Estimating measurement uncertainty in high-performance liquid chromatography methods

PRACTITIONER’S REPORT

Estimating measurement uncertainty in high-performanceliquid chromatography methods

Tatjana Tomic • Nada Uzorinac Nasipak •

Sandra Babic

Received: 6 April 2011 / Accepted: 23 December 2011 / Published online: 7 January 2012

� Springer-Verlag 2012

Abstract In every measurement procedure, it is impor-

tant to know the components of measurement uncertainty

affecting the quality of measured result and reliability of

quantified result. The procedure for recognizing measure-

ment uncertainty is not universal but depends on the

method and sample type. It has to be made according to

good laboratory practice. This paper aims at showing the

comparison of measurement uncertainty component esti-

mations for three methods using the high-performance

liquid chromatography techniques: determination of the

type and content of aromatic hydrocarbons in diesel fuels

and petroleum distillates by normal phase high-perfor-

mance liquid chromatography, determination of nitrates in

water samples by ion chromatography, and determination

of molecular weights of polystyrene by size exclusion

chromatography technique. Both similarity and differences

were found during the measurement uncertainty compo-

nent estimation, and conclusions about influences of certain

components on the result uncertainty were made.

Keywords Measurement uncertainty �High-performance liquid chromatography (HPLC) �Ion chromatography (IC) � Size exclusion

chromatography (SEC)

Introduction

Every day, crucial decisions are made based on analytical

information in any type of scientific research and in other

areas of life, such as medicine, law, and health and safety

and environmental management. Such information is based

on analytical measurements, and the assumption is that

obtained results are reliable. One of the tools of quality

assurance and quality control systems (QA/QC) that help

assure reliability of an analytical work is the estimation of

measurement uncertainty.

According to International Vocabulary of Metrology

(VIM 3) [1], measurement uncertainty is non-negative

parameter characterizing the dispersion of the quantity

values being attributed to a measured, based on the infor-

mation used. The measured refers to the particular quantity

intended to be measured. The parameter may be a standard

deviation called standard measurement uncertainty (or a

specified multiple of it), or the half-width of an interval,

having a stated coverage probability. Within this interval,

the result is regarded as being accurate, that is, precise and

true. It should be emphasized that measurement uncertainty

is different from the error. The error of an individual

analytical result is always a single value. It is the difference

between the result and the true value of the measured.

Uncertainty is expressed as a range and applies to an

analytical procedure and a specific sample type, but to

different determinations and thus measurement results

[1–4].

Uncertainty is a basic characteristic of any measure-

ment, it is quantified measurement quality. Uncertainty is

always present, at every step of an analytical procedure.

There are many possible sources of measurement uncer-

tainty, such as imprecise definition of measured, sampling,

sample preparation, matrix effects and interferences,

T. Tomic (&) � N. Uzorinac Nasipak

INA-INDUSTRIJA NAFTE d.d., Central Analytical

Laboratory, Lovinciceva b.b., 10000 Zagreb, Croatia

e-mail: [email protected]

S. Babic

Department of Analytical Chemistry, Faculty of Chemical

Engineering and Technology, University of Zagreb,

Marulicev trg 19, 10000 Zagreb, Croatia

123

Accred Qual Assur (2012) 17:291–300

DOI 10.1007/s00769-011-0872-0

environmental conditions, incorrect application of analyti-

cal procedure, uncertainties of mass and volumetric

equipment, and uncertainty associated with the instrument

calibration and random variation [2, 5]. Knowing and

defining uncertainty components that can influence mea-

surement uncertainty are very important for measuring

process quality and quantified result reliability.

Measurement uncertainty is increasingly gaining atten-

tion, in particular within the framework of accreditation. A

requirement of the international standard for testing labo-

ratories, ISO/IEC 17025 [6], is that laboratory must have

and use procedures for measurement uncertainty evaluation

for all calibrations and quantitative measurements.

