Identifying and Quantifying the Uncertainty Associated with Instrumental Analysis Vanaja Sivakumar, PhD Vice President, Manufacturing 10/27/2011 © SPEX CertiPrep 2011

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Topics Covered

Precision & Accuracy Confidence Limits Errors in Measurements Uncertainty in Measurements

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True Value (TV)

Method Instrument Skill of the Operator © SPEX CertiPrep 2011

Sources of Error (E)

Determinate (Systematic)

Indeterminate (Random)

• Improper Calibration • Personal Errors • Error in Methods

• Experimental Errors

Spurious • Human Errors • Instrument Failures

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Average (0) Average (0) =

Sum of the results of each measurement number of measurements

or

0=

Average is:

 Central tendency  Not an absolute value  Estimate of true value True value is described by average and standard deviation

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…

Standard Deviation (s) Standard Deviation

(s)

0 1

 You can narrow your estimate of True Value using standard deviation  Can also be described as sigma, 

True Value can be expressed as: 0 ± 1.96 s at 95% Confidence Interval 0 ± 2.94 s at 99% Confidence Interval © SPEX CertiPrep 2011

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 © SPEX CertiPrep 2011

Accuracy & Precision Accuracy

Precision

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

• Reproducibility of the method • Evaluated by Average Deviation, Variance and Standard Deviation

Good Accuracy (0 = 8.55)

Good Precision (0 = 8.55)

10 Values

Values

10 9 8

9 8 7

7

1

2

3

4

5

6

7

1

8

2

Measurement #

True Value ( Poor Accuracy (0 = 7.39)

10

9

9

Values

Values

7

8

) = 8.50

10

8

3 4 5 6 Measurement #

Bad Precision (0 = 8.55)

8 7

7 1

2

3 4 5 6 Measurement #

7

8

1

2

3 4 5 6 Measurement #

7

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8

Evaluation of Precision Key Word

Symbol

Equation

Description 

Average Deviation

AD

AD

0  

Variance

Standard Deviation

0 1

V

s

s

2



0 1



The smaller the AD, the more precise the measurements Not an accurate measure of precision Indicates the spread of measurements around the data Better measure of precision

The amount of variation or spread from the True Value

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Glossary Review Key Word

Symbol

Equation

Average

0

0

Standard Deviation

s

s

…

0 1

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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

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Uncertainty Estimation Determine what is to be measured Outline the various processes Identify sources of uncertainty Estimate uncertainties from each source Combine and expand all uncertainty © SPEX CertiPrep 2011

Types of Uncertainty

Type A

Type B

• Usually associated with repeated measurements • Type A uncertainty is expressed as:

• 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 o o

Rectangular Triangular Normal

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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 © SPEX CertiPrep 2011

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  Normalizing factor for converting to Standard Uncertainty is

6

X © SPEX CertiPrep 2011

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.  Normalizing factor for converting to Standard Uncertainty is



2

X © SPEX CertiPrep 2011

Combined Uncertainty (uc) We will use the following model for calculating interim uc within a process:

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

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Expanded Uncertainty (U)  The Expanded Uncertainty defines the interval within which lies the value of the measurand  To calculate, multiply the combined standard uncertainty with a Coverage Factor (k): U

ku

 Coverage factor k depends on the level of confidence and the degree of freedom:  k= 2 for 95% confidence level m1

True Value X

0

U

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Slide 19 m1

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

Glossary Review Key Word

Symbol

Equation

Average

0

0

Standard Deviation

s

s

…

0 1

Combined Uncertainty

Expanded Uncertainty

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Uncertainty in the measurement of a gold solution, analyzed by ICP © SPEX CertiPrep 2011

Review of the Steps Determine what is to be measured Outline the various processes Identify sources of uncertainty Estimate uncertainties for each source Combine all the components and expand © SPEX CertiPrep 2011

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

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Determination of concentration Crm

C



Std Std

Crm

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

Component

C

100.797 100.225 100 100.985 10,004 mg/L

Symbol

Value

NIST 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

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Uncertainty Sources The aim is to identify all major uncertainty sources and to understand their effect on the analyte and its uncertainty ICP-SRM measurement (Srmm)

Prep. SRM (Stdp)

ICP-Sample measurement (Crmm)

Tolerance Temp

Vol Mass COA

Uc(Cau) Tolerance

Temp Flask

Pipette Tolerance

Temp

Sample dil (Crmf) © SPEX CertiPrep 2011

Uncertainty from SRM - U(Stdp)  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: Weighing on the balance

