identifying and quantifying the uncertainty associated ... · process outline for analysis of au...

© SPEX CertiPrep 2011

Vanaja Sivakumar, PhDVice President, Manufacturing

10/27/2011

Identifying and Quantifying the Uncertainty Associated with Instrumental Analysis


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Topics Covered

Precision & AccuracyPrecision & Accuracy

Confidence LimitsConfidence Limits

Errors in MeasurementsErrors in Measurements

Uncertainty in MeasurementsUncertainty in Measurements


True Value (TV)

MethodMethod

InstrumentInstrument

Skill of the OperatorSkill of the Operator


Sources of Error (E)

Determinate (Systematic)

• Improper Calibration• Personal Errors• Error in Methods

Indeterminate (Random)

• Experimental Errors

Spurious

• Human Errors• Instrument Failures


Central tendency

Not an absolute value

Estimate of true value

Average (0)

Sum of the results of each measurement

number of measurementsAverage (0) = 0 =

…

True value is described by average and standard deviation

or

Average is:


Standard Deviation (s)

0 ± 1.96 s at 95% Confidence Interval

0 ± 2.94 s at 99% Confidence Interval

Standard Deviation (s)

You can narrow your estimate of True Value using standard deviation

Can also be described as sigma,

True Value can be expressed as:

0

1


Confidence Intervals

The Average gives an estimate of the true value

The Confidence Interval can pinpoint the range within which the true value lies

– Also called Interval Estimate or Confidence Limit

Confidence Intervals are usually expressed at 90%, 95% or 99%

-3 -2 - 0 2 3


Accuracy & Precision

PrecisionPrecision• Reproducibility of the method • Evaluated by Average Deviation,

Variance and Standard Deviation

AccuracyAccuracy• Correctness of the measurement

in relation to the true value• (TV=0 ± E) or Measured value ± E

True Value ( ) = 8.50

7

8

9

10

1 2 3 4 5 6 7 8

Valu

es

Measurement #

Good Accuracy (0 = 8.55)

7

8

9

10

1 2 3 4 5 6 7 8

Valu

es

Measurement #

Poor Accuracy (0 = 7.39)

7

8

9

10

1 2 3 4 5 6 7 8

Valu

es

Measurement #

Good Precision (0 = 8.55)

7

8

9

10

1 2 3 4 5 6 7 8

Valu

es

Measurement #

Bad Precision (0 = 8.55)


Evaluation of Precision

Key Word Symbol Equation Description

Average Deviation

AD

The smaller the AD, the more precise the measurements

Not an accurate measure of precision

Variance V

Indicates the spread of measurements around the data

Better measure of precision

StandardDeviation s The amount of variation or

spread from the True Value

AD

s 0

1

0

0 2

1


Glossary Review

Key Word Symbol Equation

Average 0

Standard Deviation s

0 …

s 0

1


Uncertainty

Uncertainty is a parameter associated with the result of measurement, that characterizes the dispersion of the values that could be reasonably attributed to measured value

Error vs. Uncertainty:

– Error: Usually can’t be estimated

– Uncertainty: Can be estimated


Uncertainty Estimation

Determine what is to be measuredDetermine what is to be measured

Outline the various processesOutline the various processes

Identify sources of uncertaintyIdentify sources of uncertainty

Estimate uncertainties from each sourceEstimate uncertainties from each source

Combine and expand all uncertaintyCombine and expand all uncertainty


Types of Uncertainty

Type AType A• Usually associated with

repeated measurements• Type A uncertainty is

expressed as:

Type BType B• Based on scientific judgment

made from previous experience, manufactures’ specifications, etc.

• Three common models are:

Where s is standard deviation and n is the number of replicates

o Rectangularo Triangularo Normal


Type B: Rectangular Distribution

Use this when a certificate or other specification gives limits without specifying a level of confidence

Normalizing factor for converting to Standard Uncertainty is

X

3


Type B: Triangular Distribution

This is used when distribution is symmetric and when values close to the target value are more likely than near the boundaries

Example: Uncertainty associated with volumetric glassware


6

X


Type B: Normal Distribution

This is used when an estimate is made from repeated observations of a randomly varying process and an uncertainty is associated with a certain confidence interval.