Different approaches exist for the estimation of overall

measurement uncertainty, as reviewed by several authors

[3, 5, 7, 8]. The most well-known, traditional approach is

based on identifying, quantifying, and combining all indi-

vidual contributions to uncertainty. In this bottom-up

approach, the overall uncertainty is derived from the

uncertainties of the individual components [3]. The

bottom-up approach is adopted by the International Orga-

nization for Standardization (ISO) and also initially

adapted for the analytical field by the EURACHEM/

CITAC Guide [2]. Another approach is the top-down. For

the top-down approach, methodology is considered as

whole and major sources of uncertainty (repeatability,

recovery, matrix effects, etc.) are identified and evaluated.

The top-down approach is adopted by the Analytical

Methods Committee, and it is recommended in the ISO

21748:2004, which establishes guidance for the estimation

of measurement uncertainty using data from interlabora-

tory studies performed according to ISO 5725-2 [8, 9]. The

bottom-up approach can be tedious. On the other hand, it is

illustrative and forces the analyst to understand the details

of the analytical procedure. Therefore, the bottom-up

approach is a tool that identifies the parameters or working

steps with an overly contribution to the total uncertainty,

thus offering the chance to improve them [10, 11]. Work-

ing through the whole process makes it also clear if the

standard operating procedure is written clearly, completely,

and unambiguously. Sometimes a method can be simplified

markedly, resulting in a lower uncertainty [11].

Chromatographic techniques are frequently a very

important aspect of the procedure applied to the analysis of

samples with complex and sometimes variable matrices, in

order to determine their trace constituents. Nowadays,

liquid chromatography is one of the most used methods of

analysis for compounds in solution. It is one of inevitable

techniques in quality control laboratories in oil industry.

Determination of aromatic hydrocarbon groups in oil

products is of special interest for oil refinery processes. The

determination of aromatics can help in clearly understanding

and optimizing of conversion process variables explaining

reaction pathways, physico-chemical mechanisms, and their

kinetics. Then the application and ecological properties of

final product would be determined. High-performance liquid

chromatography (HPLC) has been widely used for analysis

of crude oil, middle distillates, and liquid oil products [12].

Use of this method includes separation and identification

according to hydrocarbon composition, as well as a separa-

tion, identification, and quantification of target components

in many petrochemical samples.

Ion chromatography is a powerful tool for water sample

analysis in oil industry. Replacement of the classic ana-

lytical method leads to the use of smaller sample content,

faster analysis, and multiple information in a single anal-

ysis. In oil industry, there are numerous ion chromatography

uses, of which some are connected to oil investigations and

some to ecological demands and control. The significance

of oilfield water analyses is the analysis of liquid samples

from exploration wells, characterization of water produced

together with oil, quality check of injection water, pre-

diction of corrosion rate, prediction of scale formation rate,

monitoring of chemicals used for water treatment, and

analytical backup for projects in oil industry.

High-performance size exclusion chromatography

(HPSEC) covers determination of molecular weight aver-

ages and molecular weight distribution for linear, soluble

polystyrene polymers. The molecular weight and molecular

weight distribution are fundamental characteristics of a

polymer sample. They may be used for a wide variety of

correlations between fundamental studies, processing, or

product applications. Main source for polymer production

is oil, and 4% of oil is processed to polymer products.

This paper aims at showing estimation and comparison

of measurement uncertainty components for three methods

using the liquid chromatography techniques:

• normal phase high-performance liquid chromatography

(NP HPLC) for the determination of aromatic hydro-

carbons in diesel fuels and middle petroleum distillates

(EN 12916) [13],

• ion chromatography (IC) for the determination of

nitrates in water samples (EN ISO 10301-1) [14], and

• size exclusion chromatography (SEC) for the determi-

nation of molecular weight of polystyrene (ASTM D

5296) [15].

Both similarity and differences were found during the

measurement uncertainty component estimation and con-

clusions about influences of certain components on the

result uncertainty were made. Despite the fact that the same

technique was applied, with similar measurement proce-

dure and instrumentation, there were found significant

differences in the measurement uncertainty component.

