Determining the Heterogeneity of Reference Materials

Thomas Bagley, Dr. Cliff Stanley, Dr. John MurimbohDepts. of Earth & Environmental Science and Chemistry

Acadia University

Acknowledgements

Grants Participating Laboratories & CRM Manufacturers

• Acadia Faculty Research Funding

• Canada Summer Jobs Funding

•MAC Student Research Funding

•SEG Student Research Funding

• ACME Analytical Labs, Vancouver

• Bureau Veritas, Perth

• African Mineral Standards, Johannesburg Geostats Proprietary, Vancouver

• CDN Resource Laboratories

• Ore Research/Exploration, Melbourne

• Rocklabs, Auckland

Background

• Certified Reference Materials (CRMs): – Pulverized rock samples – With accepted element concentrations– With accepted standard deviations– Used to monitor analytical quality

Background

• How are CRMs used ?– Analyzed in a batch of samples

• CRM measured concentrations are compared with accepted concentrations to monitor accuracy– CRM measured concentrations are compared

with each other to monitor precision– These provide an assessment of data quality

for the batch

Control Chart

Background

• How are CRMs prepared ?– Pulverized– Homogenized – Sub-sampled– Chemically analyzed

(in multi-lab round robin)– Statistically analyzed

(to remove outliers)

Problem

• The variance observed in CRM concentrations is not all laboratory error

• CRMs are heterogeneous, and thus exhibit sampling error too

• But how heterogeneous are CRMs ?

Background

Outliers

Larger Samples ?

Background

• What contributes to the total variance of a CRM ?– Sampling Error– Laboratory Error– Inter-Laboratory Error

2222InterLabSampTot

Background

Outlying sample

Analysis oflarger samples ?Outlying lab

0.45

0.4

0.5

0.55

Background

• Inter-laboratory error and outliers are removed using inferential statistical tests

• Observed variation is now only the sum of lab error and sample heterogeneity

• Standard practice assumes that CRMs are homogenous; suggesting that observed variation is only lab error (NOT TRUE!)

222LabSampTot

Objective

• Need to determine the magnitude of sampling error in CRMs

Strategy

• Employ Dr. Stanley’s method to measure CRM lab and sampling errors simultaneously

– Measure CRM concentrations in small and large samples

– Solve using 3 equations / 3 unknowns

2.

2.

22.

2,

22.

2,

SmSampSmLgSampLg

LabSmSampSmTot

LabLgSampLgTot

MM

Methods – Analysis of CRMs

• Aqua Regia/ICP-MS for trace elements

• 9 * 2.00 g samples; • 36 * 0.50 g samples; • Data quality assessment samples

• Calculate large and small CRM sampling errors and laboratory error

2.

2.

,

,

SmTotSm

LgTotLg

M

M

22.

2. LabSmSampLgSamp

Sources of Error in Co

0

2

4

6

8

10

12

14

16

2P

1M 4E 7F

12

A

GS

50

20

B

BA

S-1

P4

B

BL

-10

P6

QU

A-1

(1)

QU

A-1

(2)

%R

SD

Small Sampling Error

Large Sampling Error

Analytical Error

Results%

Rel

ativ

e S

tan

dar

d

Dev

iati

on

Fundamental Sampling Constant

• Unique to pulverized material• Relates sample size to sampling error

(inversely proportional)• For a given sample size, sampling error is

known

ΨσMσM SmSampSmLgSampLg 2,

2,

ResultsVariance Ratio Plot of Co in Reference Materials

0.1

1

10

2P 1M 4E 7F

12A

GS

50

20B

BA

S-1

P4B

BL

-10

P6

QU

A-1

(1)

QU

A-1

(2)

Reference Material Batch

Lar

ge

Var

ian

ce

/ S

mal

l Var

ian

ce

02Lab σ

0and

0

2Sm.Samp

2Lg.Samp

σ

σ

Discussion

• The large & small sampling, and laboratory variances exhibit error

• Sampling and Laboratory error measurements are dependent on adequate total variance estimates

• How do we achieve adequate estimates of the total variance?

Discussion

• Standard error is dependent on the sampling & analytical variances, and the number of samples

• It is best to estimate total variances with the same standard error

1n

2s

1n

2sSE

Lg

2Lab

Lg

2Lg.Samps2 Lg.Tot

1n

2s

1n

2sSE

Sm

2Lab

Sm

2Sm.Samps2 Sm.Tot

Discussion

Discussion

Discussion

• To determine the optimal sampling strategy (equal standard errors)

1)(

)1n()1(n

2S

2

L

κλ

λ

2Lg.Samp

2Lab

Sm

Lg

s

s,

M

M λκ

Discussion

4M

M

Sm

Lg

Discussion

8M

M

Sm

Lg

Conclusions

• Reference material heterogeneity it is not zero

• Fundamental sampling constants can be used to estimate sampling error at different sample masses

• CRM manufacturers should provide fundamental sampling constants with their accepted values

• Analytical procedures should be designed to optimize standard error on the variance

Future work

• Investigate the controls on sampling strategy, to maximize precision

• Develop QAQC methods that accommodate reference material heterogeneity

References

• Stanley, C.R. 2007. The Fundamental Relationship Between Sample Mass and Sampling Variance in Real Geological Samples and Corresponding Statistical Models. Exploration and Mining Geology, 16: 109-123.

• Stanley, C.R., and Smee, B.W. 2007. Strategies for Reducing Sampling Errors in Exploration and Resource Definition Drilling Programmes for Gold Deposits. Geochemistry: Exploration, Environment, Analysis, 7: 1-12.

• Stanley, C.R, O'Driscoll, N., and Ranjan, P. 2010. Determining the magnitude of true analytical error in geochemical analysis. Geochemistry: Exploration, Environment, Analysis, 10: 355–364