Example: A calibration certificate with stated level of confidence.


X

2


Combined Uncertainty (uc)

We will be using the following model to combine overall uncertainties from all of the processes:

We will use the following model for calculating interim uc within a process:


Expanded Uncertainty (U)

The Expanded Uncertainty defines the interval within which lies the value of the measurand

U ku

To calculate, multiply the combined standard uncertainty with a Coverage Factor (k):

Coverage factor k depends on the level of confidence and the degree of freedom:

k= 2 for 95% confidence level

TrueValue X 0 U

m1

m1 should one of these be X bar?msnyder, 10/25/2011


Glossary Review

Key Word Symbol Equation

Average 0

Standard Deviation s

Combined Uncertainty

Expanded Uncertainty

0 …

s 0

1


Uncertainty in the measurement of a gold

solution, analyzed by ICP


Review of the Steps

Determine what is to be measuredDetermine what is to be measured

Outline the various processesOutline the various processes

Identify sources of uncertaintyIdentify sources of uncertainty

Estimate uncertainties for each sourceEstimate uncertainties for each source

Combine all the components and expandCombine all the components and expand


Process Outline for Analysis of Au

Components: Symbol Value

NIST SRM conc• Weighing• Flask• SRM value

Stdp 100.225 mg/L

Sample Dilution• Pipette• Flask

Crmf 100 fold

SRM Measured conc Stdm 100.985 mg/L

CRM Measured conc Crmm 100.797 mg/L


Determination of concentration

C Crm Std Crm

Std

Where:

Cau = Concentration of Au in sample

Crmm = ICP measured concentration of sample

Crmf = Dilution factor for sample

Stdp = Prepared concentration of SRM

Stdm = ICP measured concentration of standard

C100.797 100.225 100

100.985

10,004mg/L

Component Symbol ValueNIST SRM conc

• Weighing• Flask• SRM value

Stdp 100.225mg/L

Sample Dilution• Pipette• Flask

Crmf 100 fold

SRM Measured conc StdM 100.985 mg/L

CRM Measured conc CrmM 100.797 mg/L


ICP-Sample measurement (Crmm)

ICP-SRM measurement (Srmm)

Tolerance Temp

Vol

COA

Prep. SRM (Stdp)

Mass

Uncertainty Sources

The aim is to identify all major uncertainty sources and to understand their effect on the analyte and its uncertainty

Uc(Cau)

Pipette

Flask

Sample dil (Crmf)Tolerance Temp

Tolerance Temp


Uncertainty from SRM - U(Stdp)

Weighing on the balanceWeighing on the balance

500 mL Flask

SRM Certification

Used NIST SRM 3121 for calibration

To prepare the SRM:

– Weighed 5.067 g

– Diluted to 500 mL in a volumetric flask

There are 3 uncertainties associated with this process:


U(Stdp) – From the Balance

Weighing on the balanceWeighing on the balance 500 mL Flask SRM

Certification

1√3

Device Value (V) Standard u Combined u(uc)

Relative u( )

Balance 5.0670 g 8.165 x 10-5 8.165 x 10-5 1.611 x 10-5 2.597 x 10-10

UncertaintyfrombalancecalibrationListed

NormalizationFactorforRectangularDistribution

0.00013

5.774 10

StandardUncertainty TareWeight GrossWeight

5.774 10 5.774 10

8.165 10


U(Stdp) – From the Flask


SRM Certification

500 mL Flask

There are 2 uncertainties associated with the dilution:

uListedTolerance

60.26

.

.

uThermalCoefficientofWater TempVariation Volume


2.1x10 oC/mL 3oC 500mL 3

, triangular


U(Stdp) – From the Flask (cont.)

Device Value(V) uvolume utemp

Combined u(uc)

Relative u( )

Flask 500 mL 0.08165 0.1819 0.19935 0.0003987 1.5897 x 10-7


SRM Certification

500 mL Flask

CombinedUncertaintyu

0.08165 0.1819

0.19935 Relative

0.19935500

0.0003987


U(Stdp) – From the SRM Cert.