292 Accred Qual Assur (2012) 17:291–300

123

Experimental

Chemicals and standards

In NP HPLC method, n-heptane (Carlo Erba, Italy) of

chromatographic purity with \50 mg L-1 water content

has been used as mobile phase. Prior to initiating the

analysis, helium 5.0, purity 99.999% (Messer Croatia Plin,

Zapresic, Croatia) has gone through the mobile phase to

remove dissolved gasses.

The calibration standards are made by mixing the fol-

lowing components: cyclohexane p.a., o-xylene, purity

C99.5% (GC), dibenzothiophene, purity C98% (GC), and

9-methylantaracene, purity C98% (GC) all obtained from

Fluka (Buchs, Switzerland).

In ion chromatography analyses, the mobile phase is set

by eluent generator cartridge filled with potassium

hydroxide and with deionized water (18 MX) obtained

from TKA Pacific/GenPure water purification system

(TKA Water Purification Systems GmbH, Germany). The

mobile phase ion strength is set by computer program. A

standard solution of nitrates concentration 101 mg L-1

(Dionex, USA) was used for calibration.

In SEC method, tetrahydrfuran p.a (Kemika, Croatia) is

used as mobile phase. Before analysis, solvent is put in an

ultrasonic bath to remove dissolved gasses. For calibration,

ten polystyrene standards with narrow molecular weight

distribution and of known molecular weight were used

(Polymer Laboratories, USA).

NP HPLC procedure

Liquid chromatography system used for analyses is a

complex modular system consisting of the following units:

9012Q HPLC pump (Varian Chromatography System,

California, USA), injection system (ProStar 410 Autosam-

pler, Varian), column oven (Mistral Column Oven, Varian),

refraction index detector (Varian 9040), and STAR soft-

ware program (Varian) for data processing. Aromatic

hydrocarbons are separated from saturates and form distinct

peaks according to their ring structure as mono-aromates

(MA), di-aromates (DA), and tri-aromates (TA) on the

Zorbax polar amino-bonded silica column, 4.6 9 250 mm

i.d., particle size 5 lm (Agilent, CA, USA). n-heptane was

used as mobile phase with 0.8 mL min-1 flow rate. The

column temperature was set at 30 �C, and injection volume

was 10 lL. Samples were analyzed on RI-detector capable

of being operated over the refractive index range 1.3–1.6.

Samples analyzed were commercial products known as

euro diesel, collected from different oil stations. Before the

chromatographic analysis, samples were diluted in n-hep-

tane (0.4 g/5 mL) and filtrated through 0.45 lm membrane

filter. A volume of 10 lL of prepared solution was injected

using a partial filling of the sample loop.

Determination of concentrations of each aromatic species

is based on calibration curves prepared using cyclohexane,

o-xylene, 1-methylnaphtalene, and phenanthrene. Calibra-

tion is performed with four calibration standards in

concentration range from 0.05 to 4.0 g mL-1 for MA, from

0.02 to 2.0 g mL-1 for DA, and from 0.01 to 0.4 g mL-1 for

TA. Coefficients of correlation were higher than 0.99.

IC procedure

IC measurements were performed on Dionex ICS 3000

modular ion chromatography system (Dionex, USA)

comprising high pressure pump, eluent generator module

filled with KOH cartridge, AG 19 guard column and AS 19

analytical column, and suppressor and conductivity detec-

tor. Flow rate was set on 1.0 mL min-1, temperature of

column department was set at 30 �C, and of detector

department was set at 20 �C. For data processing, Chro-

meleon software from Dionex was used. Before analysis

groundwater, samples were filtrated through 0.45 lm

regenerated cellulose membrane filter.

For calibration, stock standard solution (Dionex, USA)

was diluted to appropriate concentrations, and calibration

graph was constructed with five calibration points in con-

centration range from 0.1 to 20 mg L-1.

