Calculaion of Measurement Uncertainty in
Pipette Calibration with QMSYS GUM Software
Stefan Golemanov
QUALISYST Ltd., Bulgaria
Email: Stefan@qsyst.com
Abstract
Pipette calibration is the most essential part of every laboratory, and as such, requires careful understanding and selection of calibration
methods and procedures in order to prevent errors, deviations and inconsistencies. This article provides an example for uncertainty
analysis in pipette calibration with all relevant steps in equation modeling and uncertainty calculations. Furthermore, the paper examines
two approaches for the relevant calculations: by means of a professional software and an excel add-in program. The presented data is
supported with graphical representations and illustrations.
Keywords uncertainty analysis, pipette calibration, gravimetric method, measurement uncertainty calculation, GUM, GUM Uncertainty
Framework, Monte-Carlo method
analytical balance the amount of pure water that is purged from
the piston pipette to a specific vessel and to convert the
obtained measuring result in mass unit into volumetric unit. In
practice, a number of 5 to 10 repeat measurements are made to
determine the systematic and random errors of the calibrated
pipette [1]. To the semeasurements, corrections must be
applied in order to compensate for any deviation from standard
temperature, atmospheric conditions and any significant
evaporation of the water during the calibration. Variable
volume pipettes should be tested at three points over their
designated range: maximum (nominal) volume, 50% of
nominal volume and the lower limit of the volume range or
10% of the nominal volume, whichever is the greater.
1. Introduction
Volume measurement is an important step in most industrial
and analytical measurement operations. Piston pipettes are
used for very precise measurements in many fields such as
chemistry, health, biology and pharmacy. In order to reduce
and identify possible errors in liquid handling, it is necessary to
calibrate pipettes using the correct methods. National
Metrology Institutes and accredited laboratories are using as
standard method the gravimetric method to calibrate volume
instruments. In this article, this method will assist the
calibration procedures as well.
The basic principal of the gravimetric method is to weigh with
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It is also necessary to evaluate the measurement uncertainty,
since this must be stated with the result of the measurement to
give the end use confidence in the measurement. Some
uncertainty contributions vary with each tested volume,
pipette channel, calibration equipment and environmental
conditions. Therefore, each pipette calibration must be
accompanied by individual calculations of the measurement
uncertainty for every testing volume and pipette channel.
In practice, the measurement uncertainty calculation is a
complex process, because when calibrating a variable onechannel pipette, the measurement uncertainty should be
calculated 3 times separately for every measured volume. For
multi-channel pipettes, the number of uncertainty calculations
is multiplied by the number of the channels – 24 times for 8channel pipette, 36 times for 12-channel pipette.
The complexity of the measurement uncertainty analysis often
requires the additional use of a software product. The example
analyzed further in this article is aided with the QMSys GUM
software, which demonstrates practical and easy, yet accurate
and fully reliable method for calculation of the measurement
uncertainty in pipette calibration.
Figure 1. Developing the measurement model
and specifying the input quantities
In the basis equation of the gravimetric method [1], [2] are
included two additional quantities, describing the influence of
the temperature differences between water, air and pipette
(δVT_DIFF), and the operator contributions (δVOP).
2. Measurement Uncertainty of the Pipette
Calibration by the Gravimetric Method
The correction for the thermal expansion of the pipette is
given by
The systematic procedure of building measurement
uncertainty analysis according to the GUM Uncertainty
Framework consists of the following stages:
●
Formulation: this stage consists of defining the output
quantity, identifying the input quantities, developing a
measurement model that relates the output quantity to the
input quantities and assigning probability distributions to
the input quantities on the basis of available knowledge
●
Calculation: the measurement uncertainty calculation is
performed by using the appropriate methods:
-1
where αc is the cubic expansion coefficient in °C , tp is
the pipette temperature in degrees Celsius;tp20 is equal to
20 °C.
The cubic expansion coefficient is not generally valid
determinant due to the different design and construction of each
pipette types. Material properties, material combinations and
different geometries and types affect the cubic expansion
coefficient. However, these influences are not mathematically
representable and therefore cannot be defined by all
manufacturers. It should also be emphasized that the
temperature of the piston-pipette is not constant because of
hand-warming at the middle and the top, and because of the
evaporation-cooling process at the bottom of the pipette. The
cubic expansion coefficient is approximated with the value
"zero" and the correction for the thermal expansion is not
included in the model equations.
-GUF method for linear models (GUM Uncertainty
Framework)
-GUF method for nonlinear models with symmetric
distribution of output quantity
-Monte-Carlo method for all models and distributions of
the result quantities
Summarizing and Validation: this stage summarizes and
validates the results and prepares the uncertainty budget.
