Abstract
The nominal volume of a variable volume micropipette is the maximum volume of the range specified by the micropipette manufacturer. The purpose of the present study is to quantify the reproducibility standard uncertainty for nonnominal volume micropipette calibrations for variable volume micropipettes. An interlaboratory study was implemented for this purpose, using variable volume micropipettes with nominal volumes of 10 μL, 200 μL, and 1000 μL and setting the measurement volumes as 10 %, 50 %, and 100 % of the nominal volumes. In our previous paper (Accredit Qual Assur 19:377, 2014), we quantified the uncertainty due to reproducibility using only data obtained for nominal volume calibrations. From the results obtained, we found that the reproducibility uncertainties quantified for nominal volumes are too small to cover interlaboratory variations for nonnominal volume calibrations. For calibrations of 10 % and 50 % of the nominal volumes, the reproducibility uncertainties must be increased by the factors of 2.7 and 1.4, respectively, for micropipettes with nominal volumes of 10 μL to 1 000 μL, based on the results of this study.
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References
 1.
ISO (2002) ISO 8655–6 Pistonoperated volumetric apparatus—part 6: gravimetric methods for the determination of measurement error. ISO, Geneva
 2.
Shirono K, Shiro M, Tanaka H, Ehara K (2014) Evaluation of “method uncertainty” in the calibration of piston pipettes (micropipettes) using the gravimetric method in accordance with the procedure of ISO 86556. Accred Qual Assur 19(5):377–389
 3.
German Calibration Service (DKD) (2011) DKDR 81:2011 Calibration of pistonoperated pipettes with air cushion. German Calibration Service, Braunschweig
 4.
Batista E, Arias R, Jintao W (2013) Final report on BIPM/CIPM key comparison CCM.FFK4.2.2011: Volume comparison at 100 µL—calibration of micropipettes (piston pipettes). Metrologia 50(1):07003
 5.
Batista E, Matus M, Metaxiotou Z, Tudor M, Lenard E, Buker O, Wennergren P, Piluri E, Miteva M, Vicarova M, Vospĕlová A, Turnsek U, Micic L, Grue LL, Mihailovic M, Sarevska A (2017) Final report on the EURAMET.M.FFK4.2.2014 volume comparison at 100 µL—calibration of micropipettes. Metrologia 54(1):07016
 6.
Batista E, Almeida N, Filipe E (2015) A study of factors that influence micropipette calibrations. NCSLI Measure 10(1):60–66
 7.
Batista E, Alberini M (2018) Results of the supplementary comparison (bilateral) of a fixed volume micropipette of 50 µl and a variable volume micropipette of 1000 µl. Metrologia 55(1):07009–07009
 8.
ISO (2005) ISO/IEC 17025 General requirements for the competence of testing and calibration laboratories. ISO, Geneva
 9.
ISO (1994) ISO 5725–2 Accuracy (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method. ISO, Geneva
 10.
National Institute of Technology and Evaluation (NITE). Website of International Accreditation Japan (IAJapan). https://www.nite.go.jp/en/iajapan/index.html. Accessed 10 Jan 2019
 11.
Japan Accreditation Board (JAB). Website of the Japan Accreditation Board. https://www.jab.or.jp/en/. Accessed 10 Jan 2019
 12.
National Institute of Standards and Technology (NIST). Website of the National Voluntary Laboratory Accreditation Program (NVLAP). https://www.nist.gov/nvlap. Accessed 10 Jan 2019
 13.
American Association for Laboratory Accreditation (A2LA). Website of the American Association for Laboratory Accreditation. https://www.a2la.org/. Accessed 10 Jan 2019
 14.
Tanaka M, Girard G, Davis R, Peuto A, Bignell N (2001) Recommended table for the density of water between 0 °C and 40 °C based on recent experimental reports. Metrologia 38(4):301–309
 15.
Kell G (1975) Density, thermal expansivity, and compressibility of liquid water from 0.deg. to 150.deg. Correlations and tables for atmospheric pressure and saturation reviewed and expressed on 1968 temperature scale. J Chem Eng Data 20(1):97–105
 16.
ISO (2002) ISO 8655–2 Pistonoperated volumetric apparatus—part 2: Piston pipettes. ISO, Geneva
 17.
ISO (2010) ISO/IEC 17043 conformity assessment—general requirements for proficiency testing. ISO, Geneva
 18.
Cox M (2002) The evaluation of key comparison data. Metrologia 39(6):589–595
 19.
Euramet (2018) Euramet Calibration Guide No. 19 Guidelines on the determination of uncertainty in gravimetric volume calibration (version 3.0). Euramet, Braunschweig
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Appendices
Appendix 1: Approximate uncertainty evaluation
We would like to quantify the reproducibility standard uncertainty independent of the other uncertainty sources. Although we have gathered some information on uncertainty in the interlaboratory study, the evaluated uncertainty sources based on the individual laboratories’ SOPs were not necessarily assessed independently of the reproducibility uncertainty considered in this study. Therefore, apart from the obtained uncertainty assessment information from the participants, we implemented the computation of the standard uncertainty of V_{i,j} in an approximate manner so carefully that the reproducibility uncertainty is independent of other uncertainty sources. Since the uncertainties other than the reproducibility uncertainty were marginal compared to the interlaboratory variations as mentioned in the main manuscript, only brief explanations are given. Uncertainties are computed basically in accordance with the EURAMET Calibration Guide No. 19 [19] using the information in ISO 8655 Parts 2 [16] and 6 [1].
Uncertainty sources

