4.2 Bias, Standards and Standardization

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4.2 Bias, Standards and Standardization

• bias and accuracy, estimation of bias

• origin of bias and the uncertainty in reference values

• quantifying by mass, chemical reactions, and physical methods

• standard reference materials

• primary, secondary and working standards

• methods of standardization: direct

calculation, direct calibration, standard

addition, internal standard, and calibration

graphs

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Bias and Accuracy

Bias is defined as the difference between the "true" value and the mean of the pdf of the measurement. An accurate measurement is one that is more or less free of bias.

An accurate measurement can have experimental error. However, the precision of the measurement affects our ability to estimate the mean of the pdf.

It is often impossible to determine accuracy because the true value is unknown. To avoid this dilemma, true values are replaced by reference values.

a) a real, although unknown, value such as Avogadro's number b) an assigned or agreed-upon value such as standard reference

materials from NIST

c) a value based on a definition followed by a series of experiments, such as the length of a meter.

Only type (b) can be used to determine bias.

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Mathematical Definition of Bias

Let x be a measurement and R be a reference value. The total error is (x-R) which can be written as,

(x-R) = (x-μ) + (μ-R)

where (x-μ) is the random error and (μ-R) is the bias. Accuracy implies that μ = R.

For linear equations, the dependent random variable has a bias which is the sum of the individual biases.

S = aA + bB + cC μ

_S

= aμ

_A

+ bμ

_B

+ cμ

_C

μ

_S

-S = a(μ

_A

-A) + b(μ

_B

-B) + c(μ

_C

-C)

Note that bias can be signed, thus it is possible for biases to cancel.

If the individual measurements are free of bias, the linear dependent

random variable is free of bias.

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Estimation of Bias

The t-test for one average can be used to identify the presence of bias. As an example consider the following data for the

spectrophotometric determination of beryllium [John Mandel, The Statistical Analysis of Experimental Data, Interscience, New York, 1964, p. 123]. The sample was a NIST standard known to contain 3.179 mg beryllium.

{3.167,3.177,3.177,3.169,3.173,3.177,3.177,3.177,3.171,3.169}

avg = 3.1734 mg; std.dev. of data = 0.00409

( )

3.1734 3.179 10

4.33 0.95,9 2.26

0.00409

calc table

t − t

= = =

Since t

_calc

> t

_table

, the measurement is biased. To estimate the range of the true bias, compute the t-interval about the bias mean.

2.26 0.00409

0.0056 0.0056 0.0029

10 0.0085 0.0027

B

B x R B ts μ N

μ

= −

= ± = − ± × = − ±

− ≤ ≤ −

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Origin of Bias

The most common source of bias for an inexperienced analyst is an error in understanding, or not following the procedure. A one-time procedural error is caught by analyzing replicates. A systematic procedural error is caught by analyzing standards.

For an experienced analyst the most common sources of bias are uncalibrated instrumentation, impure reagents, and the application of a familiar technique to a new type of sample.

Classification of bias into constant or proportional categories can often lead to identification of the error. Consider a spectrophoto- metric calibration curve.

• A constant error appears as a non-zero intercept. It might be due to an impurity in a buffer or an instrumental offset.

• A proportional error appears as an incorrect slope. It might be due to an impurity in the sample, or an incorrectly calibrated

photometric scale.

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Uncertainty in Reference Values

The reference value is determined using two or more methods. The methods are chosen such that the expected sources of bias will be different for each method. The values are then compared with the t- test for two averages. If they are statistically indistinguishable it is doubtful that bias is influencing any of the values.

Standards are also designed for specific methods of measurement, e.g. atomic absorption or ICP mass spectrometry. This helps

ensure that the instrumentation is not biasing the result.

Standards are prepared in matrices which are as close as possible to that found in the samples. For example, it is possible to

purchase from NIST salmon containing known levels of mercury.

Reference values will be provided with an upper bound for the total uncertainty: precision + bias. The analysis method will also be

described in sufficient detail to minimize bias originating from

changes to the procedure by the analyst.

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Quantifying by Mass

Large amounts of pure substances are expressed in moles, where 1 mole = 6.02214199×10

²³

±0.00000047×10

²³

molecules. The number of moles is related to mass by the expression, moles = mass/molecular mass. The mole depends upon both the definition of atomic mass and the kilogram.

Sample purity is a concern if a high level of accuracy is required.

For example, an analytical balance with Class-S weights can determine 100.00000 g to an accuracy of 0.00025 g. A

measurement limited by the balance would require a sample at least 99.99975% pure. At this level of accuracy sample-to-sample variations in isotope abundance can produce an error.

The key mole standard for much of analytical chemistry is silver

with an average mass of 107.8682 ±0.0002 g/mol. Silver has two

important properties. (1) It has an isotope distribution which is

virtually independent of mineral source. (2) It can be used to

standardize HCl, reducing reagents, and precipitating reagents.

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Quantifying by Other Methods

If a substance is impure or otherwise un-weighable, it can often be quantified by a specific reaction and identification of the equivalence point.

analyte + standard → products

Equivalence point methods are not useful for low concentrations, or the reaction may not be sufficiently specific. The method is

ultimately based on weighing the standard.