SEC measurements

SEC measurements were performed on modular system

comprising a Waters 6000A high pressure pump equipped

with U6 K manual injector capable for injection up to

2.0 mL of sample solution, three PL-gel mixed B columns,

and RI-detector. Flow rate was 1.0 mL min-1, and temper-

ature was ambient. Data acquisition, reduction, processing,

and reporting were performed using homemade software and

composition hardware. Polystyrene sample solution was

prepared by dissolving sample in freshly distillated tetra-

hydrofuran and staying over night. Before injection, solution

was filtered through 0.5 lm membrane filter, and 50 lL of

solution was injected.

Calibration standards have molecular mass in range

from 1320 to 2880000 g mol-1.

Results

The usual analytical procedure consists of the following

main steps: sampling, sample preparation, measurement, and

quantification. Uncertainty of result may arise from each

of these steps including examples such as sample repre-

sentativeness, sample contamination, storage conditions,

Accred Qual Assur (2012) 17:291–300 293

123

matrix effects and interferences, environmental conditions,

reagent purity, uncertainty of masses and volumetric

equipment, reference values, separation, identification, cal-

ibration, approximations incorporated in the measurement

method and procedure, and random variation. Not all of the

components will make a significant contribution to the

combined uncertainty. Therefore, a detailed estimation of

the contribution of components to the uncertainty should

be made, and it should be evaluated which of them has a

significant contribution to the overall measurement

uncertainty.

Usually, measurement uncertainty estimation includes

the following steps:

• description of measurement procedure and measured

specification,

• identification of components that influence measure-

ment uncertainty of final results,

• quantification of measurement uncertainty components,

• calculation of combined standard measurement uncer-

tainty, and

• calculation of extended measurement uncertainty.

Measurement procedures and specification

of the measured

For the correct estimation of measurement uncertainty, it is

essential for the analyst to understand measurement pro-

cedure. It should be preciously described in standard

operating procedure, and if it is done correctly, it is easy to

draw flow diagram describing measurement procedure.

Drawing flow diagram is very helpful, and it is a first step

in estimation of measurement uncertainty.

Flow diagrams of measurement procedures shown in

Fig. 1 are suitable for chromatographic methods investi-

gated in this paper.

For aromatic hydrocarbons (NP HPLC) and anions (IC)

determination, analyte content was determined from the

calibration equation:

c0 ¼y� a

bð1Þ

where

c0 analyte concentration determined from calibration

curve (mg L-1)

y peak area (number of pixel)

a intercept of calibration curve

b slope of calibration curve

For NP HPLC method, c0 is multiplied by V/m, where

V is sample dilution volume (the volume to which the

sample is diluted), and m is sample mass. That way, final

result was shown as mass fraction (g/100 g).

w ¼ y� a

b� Vm

ð2Þ

For IC, the analyte concentration in the sample is

obtained from the equation:

c ¼ y� a

b� f ðmg L�1Þ ð3Þ

where f is the dilution factor.

For size exclusion chromatography, the measured is the

molecular mass of polystyrene given by

lg Mr ¼ aMM þ bMM � Ve ð4Þ

where

Mr relative molecular mass of polystyrene

Ve sample elution volume (mL)

(a)

(b)

Result

Sample weighting or

ppetting

Standard weighting or

pipetting

Standard preparation

Instrument (Calibration)

Sample preparation

Instrument (Measurement)

Result

Sample dilution pipette 5 mL flask 50 mL

Standard dilution pipette 100 L flask 100 mL

Standard filtration

Sample filtration

Instrument (Measurement)

Instrument (Calibration)

µ

Fig. 1 Flow diagrams of the measurement procedure for NP HPLC

and IC methods


123

aMM intercept of the calibration curve

bMM slope of the calibration curve

Uncertainty sources identification

Before identifications of the uncertainty sources, method

validation was performed. On that way, different influences

on results were determined such as selectivity, reproduc-

ibility, drift of the system parameters between calibration

and measurement, dilution of the stock standard solution,

linearity, etc. Upon validation results, conclusions have

been made about which of them have a significant influ-

ence on result uncertainty. Laboratory follows a procedure

for everyday certificated working material (CRM) control

and by that bias of analytical procedure is not significant.