500 mL Flask

SRM Certification

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U(Stdp) – From the Balance Weighing on the balance Uncertainty from balance calibration

SRM Certification

500 mL Flask Listed Normalization Factor for Rectangular Distribution

0.0001 3

1 √3

5.774 Standard Uncertainty

Tare Weight 5.774 8.165

Device Balance

Value (V) 5.0670 g

10

10

Gross Weight 5.774

10

10

Standard u

Combined u (uc)

Relative u ( )

8.165 x 10-5

8.165 x 10-5

1.611 x 10-5

2.597 x 10-10 © SPEX CertiPrep 2011

U(Stdp) – From the Flask Weighing on the balance

500 mL Flask

SRM Certification

There are 2 uncertainties associated with the dilution: u

Listed Tolerance , triangular

6 0.2 6 .

u



Thermal Coefficient of Water Temp Variation Volume Normalization Factor for Rectangular Distribution 2.1 x 10

oC/mL

3oC 500 mL

3 . © SPEX CertiPrep 2011

U(Stdp) – From the Flask (cont.) Weighing on the balance

Combined Uncertainty u

SRM Certification

500 mL Flask



0.08165

0.1819 Relative

0.19935



0.19935 500 0.0003987

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 © SPEX CertiPrep 2011

U(Stdp) – From the SRM Cert. Weighing on the balance

Standard u

SRM Certification

500 mL Flask



Expanded Coverage Factor



0.02 2

Standard V 0.01 9.89

0.01

1.011x 10

Relative u



Device

Certified 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

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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

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Uncertainty from Sample Dilution U(Crmf)  To prepare the sample: – A 5 mL of the sample was diluted to 500 mL

There are 2 uncertainties associated with Crmf: Pipette

500 mL Flask

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U(Crmf) – From the Pipette 500 mL Flask

Pipette

u

Listed u Normalization Factor for Triangular Distribution 0.01 6 .

u

Thermal Coefficient of Water Temp Variation Volume Normalization Factor for Rectangular Distribution 2.1 x 10

oC/mL

3oC 5 mL

3 .

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U(Crmf) – From the Pipette (cont.) 500 mL Flask

Pipette

Combined Uncertainty 0.004082

0.1819 x 10

0.004469

Relative u

Combined V 0.004469 5 . © SPEX CertiPrep 2011

U(Crmf) – From the Flask Pipette

u

Flask

Listed u Normalization Factor Type B, triangular 0.2 6 0.08165

Thermal Coefficient of Water Temp Variation Volume Normalization Factor for Rectangular Distribution 2.1 x 10

oC/mL

3oC 500 mL

3 . © SPEX CertiPrep 2011

U(Crmf) – From the Flask (cont.) Pipette

Flask

Relative u

Combined Uncertainty 0.08165

0.19935 500

0.1819

0.19935

0.0003987

utemp

Combined u (uc)

Relative u

0.004082

1.819 x 10-3

0.004469

0.0008938

7.989 x 10-7

0.08165

0.1819

0.19935

0.0003987

1.5896 x 10-7

Value (V)

utol

Pipette

5 mL

Flask

500 mL

Total

u V

(

)

9.5793 x 10-7

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ICP Measurement for standardU(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

s

…

100.985 mg/L

0 1

0.29397 mg/L

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Standard Concentration MeasurementU(Stdm)

0.29397

0.09799 100.985

9 0.09799

Stdm

0.00097035

Value (V)

sd

# reps

uc

100.985

0.29397

9

0.09799

0.00097035

9.4158 x 10-7

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ICP Measurement for sampleU(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

s

…

x n

100.797 mg/L

0 1

0.3065 mg/L

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Sample Concentration MeasurementU(Crmm)

u

sd

Relative u

9 0.3065

0.102165 100.797

9 0.102165

Crmm

u V

0.0010136

Value (V)

sd

# reps

Uc

100.797

0.3065

9

0.102165

0.0010136

1.0273 x 10-6

© SPEX CertiPrep 2011

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

Stdm

Uncertainty SRM measurement

9.4158 x 10-7

Crmm

Uncertainty sample measurement

1.0273 x 10-6

Total uc

4.10843 x 10-6

0.002026927 Uc (Cau)

0.002027 x (10004)

20.2771 © SPEX CertiPrep 2011

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



Coverage Factor for 95% CI 2

Total Combined Uncertainty uc

20.277

40.554 mg/L

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

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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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