Weighing on the balanceWeighing on the balance 500 mL Flask

DeviceCertified

SRM Value(V)

Expanded u Coverage Factor Standard u Relative u

COA 9.89 mg/g 0.02 2 0.01 1.0111 x 10-3 1.0224 x 10-6

SRM Certification

Standardu Expanded

CoverageFactor

0.022

0.01

Relativeu Standard

V

0.019.89

1.011x10


Total Uncertainty Due to Stdp

Add each uncertainty component from the preparation of the standard…

Balance 1.61 x 10-5 2.597 x 10-10

500 mL Flask 0.0003987 1.5897 x 10-7

SRM Certification 0.001011 1.0224 x 10-6

Total: 1.1816 x 10-6


Uncertainty from Sample Dilution -U(Crmf)

500 mL FlaskPipette

To prepare the sample:

– A 5 mL of the sample was diluted to 500 mL

There are 2 uncertainties associated with Crmf:


U(Crmf) – From the Pipette

uListedu

NormalizationFactorforTriangularDistribution

0.016

.

.

Pipette 500 mL Flask

uThermalCoefficientofWater TempVariation Volume


2.1x10 oC/mL 3oC 5mL 3


Pipette 500 mL Flask

CombinedUncertainty

0.004082 0.1819x10

0.004469

RelativeuCombined

V

0.0044695

.

U(Crmf) – From the Pipette (cont.)


Pipette

U(Crmf) – From the Flask

Flask

uListedu

NormalizationFactor

0.26

0.08165

.

ThermalCoefficientofWater TempVariation Volume NormalizationFactorforRectangularDistribution

2.1x10 oC/mL 3oC 500mL 3

TypeB, triangular


U(Crmf) – From the Flask (cont.)

Pipette Flask

Value(V) utol utemp

Combined u(uc)

Relative u( )

Pipette 5 mL 0.004082 1.819 x 10-3 0.004469 0.0008938 7.989 x 10-7

Flask 500 mL 0.08165 0.1819 0.19935 0.0003987 1.5896 x 10-7

Total 9.5793 x 10-7

CombinedUncertainty

0.08165 0.1819

0.19935

RelativeuuV

0.19935500

0.0003987


ICP Measurement for standard-U(Stdm)

Measurement Stdm conc (mg/L)

1 101.260

2 100.853

3 100.985

4 101.542

5 100.914

6 100.685

7 101.251

8 100.720

9 100.654

0 100.985

s 0.29397

0

1s 0.29397mg/L

0… 100.985mg/L


Standard Concentration Measurement-U(Stdm)

0.293979

0.09799

0.09799100.985

0.00097035

Value(V) sd # reps uc

Stdm 100.985 0.29397 9 0.09799 0.00097035 9.4158 x 10-7


ICP Measurement for sample-U(Crmm)

Measurement Crmm conc (mg/L)

1 100.200

2 100.781

3 101.054

4 101.108

5 100.775

6 100.854

7 100.578

8 101.075

9 100.752

0 100.797

s 0.3065

0.3065mg/Lx 0

n 1s

0…

100.797mg/L


Sample Concentration Measurement-U(Crmm)

Value(V) sd # reps Uc

Crmm 100.797 0.3065 9 0.102165 0.0010136 1.0273 x 10-6

usd9

0.30659

0.102165

RelativeuuV

0.102165 100.797

0.0010136


Concentration And Uncertainty of Au Solution By ICP-OES

Symbol Input

Stdp Uncertainty SRM prep 1.1816 x 10-6

Crmf Uncertainty sample prep 9.5793 x 10-7

StdmUncertainty SRM

measurement 9.4158 x 10-7

CrmmUncertainty sample

measurement 1.0273 x 10-6

Total uc 4.10843 x 10-6

0.002026927

Uc (Cau) 0.002027 x (10004) 20.2771


The Certified Value And The Expanded Uncertainty U for Gold Solution By ICP Analysis

The ICP value for Gold = 10,004 ± 41 mg/L

2 20.27740.554mg/L

CoverageFactorfor95%CI TotalCombinedUncertainty uc


References

Kocherlakota, N, Obenauf, R, “A statistical approach to reporting uncertainty”

Quantifying Uncertainty Measurements, Eurachem/CETAC Guide

Kocherlakota, N, Obenauf, R, “Quantitative Calculation of Uncertainty Associated with Gold Reference Material,” To be published by NIST

British Standards 1797:1952


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