Ishikawa diagram, also termed fishbone diagram, is a very

useful tool to identify the influence parameters, that is, the

sources of the measurement procedure uncertainty. There-

fore, Ishikawa, or causes and effects diagrams, were created

(Figs. 2, 3) in order to determine measurement uncertainty

components. It helps in simplified review of all registered

measurement uncertainty causes and shows other potential

uncertainty sources, which need to be investigated.

Based on Ishikawa diagrams, possible influences on

measurement uncertainty have been identified for chro-

matographic methods investigated (Table 1).

By validation, it was confirmed that for molecular

weights determination of polystyrene by SEC method,

different sample concentrations in recommended field do

not influence the result, so there is no need for dilution

factor measurement uncertainty determination. For this

method, just two measurement uncertainty components

have been found important—calibration curve and mea-

surement repeatability.

Uncertainty due to sampling was not taken into account

because samples in the laboratory come from the customer

who orders the analyses.

Uncertainty components quantification

Having identified the uncertainty sources, the next step is

to quantify the uncertainty arising from these sources. In

this paper, uncertainty sources are combined in three main

groups: (1) uncertainty associated with standard solution

and sample preparation, (2) uncertainty associated with

calibration curve, and (3) uncertainty associated with

method repeatability. Uncertainty arising from each group

is evaluated.

Uncertainty associated with standard and sample

preparation, uprep

For standard stock solutions preparations in NP HPLC

method, the main sources of uncertainty were as follow:

standard mass (mst), purity of the standard (P) and the

standard solution volume (Vst); in IC method uncertainty

sources were purity of the standard (P), pipette volume (Vp)

and flask volume (Vflask).

AROMATE MASS FRACTION (w, g/100 g)

standard preparation

ANALYTE CONCENTRATION(γ 0)

REPEATABILITY

calibration curve

DILUTION FACTOR ( f )

sample mass(msample)

standard purity (P) standard mass (mst)

standard volume (Vst)

sample preparation

sample volume (Vsample)

Fig. 2 Ishikawa diagram for

aromatic hydrocarbon

determination in eurodiesel

fuels by NP HPLC method


123

For sample preparation in NP HPLC method, the fol-

lowing two factors were significant: sample mass (msample)

and flask volume (Vsample). For IC method, the significant

factors were pipette volume (Vp) and flask volume (Vflask).

Uncertainty of weighing was calculated for each stan-

dard and sample according to Eq. 5:

uðmstÞ ¼sffiffiffi

np ð5Þ

where s represents standard deviation, and n is number of

measurements of weighing 1 g.

Since the uncertainty of purity of analytical standards

was not declared by supplier, it was estimated from Eq. 6:

u Pð Þ ¼ 0:5 1� Pð Þffiffiffi

3p ð6Þ

where P represents the purity of the standard expressed as a

mass fraction and given on the certificate. For standards

used in this work, P is between 0.9 and 1.

It should be noted that concentration of analytes in stock

and calibration solutions was not corrected for standard purity.

The manufacturer’s tolerance for the 5 mL pipette is

(5 ± 0.03) mL, at 20 �C. The value of uncertainty is given

without confidence level or distribution information, so the

standard uncertainty associated with the pipette volume is

estimated assuming triangular distribution (Table 2).