(1)
Repeatability uncertainty \(u_{\text{rep}} \left( {V_{i,j} } \right)\)
This source was evaluated in accordance with the EURAMET Calibration Guide No. 19.

(2)
Uncertainty relating to the mass scale calibration \(u_{\text{scale}} \left( {V_{i,j} } \right)\)
This source was evaluated conservatively using the maximum values for “repeatability and linearity” (uncertainty), and “standard uncertainty of measurement” (in the calibration) given in ISO 8655 Part 6. Only for the 200 μL test case, 0.05 mg was used for the both uncertainties instead of 0.2 mg in ISO 8655 Part 6.

(3)
Uncertainty of the water density \(u_{\text{wat}} \left( {V_{i,j} } \right)\)
This source was evaluated in accordance with the EURAMET Calibration Guide No. 19 with slight modifications.

(4)
Uncertainty of the air density \(u_{\text{air}} \left( {V_{i,j} } \right)\)
This source was evaluated in accordance with the EURAMET Calibration Guide No. 19 with slight modifications. The guide offers the relative standard uncertainty due to using a formula [Eq. (5) in the guide] as 2.4 × 10^{−4}. While that formula was employed, the uncertainty was not used in this study. Although this uncertainty is given for the case where the air temperature, the air pressure, and the relative air humidity are not well specified, we could specify these by their measured values in this study.

(5)
Uncertainty of the cubic expansion coefficient \(u_{\exp } \left( {V_{i,j} } \right)\)
This source was evaluated in accordance with the description in the EURAMET Calibration Guide No. 19 that “data from the literature or manufacturer should be used and this would be expected to have an (standard) uncertainty of the order of 5 % to 10 %.” We assumed the relative standard uncertainty of 10 %.

(6)
Uncertainty of the temperature of the micropipette \(u_{\text{pip}} \left( {V_{i,j} } \right)\)
The EURAMET Calibration Guide No. 19 offers an approach to determine this uncertainty based on the difference between measured air and water temperatures, which is recommended to be no more than 2 K in the guide. We hence conservatively assumed the uniform distribution with the width of ± 2 K for the uncertainty of \(t_{i,j}^{\text{pip}}\). Although \(t_{i,j}^{\text{pip}}\) was determined to be identical to \(t_{i,j}^{\text{wat}}\), the uncertainty of \(t_{i,j}^{\text{wat}}\) was neglected for \(u_{\text{pip}} \left( {V_{i,j} } \right)\), because it was quite minor.