When calculating the uncertainty with the software, the user
needs to furnish only the formulation stage, while the program
performs all other stages of the uncertainty analysis.
●
The equation of the weighing result consist of input quantities,
describing the repeated measurements of the delivered water,
the correction for evaporation loss, the calibration of the
balance and the rounding error due to balance resolution. The
uncertainty contributions due to the finite resolution of the
balance are listed twice in the model –once for the zero-setting
and once for the measurement at load.
2.1. Developing the Measurement Model
The formulation stage of uncertainty evaluation involves
developing a measurement model, incorporating corrections
and other effects as necessary. The model equations (Figure 1)
are the starting point for all subsequent calculations by the
software and are developed in accordance with the DKD guide
for pipette calibration [2], which offers the most detailed
analysis of the factors, influencing the measurement result.
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Air and water densities (ρA,ρW) are calculated by the
approximation formulas by Jones and Haris, given in [4]. The
equations for the environmental conditions (air temperature,
air pressure, water temperature and relative humidity) follow
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the same model and consist of input quantities, describing the indication of the measuring instrument, the calibration of the
instrument, the rounding error due to finite resolution and possible drift during the pipette calibration. The following table presents
all quantities of the measurement model:
* Quantity type: R – result, IR – interim result, A – type A uncertainty estimation, B – type B uncertainty estimation
The following graph (Figure 2) represents an example of determining the experimental standard uncertainty associated with the
repeatability of the balance measurements (GUM: type A evaluation) for testing volume 10 µl.
Figure2. Evaluation of type A input quantities
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If an outlying value can be traced back to a pipetting error, then it is permissible to repeat the measurement in question. If such a
retrace does not come up with any failure, then the outlying value should be considered as a real but rare incident and kept in the
data set.
2.2. Analysis of the measurement model and calculation of the measurement uncertainty
According to the GUM Supplement 1 [6] several conditions must be fulfilled for the valid implementation of the GUM
Uncertainty Framework in the process of the measurement uncertainty calculation. Figure 3 presents the results of the
measurement model analysis, which includes the following tests and calculations:
●
Linearity test for each input quantity in sixth areas of the distribution interval
●
Calculation and validation of the results of the equivalent linear model
●
Determination of the symmetry and the distribution type of the result quantities
●
Check for correlated input quantities with a finite degree of freedom
●
Check for non-linear correlated or non-normally distributed input quantities
Figure3.Expert analysis of the measurement model
The linearity of the model and the normal distribution of the result quantity are confirmed by conducting several analyses of the
measurement model with different input.
The recommended GUF method, corresponding to the standard GUM Uncertainty Framework, is applied for calculation of
the measurement uncertainty. The QMSys GUM software calculates the partial derivatives (the first term of a Taylor series) to
determine the sensitivity coefficients of the equivalent linear model and then calculates the combined standard uncertainty in
accordance with the Gaussian error propagation law. The value of the coverage factor is determined automatically to the
selected distribution and the specified coverage probability.
2.3. Measurement uncertainty budget
The result of the analysis is presented as a clearly structured measurement uncertainty budget (Fig. 4) in a table form. This table
holds all model quantities with their short names, value, associated standard uncertainty, distribution type, degrees of freedom,
sensitivity coefficient and contribution to the combined standard uncertainty of the output quantity. Interim results are shown
with the value and the standard uncertainty.
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Figure4.Measurement uncertainty budget (GUF method)
The result quantity is displayed in the bottom line with its value, corresponding combined standard uncertainty and the effective
degrees of freedom. Finally, the complete result of the examination is presented as a value with associated expanded uncertainty,
coverage factor, coverage probability and distribution type.
It should be kept in mind that some of the numerical values of the sensitivity coefficients are volume dependent; therefore, it is not
possible to use the values given in the example for other volumes.
When the uncertainty calculation by the Monte-Carlo method is activated, the software automatically validates the results of the
GUF Method by comparing the values, the combined standard uncertainties and the limits of the coverage intervals. The Monte
Carlo method (Fig. 5) displays a histogram, statistical parameters of the estimated distribution of the result quantities and validation
of the results.
The result of the uncertainty analysis together with all input data can be printed with the help of configurable templates in MS Word
or Excel format. Each analysis can be completely saved in a file with a selectable name. In this way, the examination is available at
any time for a later review or editing. Each saved analysis can be used as a starting point for new uncertainty analyses using the same
model, but with new or changed data sets.