Flask volume uncertainty consists of uncertainty from

imperfect flask calibration (u(VC)), and uncertainty of

temperature difference between working temperature and

temperature on flask was calibrated (u(Vt)). Total flask

volume uncertainty was calculated from Eq. 7:

u Vflaskð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u2 VCð Þ þ u2 Vtð Þp

ð7Þ

Uncertainty from imperfect flask calibration (u(VC))

was calculated assuming a triangular distribution (the

manufacturer’s tolerance for the flask is ± 0.025 mL, at

20 �C). Uncertainty of temperature effect (u(Vt)) is calculated

using the following equation:

u Vtð Þ ¼DT � a � V

ffiffiffi

3p ð8Þ

where DT represents difference between temperature at

which flask was calibrated (20 �C) and expected laboratory

temperature (25 �C), and a is coefficient of volume

expansion for water 2.1910-4 �C-1:

Table 2 summarizes factors and their standard and rel-

ative uncertainties associated with standard and sample

preparation for NP HPLC and IC methods.

As can be seen from Table 2 uncertainty of mass mea-

surement has neglected influence on total measurement

uncertainty. In future calculation, this influence could be shut

off. Uncertainty of volume measurement has the highest

influence on sample and standard preparation uncertainty.

NITRATE MASS CONCENTRATION

(γ , mg·L-1)

standard preparation

ANALYTE CONCENTRATION (γ ) REPEATABILITY

calibration curve

DILUTION FACTOR ( f )

pipette volume (Vp)

standard purity (P)

sample preparation

flask volume (Vflask)

0

Fig. 3 Ishikawa diagram for

nitrates determination in water

samples by IC method

Table 1 Uncertainty sources

Influence NP HPLC IC SEC

Sample and standard mass H Ø Ø

Sample and standard volume H H Ø

Standard purity H H Ø

Calibration curve H H H

Measurement repeatability H H H

NP HPLC Normal phase high-performance liquid chromatography,

IC ion chromatography, SEC size exclusion chromatography


123

Single uncertainty components were combined by the

error propagation rules in uncertainties associated with

standard and sample preparation:

uprep;IC ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u Pð ÞP

� �2

þu Vp

� �

Vp

� �2

þ u Vflaskð ÞVflask

� �2s

ð10Þ

As stated before, different sample concentrations do not

influence the result in SEC method, so the uncertainty

associated with standard and sample preparation is not

determined for this method.

Uncertainty associated with calibration curve, ucalib

Based on relative (comparative) nature of chromatographic

methods, they involve a calibration step. Calibration curves

were prepared by plotting the peak area versus the analyte

concentration. For construction of calibration curve, cali-

bration standards were prepared as already stated in

experimental section for each method. Each standard and

sample was measured in triplicate. Prepared calibration

curves were used to estimate the analytes concentration in

samples (Table 3).

This step of analytical procedure influences the uncer-

tainty of final results and therefore must be included in the

uncertainty budget. The standard uncertainty for x (analyte

concentration in the sample as a result of the calibration

correlation being applied) due to the uncertainty of the

calibration and linear regression method can be calculated

using the regression parameters determined from the fol-

lowing relationship:

u xð Þ ¼ sresidual

b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

pþ 1

nþ x� �xð Þ2Pn

i¼1 xi � �xð Þ2

s

ð11Þ

where sresidual is residual standard deviation of the

calibration curve:

sresidual ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pni¼1 Ai � ðbxi þ aÞ½ �

n� 2

2

s

ð12Þ

Ai peak area

b slope of calibration curve

a intercept of calibration curve

p number of repeated measurements carried out for a

given sample

n the total number of calibration standards used for

plotting calibration curve

x analyte concentration (mass fraction or mass

concentration depending on chromatographic method

used)

�x mean value of all the concentrations of calibration

standards (x1, x2, …, xn)

xi concentration of i-th calibration standard

Table 2 Uncertainty associated with standard and sample preparation in NP HPLC and IC methods