(7)
Reproducibility uncertainty \(u_{\rm m}\left(V_i{\rm set}\right)\), \(u_{\rm mod}\left(V_i{\rm set}\right)\)
For the computation of \(\chi_i^2\) in the Applicability of quantifying reproducibility standard uncertainty for nominal volumes subsection in the Discussion section, the standard uncertainties of reproducibility \(u_{\rm m}\left(V_i^{\rm set}\right)\) in Eq. (5) is employed. For the computation of \(\chi_{\text{mod}, i}^2\) in the Modified reproducibility standard uncertainties for nonnominal volumes subsection in the Discussion section, the standard uncertainties of reproducibility \(u_{\rm mod}\left(V_i^{\rm set}\right)\) in Eq. (6) is employed.
Combination of the standard uncertainties
The combined standard uncertainty of V_{i,j}, u_{i,j}, is computed by the following equation:
It is found that the values are almost the same as the corresponding reproducibility standard uncertainty \(u_{\rm m}\left(V_i{\rm set}\right)\). For the nominal volumes, u_{i,j} was only 12 % larger than \(u_{\rm m}\left(V_i{\rm set}\right)\) at the maximum. Although our assessment of the uncertainties shown in this appendix is approximative, the approximation may not be so crucial, because the reproducibility standard uncertainty is the dominant source of the uncertainty. For the nonnominal volumes, u_{i,j} was 41 % larger than \(u_{\rm m}\left(V_i{\rm set}\right)\) at the maximum. The reproducibility standard uncertainty was still the main source of the assessed standard uncertainty of the calibrated volume.
The modified combined standard uncertainty of V_{i,j}, \(u_{i, j}^{\bmod }\), is computed by the following equation:
The computed \(u_{i, j}^{\bmod }\) is only 9 % larger than \(u_{\rm mod}\left(V_i{\rm set}\right)\) at the maximum. It can be said that the uncertainties except for the reproducibility uncertainty are negligibly small for the purpose of this study.
Appendix 2: Oneway ANOVA in a heteroscedastic condition
We found inhomogeneity in repeatability reported from participants. In other words, repeatability is found to be heteroscedastic in most cases. However, even in the situation with the heteroscedastic variance, the oneway ANOVA in this study is still a way to estimate reproducibility variance unbiasedly. The explanation is given in this appendix.
Instead of the complicated symbols employed in the main manuscript, we define some symbols only for this appendix as follows:

a: Number of the levels of the factor in the oneway ANOVA,

i: Indication of the level of the factor when Condition i is the ith level of the factor (i = 1, …, a),

n: Number of the repetitions in a condition level,

x_{i,j}: measured value in the jth repetition of Condition i (j = 1, …, n),

\(\bar{x}_{\text{i}}\): Average value of x_{i, 1} to x_{i, n},

\(\bar{\bar{x}}\): Average value of \(\bar{x}_{ 1}\) to \(\bar{x}_{\text{a}}\).
Moreover, Exp[∙] and Var[∙] denote the operators for the expectation and the variance, respectively.
We assume the following statistical model:
where A_{i} is the effect of the factor for Condition i, and E_{i,j} is the error for the repeatability.
When homoscedasticity is considered, the variance of E_{i,j} does not depends on Condition i. We hence define the variances of A_{i} and E_{i,j} as follows:
In this case, the unbiased estimator of \(\sigma_A^2\) essentially employed in ISO 5725 Part 2 [9], \(\hat{\sigma }_{A}^{ 2}\), is given as follows:
When heteroscedasticity is considered, the variance of E_{i,j} depends on Condition i. We hence define he variances of A_{i} and E_{i,j} as follows:
The expectation of the square sum of \(\left( {\bar{x}_{i}  \bar{\bar{x}} } \right)\) is given as follows:
With
and
the following relation can be yielded:
Since
\(\hat{\sigma }_{A}^{ 2}\) satisfying the following equation is an unbiased estimator of \(\sigma_A^2\):
The solution of \(\hat{\sigma }_{A}^{ 2}\) is, in fact, given by Eq. (22). Therefore, it is concluded that even when heteroscedasticity is considered, Eq. (22) gives an unbiased estimator of the variance of the factor in the oneway ANOVA. Although other analysis may be possible, no essential differences can happen as long as we apply an unbiased estimation method, because \(s_{i,j}^{\text{lab}}\) is much larger than \(s_{i,j}^{\text{rep}} /\sqrt {10}\) in this study.
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Shirono, K., Tanaka, H. Evaluation of reproducibility uncertainty in micropipette calibrations for nonnominal volumes through an interlaboratory study. Accred Qual Assur 26, 27–39 (2021). https://doi.org/10.1007/s00769020014535
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Keywords
 Piston pipette
 Micropipette
 ISO 8655
 Calibration
 Nonnominal volume
 Gravimetric method