3. Automated Uncertainty Calculation for Several Measurement Seriesin MS Excel
In this section, an additional application of the developed model for calculation of measurement uncertainty is suggested, namely,
with an Excell Add-In program. The full integration of calculating the measurement uncertainties in MS Excel is implemented by
using the software QMSys GUMX (Excel Add-In). This program also enables calculation of measurement uncertainty in MS Excel
for unlimited number of measurement series (result quantities with identical measurement model), using a model file for only one
set of measurements. This functionality is particularly useful when calibrating in several points of the measuring range. In addition,
it simplifies the modelling of the measurement process. The software QMSys GUMX reads the parameters of the input quantities
from the current Excel file, calculates the measurement uncertainty and then exports the results in the same Excel file. The cells with
the parameters of the input quantities are assigned in the model file with cell array names, defined in the Excel file. The parameters
of the result quantities and the statistical evaluation type A input quantities can be exported by columns or rows.
The following steps represent the procedure for calculating the uncertainty in MS Excel:
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●
Development of Excel files containing the data to be imported to the input quantities and coded cells for the export of the
results of the uncertainty calculation
●
Development of model file for calculating the measurement uncertainty with the QMSys GUM software, selecting the
appropriate cells or cell ranges of the input quantities in the model file.
●
Assigning of the model file to the Excel file; up to three different model files can be selected in one Excel file
On the Figure 6 are shown the worksheets for entering the calibration data, calculating the measurement results and generating the
calibration certificate.
Figure5.Measurement uncertainty budget (Monte-Carlo method)
The blue cells indicate entries of the volume measurement series and the environmental conditions. In the yellow cells are
calculated the volume-dependent parameters of the additional input quantities. Following the described procedure, Excel files for
calibrating multi-channel pipettes that are using the same model file for calculating the measurement uncertainty are also developed
for each tested volume and channel.
4. Summary
The developed measurement model for pipette calibration by the gravimetric method includes all relevant factors that influence the
calibration results. The linearity of the model and the normal distribution of the result quantity are confirmed by conducting the
model analysis according to the GUM Supplement 1. The significant uncertainty contributions, causing over 99% of the combined
standard uncertainty of the pipette calibration, are:
●
Repeatability of the balance measurements of the delivered volume
●
Calibration of the balance
●
Temperature difference water - pipette – air
●
Operator contributions
●
Rounding error due to balance finite resolution
Furthermore, excluding the balance resolution, all significant uncertainty contributions depend on the measured volume. The
repeatability of the balance measurements is specific for each channel of the multi-channel pipettes. Therefore, when
calibrating piston pipettes by the gravimetric method, the measurement uncertainty should be calculated separately for every
measured volume and channel.
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Finally, the article acknowledges the complexity of measurement uncertainty analysis and the importance of valid results and
reliable interpretations, thus, suggests two practical solutions by means of a professional software product.
Figure 6. Calibration worksheet in MS Excel
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Figure 7 . Calibration certificate in MS Excel
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About the Software:
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
ISO 8655-6:2002 Piston-operated volumetric apparatus - Pt 6:
Gravimetric methods for the determination of measurement
error.
Guideline DKD-R 8-1 Kalibrierung von Kolbenhubpipetten
mit Luftpolster(12/2011)
Blues, J., Bayliss, D.J., Buckley, M.: The calibration and use
of piston pipettes, Measurement Good Practice Guide No.
69(07/2004)
ISO/TR 20461:2000Determination of uncertainty for volume
measurements made using the gravimetric method
JCGM 100:2008 Evaluation of measurement data – Guide to
the expression ofuncertainty in measurement
JCGM 101:2008 Evaluation of measurement data Supplement 1 to the “GUM” - Propagation of distributions
using a Monte Carlo method
EURAMET/cg-19 Guidelines on the determination of
uncertainty in gravimetric vol. calibration, Ver. 2.1 (03/2012)
QMSys GUM software is a comprehensive tool for
analysis of the measurement uncertainty of physical
measurements, chemical analyses and calibrations. The
program supports the systematic procedure in building an
uncertainty analysis, as requested in the corresponding
standards and guides, and reduces significantly the
analytical and computational effort. The user of the
software needs to furnish only the formulation stage,
while the program performs automatically all other stages
of the measurement uncertainty analysis. The software
allows the user to freely enter or modify the model
equation. With this feature, the application can be used to
evaluate measurement uncertainty of almost any
measurement process.
For further information, please visit: www.qsyst.com.
The Author
About the Author: Stefan Golemanov graduated with a Master of Science in Mechanical and
Precision Engineering (M.Sc.Eng.) at the Technical University of Gabrovo, Bulgaria. After
performing specializations in Austria and Germany focused on metrology and quality
management, he founded in 1995 with partners the company Qualisyst Ltd., which is
specialized in the development of metrology, quality assu rance and quality management
software.
Contact
Email: Stefan@qsyst.com
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