Method Symbol Component x u(x) u(x)/x

NP HPLC PMA MA standard purity 0.995 g/g 0:0025ffiffi

3p g/g = 1.4 9 10-3 g/g 1.5 9 10-3

PDA DA standard purity 0.97 g/g 0:015ffiffi

3p g/g = 8.7 9 10-3 g/g 8.9 9 10-3

PTA TA standard purity 0.97 g/g 0:015ffiffi

3p g/g = 8.7 9 10-3 g/g 8.9 9 10-3

mst Standard mass 1 g 9:5�10�6ffiffiffiffi

10p g = 3.0 9 10-6 g 3 9 10-6

Vst Standard solution volume 10 mL 2.0 9 10-2 mL 2 9 10-3

msample Sample mass 50 g 1:2�10�5ffiffiffiffi

10p g = 3.8 9 10-6 g 8 9 10-8

Vsample Sample volume 5 mL 3.0 9 10-2 mL 6 9 10-3

IC P Standard purity 101 mg L-1 1:01ffiffi

3p mg L-1 = 5.8 9 10-1 mg L-1 5.7 9 10-3

VP Automatic pipette volume 5 mL 0:03ffiffi

6p mL = 1.2 9 10-2 mL 2.4 9 10-3

Vflask Flask volume 50 mL 3.0 9 10-2 mL 6 9 10-4

PMA purity of mono-aromatic standard, PDA purity of di-aromatic standard, PTA purity of tri-aromatic standard

uprep;NPHPLC ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u Pð ÞP

� �2

þ u mstð Þmst

� �2

þ u Vstð ÞVst

� �2

þu msample

� �

msample

� �2

þu Vsample

� �

Vsample

� �2s

ð9Þ


123

Relative standard uncertainty associated with calibration

curve is ucalib(x) = u(x)/x. Uncertainties associated with

calibration curve are shown in Table 4.

Uncertainty associated with method repeatability, urep

On of the main factor influencing measurement uncertainty

is uncertainty associated with repeatability of measure-

ments. All the repeatability contributions are combined

into one contribution for the overall experiment, and the

values from the method validation are used to quantify

their size. It is quantified in terms of the standard deviation

(s) of the measured value. Uncertainties associated with

method repeatability (urep = s) are shown in Table 4.

Combined uncertainty calculation

From quantified uncertainty components of each identi-

fied uncertainty source, combined standard measurement

uncertainty of overall analytical method can be calcu-

lated. Before combination, all uncertainty contributions

must be expressed as standard uncertainties, that is, as

standard deviations. For NP HPLC and IC method com-

bined uncertainty was evaluated from the following

expression:

uCðxÞ ¼ x �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u2prep þ u2

calib þ u2rep þ u2

bias

q

ð13Þ

For SEC method, uncertainty associated with sample

preparation is neglected; therefore, expression for the

determination of combined uncertainty for size exclusion

method is Eq. 13 reduced for factor uprep.

As already stated laboratory follows a procedure for

everyday certificated working material (CRM) control and

by that bias of analytical procedure is not significant. In

Fig. 4 is given an example of the control chart for aromatic

standard o-xylene. As can be seen from Fig. 4, deviation of

measured value and assigned value of CRM not exceed 5%

so one can conclude that bias is not significant.

Expanded uncertainty calculation

The final step is to derive expanded uncertainty based on

combined uncertainty determined in previous step. Expan-

ded measurement uncertainty is dimension which represents

the measurement result range inside which a result is with

certain level of confidence. Expanded uncertainty, U, can be

provided by multiplying combined standard measurement

uncertainty uC(x) with coverage factor k: U ¼ k � uC xð Þ. So,

measured result can be expressed as X ¼ x� U.

Selection of the coverage factor depends on confidence

level. Usually, k is 2 or 3 but it can have other values as

well. If it is case of normal distribution, selection of k = 2,

will have level of confidence 95% and with k = 3 level of

confidence is 99%.

Table 4 shows the obtained results which considered the

uncertainty associated with methods investigated in this

paper.

As can be seen from obtained results (Table 4.), uncer-

tainty associated with sample and standard preparation has

the smallest influence on the combined uncertainty. The

contribution of calibration procedure and method repeat-

ability shows the same order of magnitude, except for the

size exclusion method, where uncertainty associated with

repeatability is an order of magnitude higher. In real sample

analysis, it is quite common that the uncertainty associated

with the repeatability of measurement is frequently the main

element of the uncertainty budget.

