(V)
99,5
95
70
40
10
1
28
30
32
34
36
V/mL
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
U. Pyell
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Basic Course Experiments to Demonstrate Intercomparisons
Probability Equation: Mathematical Description
1
F ( z) 
 2
z
e
( x   )2

22
dx
0
Probability = (z, , )
Density of probability = (x, , )
Distribution function
1
p( x ) 
e
 2
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
( x   )2

2 2
U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
The Gaussian Distribution Function
1
p( x ) 
e
 2

( x   )2
22
two parameters:
 = maximum
 = spread
In most cases repeated measurements of a single quantity are
normally distributed (after elimination of outliers).
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
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Basic Course Experiments to Demonstrate Intercomparisons
Calculation of Estimates from Experimental Data

x
n
arithmetic mean

n
x =
i 1 i
n =
x = estimate of 
x
i 1 i
n
sum of all measurements;
number of measurements
2
(
x

x
)
i  1 i
n
standard deviation s = estimate of  s 
relative standard deviation (RSD) = (s/
(n  1)
x )  100 
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Influence of Experimental Parameters on
Standard Deviation
(manual 50 mL-burette)
Concentration of standard solution:
s
RSD
N
c1 = 0.1 mol L-1
3.21 mL
0.118 mL
3.67 %
54
c2 = 0.0100 mol L-1
25.71 mL
0.249 mL
0.97%
54
The RSD is a measure of the precision of a method.
The precision of a method can be improved by variation of experimental parameters.
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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Basic Course Experiments to Demonstrate Intercomparisons
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Improving Precision by Repeated Measurements
18 students, each student performed three titrations (N=54)
 arithmetic mean of three results
c
0.1 mol L-1
0.0100 mol L-1
x
3.22 mL
0.118 mL
3.66 %
25.70 mL
0.249 mL
0.97%
3.22 mL
0.0862 mL
2.68 %
25.70 mL
0.215 mL
0.84%
s
RSD
xM
sM
RSDM
The RSD is a measure of the precision of a method. The precision of a method can be
improved by forming the arithmetic mean of the results of repeated measurements.
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Standard Deviation of the Mean
mean of a sample of measurements
 estimate of the true value 
in case of no systematic deviation:
 = quantity to be measured
standard error of the mean = standard deviation of the mean sM

n
sM 
i 1
( xi  x )
n (n  1)
2
s
sM 
n
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Normal Distribution with Different Spread
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Analyzing the Measurement Uncertainty
(1) random deviation:
volume uncertainty volumetric flask, volume uncertainty burette,
reading uncertainty volumetric flask, reading uncertainty burette
(dominating when using not-appropriate standard solution),
individual uncertainties
(2) systematic deviation:
i.e. uncertainty of concentration of standard solution, cannot be
reduced by forming the arithmetic mean of the results of repeated
measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Definition: Precision
Precision = closeness of agreement between independent test
results obtained under stipulated conditions
(ISO 3534-1, 1993)
• high precision  low standard deviation
• low precision  large standard deviation
• estimate of the precision does not consider the
deviation of the arithmetic mean of a series of
results from the true value
• precision can be estimated if the true value
is not known
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Definition: Accuracy
Accuracy = closeness of agreement between the result of a
measurement and the true value of the measurand
(International vocabulary .....,1984)
• accuracy is not only given by the spread of a normal
distribution, but also by the deviation of the arithmetic
mean of a series of results from the true value
• accuracy can only be determined if the true value
is known
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Confidence Limits of the Mean
confidence interval
= range within, with given probability, the true value lies
confidence limits
= extreme values of the confidence range
x t
s
x t
n
s
n
t is a factor that depends both on the degree of confidence required
and the degrees of freedom (n - 1)
The confidence limit is a measure of the precision of a result. The precision of a
result can be improved by repeated analysis.
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Confidence Limits of the Mean
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
Presentation of Analytical Results (1)
No quantitative experimental value is of any value unless it is accompanied
by an estimate of the uncertainty involved in its measurement.
Possibilities of presentation
(1)
x as estimate of the quantity measured,
s as estimate of the precision
(2)
x as estimate of the quantity measured,
95% confidence limit as estimate of the
precision of the measurement
no universal convention
 the form used has to be stated, n has to be given
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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U. Pyell
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Basic Course Experiments to Demonstrate Intercomparisons
Presentation of Analytical Results (2)
The number of significant figures given indicates the precision of a value.
significant figures: all digits which are certain plus the first uncertain one
number of
measurements
In our case
result of titration: c(NaOH) = 0.00972 mol L-1 (0.000010 mol L-1) n = 54
arithmetic mean
three significant figures
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
standard
deviation
U. Pyell
Basic Course Experiments to Demonstrate Intercomparisons
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Summary
Learning objectives
• statistical deviation  systematic deviation
• Gaussian normal distribution, statistical evaluation
of data
• improving the precision of a method
• presentation of data
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000