Physical methods are preferred for small concentrations of analyte.

As an example, Beer's Law relates concentration to sample

absorbance. To use Beer's Law, the molar absorptivity needs to be known. The value of molar absorptivity is ultimately based on

weighing a standard sample of the analyte. In this case the

standard must be the compound of interest. This can be a severe

restriction.

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NIST Standard Reference Materials

The National Institute for Standards and Technology (NIST) provides many services to the analytical chemistry community.

• calibration services for many types of instruments

• physical standards such as mass and length

• generation and archiving of standard reference data such as atomic masses

• standard reference materials (SRMs) that can be used to calibrate instrumentation

There are four major considerations involved in certifying an SRM.

• homogeneity - every sub-portion of a lot has to be statistically representative of the whole

• stability - the reference value should not change with time

• handing procedures - special handling such as temperature

• certified values - the stated uncertainties include possible sources of

bias as well as the propagation of precision

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Standards and Certified Reagents

A primary standard is a solid weighable compound which is used in equivalence point techniques. It has the following characteristics.

• the purity must be known and higher than the required accuracy of the analysis - usually 99.9% is sufficient

• it must be of known composition and react quantitatively with the analyte

• the composition should not change upon drying

• it and its salts should be soluble in water

• the molecular mass should be high

Certified reagents have a specified purity and come with a listing of known contaminants. The ACS provides specifications for hundreds of reagents, and companies manufacture reagents to these

specifications.

This link shows the scope of ACS involvement in the certification

activity: http://pubs.acs.org/reagent_demo/

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Example Reagents from Sigma

• standard base

sodium carbonate produced by the thermal decomposition of sodium oxalate; sodium oxalate, ACS certified reagent, 99.5%

tris-(hydroxymethyl) aminomethane (THAM or TRIS), primary standard, 99.9%

• standard acid

potassium hydrogen phthalate, primary standard, 99.95%

• standard reductant

sodium oxalate, ACS certified reagent, 99.5%

arseneous acid, As

₂

O

₃

, primary standard, 99.95%

• standard oxidant

potassium dichromate, ACS certified reagent, 99.0%

potassium bromate, primary standard, 99.8%

• standard precipitant

sodium chloride, ACS certified reagent, 99.0%

silver nitrate, ACS certified reagent, 99.0%

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Secondary and Working Standards

Secondary standards are compounds that do not have all of the attributes of a primary standard but are still useful in equivalence point techniques or physical methods.

• sodium hydroxide cannot be quantified by weighing and reacts with CO

₂

in air and glass surfaces - requires repetitive

standardization

• potassium permanganate is so reactive it needs repetitive standardization

• thiosulfate is eaten by air-borne bacteria

Working standards have a low or unknown accuracy, but a stable

response. Such standards are used to monitor instrumental drift,

where the response is used to normalize time-dependent signals.

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Direct Calculation

This method of standardization is used when there is a known

mathematical relationship between the measured response and the concentration.

R = kC

where R is the measured response, k is the known proportionality constant, and C is the analyte concentration. A good example is the titration of an unknown acid with a known base,

base acid acid

base

mL mL M

= M ×

where, mL

_acid

and M

_base

are known and mL

_base

is measured.

• one measurement per sample

• the standard and the analyte are different compounds

• cost effective for any number of samples

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Direct Calibration

This method of standardization is used when the measured response is known to be proportional to concentration, but the proportionality constant is unknown.

R = kC

An example would be spectrophotometry using Beer's law, A = εlC

where A is absorption, ε is the molar absorptivity, l is the

pathlength, and C is the molar concentration of analyte. The value of l is known, but ε needs to be determined in a separate

experiment.

• two measurements are required - standard and unknown

• the standard and unknown are the same compound

• cost effective when many determinations of the same compound

need to be made

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Standard Addition

This method of standardization is used when the proportionality constant is unknown and matrix dependent. The analyte

concentration is determined by a two-step process. (1) Run the unknown by itself.

R

_u

= kC

_u

(2) Run the unknown plus a small, known amount of analyte called the "spike."

R

_u+s

= k(C

_u

+ C

_s

)

The concentration of the unknown is obtained by solving the two equations.

u u s

u s u

C R C

R

₊

R

= −

• two measurements are required

• the standard and analyte are the same compound

• not cost effective with many samples (it would be preferable to

dilute into a solvent where a direct calibration could be used)

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Internal Standardization

This is a form of standardization used when the proportionality

constant is time dependent. The temporal dependence is eliminated by using a ratio of responses measured simultaneously. The second compound must have the same temporal response as the analyte.

R

_u

= k

_u

f(t)C

_u

R

_s

= k

_s

f(t)C

_s

First, known concentrations of both the analyte and standard are used to determine the time-independent ratio, K = k

_u

f(t)/k

_s

f(t) = k

_u

/k

_s

. Then the unknown and standard are measured

simultaneously using the known value K.

u s u

s

R R K C

= C

• four measurements are required

• the standard and analyte are different compounds

• internal standards can be inefficient unless there is an easy way to

make the simultaneous measurements

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