Calibration curve uncertainty has the highest contribu-

tion on the total uncertainty in nitrate determination by ion

chromatography and after that the measurement repeat-

ability uncertainty occurs. Because of the measured lower

Table 3 Calibration curves and coefficients of correlation for NP

HPLC and IC methods

Analyte Calibration curve r2

Mono-aromatics (MA) y = 485590 9 x ? 3586.7 0.9999

Di-aromatics (DA) y = 851510 9 x ? 14331 0.9997

Tri-aromatics (TA) y = 1075200 9 x ? 1485 0.9989

Nitrates y = 0.20504 9 x ? 0.218434 0.9995

y peak area (number of pixel), x analyte mass fraction (g/100 g) for

mono-, di- and tri-aromatics and mass concentration (mg L-1) for

nitrates

Table 4 Combined and expanded uncertainty for investigated chromatographic methods

Method Analyte uprep ucalib urep Analyte content uC(x) U(k=2)

NP HPLC MA 0.007 0.007 0.009 22.43 g/100 g 0.300 g/100 g 0.600 g/100 g

DA 0.011 0.013 0.020 1.18 g/100 g 0.031 g/100 g 0.062 g/100 g

TA 0.011 0.014 0.020 0.49 g/100 g 0.013 g/100 g 0.026 g/100 g

IC NO3- 0.006 0.035 0.028 0.22 mg L-1 0.010 mg L-1 0.020 mg L-1

SEC Polystyrene – 0.023 0.167 1320 g mol-1 222.5 g mol-1 445 g mol-1

NP HPLC normal phase high-performance liquid chromatography, IC ion chromatography, SEC size exclusion chromatography, MA mono-

aromatics, DA di-aromatics, TA tri-aromatic


123

concentration range, mistakes during the calibration are

bigger, so final results for uncertainties are expected.

As a result, in order to improve the method (reduced

uncertainty), the efforts should be directed toward decreasing

the uncertainty of sample measurement, and it is not helpful to

increase the number of data points on the calibration line.

Increase of the number of calibration points does not lead to

noticeable decrease in uncertainty [16]. In order to obtain

smaller calibration uncertainty, analyte concentration should

be in the middle of the concentration range of calibration

standards as can be concluded from Eq. 11.

Conclusion

In this paper, measurement uncertainties of three liquid

chromatography methods were evaluated (NP HPLC, IC,

and SEC) based on EURACHEM/CITAC guide. Detailed

analysis of contributions of the various uncertainty sources

in three chromatographic methods was carried out. Three

main components considered were as follows: uncertainty

associated with sample and standard preparation, uncertainty

associated with calibration curve, and uncertainty associated

with method repeatability. It was demonstrated that different

sources had different influences on measurement uncertainty

as well as same sources had different contribution in dif-

ferent technique despite the fact that all of them are

chromatographic technique. The sample preparation has a

minor influence on overall measurement uncertainty.

In all investigated liquid chromatography methods,

measurement repeatability uncertainty contribution is high.

The reason for that is in the fact that measurement

repeatability includes whole measurement process from

sample preparation over calibration process to final result

calculation.

The main conclusions from the analysis of uncertainty

budgets are the following:

• The uncertainty originating from sample and standard

preparation was small because it contains only standard

purity, weighting, and volume measurement.

• The uncertainty associated with calibration was

significant.

• The contribution of repeatability component gives a

large contribution to the combined uncertainty in all

investigated methods.

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-6,0

-5,0

-4,0

-3,0

-2,0

-1,0

0,0

1,0

2,0

3,0

4,0

5,0

6,0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

sample number

dev

iati

on

(%

)

Fig. 4 Control chart for aromatic standard, o-xylene


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Estimating measurement uncertainty in high-performance liquid chromatography methods

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