ANALYTICAL PROPERTIES

PART I

ERT 207 ANALYTICAL CHEMISTRY

SEMESTER 1, ACADEMIC SESSION 2015/16

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Overview

INTRODUCTION THE CHEMICAL METROLOGICAL

HIERARCHY: UNCERTAINTY AND TRUENESS

SYSTEMATIC ERRORS RANDOM ERRORS DISTRIBUTION OF EXPERIMENTAL

RESULTS STATISTICAL TREATMENT OF

RANDOM ERRORS bblee@unimap

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Overview

THE SAMPLE STANDARD DEVIATION

STANDARD ERROR OF THE MEAN

VARIANCE AND OTHER MEASURES OF PRECISION

REPORTING COMPUTED DATA

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INTRODUCTION

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Quality: is defined as the totality of features

(properties, attributes, capabilities) of an entity that make it equal to, better or worse than others of the same kind.

Figure1 provides an overview of analytical properties and classifies them into three hierarchical categories: (i) capital properties, (ii) basic properties

and (iii) accessory properties

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INTRODUCTION

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Figure 1: Types of analytical properties and relationships among them and with analytical quality (results and processes).

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INTRODUCTION

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Capital analytical properties are typical of results (analytical information). The quality of the analytical process is

obviously related to that of its results; consequently, basic properties support capital properties.

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THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS

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Subjecting n aliquots of the same sample to an analytical process provides n results. Obviously, the quality of the information

obtained will increase with increase in n. The lowest information level corresponds

to an individual result (Xi) provided by a single sample aliquot.

The mean obtained with n > 30, μ', will be of higher quality than the previous one.

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Figure 2:The chemical metrological hierarchy and its relationships to analytical properties

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THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS

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The theoretical upper quality limit for the series being the mean for a statistical population (n = α), which is denoted by μ.

When n sample aliquots are analysed by different laboratories using various analytical processes and a consensus is reached from a thorough technical study, a result X' is obtained that is held as true.

Generic and specific uncertainty in Analytical Chemistry can be efficiently defined via the chemical metrological hierarchy as schematized in Figure 3.

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THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS

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Figure 3: Graphical depiction of generic and specific uncertainty, and of their relationships to the chemical metrological hierarchy

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THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS

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The maximum possible uncertainty about the relative proportion (concentration, content) of an analyte in an unknown sample corresponds to absolute specific uncertainty, which ranges from 0.00% to 100.00% in percentage terms.

The true value, , is subject to zero specific uncertainty, which coincides with the absolute absence of uncertainty in the analyte proportion in the sample.

This is intrinsic information that corresponds to ideal quality.

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THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS

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Absolute trueness coincides with the sample's intrinsic information with the true value (X), and represents the maximum possible (ideal) quality level in Figure 4 and the top level in the metrological hierarchy of Figure 2.

Figure 5 illustrates different approaches to clarify the meaning of trueness in Analytical Chemistry.

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THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS

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Figure 4: Analytical chemical information levels ranked according to quality and to their relationship to trueness, accuracy and uncertainty

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THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS

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Intrinsic information: It is subject to no uncertainty,

represents the top level of analytical quality and is characterized via an ideal, unattainable property: trueness. At a lower level is

Referential information: It is factual, reasonably accurate -

nearly as much as intrinsic information - and scarcely uncertain.

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THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS

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It is obtained under uncommon conditions (e.g. in intercomparison tests involving many participating laboratories using different analytical processes)

It is basically used as a reference to extract ordinary analytical information or assess its goodness.

Actual information: it is also based on facts but is less

accurate and more uncertain than referential information.

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THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS

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Figure 5: Connotations of the word "trueness" in Analytical Chemistry

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THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS

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Figure 5 a: Expression of

a result (and'its

uncertainty) and

relationships to the

different types of error and to the analytical

properties accuracy and

precision

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SYSTEMATIC ERRORS

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Error: the uncertainty in a measurement.

Errors are often associated with mistakes or with something that is definitely wrong but scientists tend to think of them as either:(i) The discrepancy between a measured

value and some generally accepted true value (the accuracy of the measurement), or

(ii)The degree to which repeated measurement of a particular quantity gives the same result (precision).

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SYSTEMATIC ERRORS

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An experimental method may be accurate but not especially precise if it suffers from random errors, but equally it may yield inaccurate results with high precision if it is subject to systematic errors.

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SYSTEMATIC ERRORS

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In an experiment free from serious systematic errors, we can improve the precision considerably by increasing the sample size.

Systematic errors always affect the accuracy of the result and will tend to do so in the same direction the answer will always tend to be too high or too low rather than fluctuating about some central value following successive measurements.

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SYSTEMATIC ERRORS

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As a consequence of the fact that they derive from deficiencies of one sort or another in the experimental apparatus, they cannot be quantified by a statistical analysis of repeated observations.

Figure 6: The frequency

distribution of twenty boiling point

measurements of an ester made with twenty different thermometers.

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RANDOM ERRORS

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Random errors are present in every measurement no matter how careful the experimenter.

The somewhat uncomfortable reality of experimental measurement is that two measurements of the same physical property with identical apparatus, using a nominally identical procedure, will almost invariably yield slightly different results.

Repeating the measurement further will continue to yield slightly different values each time.

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RANDOM ERRORS

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Such variations result from random fluctuations in the experimental conditions from one measurement to the next and from limitations associated with the precision of the apparatus or the technique of whoever is conducting the experiment.

The statistical nature of these fluctuations means that the discrepancies with respect to the ‘true’ value are equally likely to be positive or negative.

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DISTRIBUTION OF EXPERIMENTAL RESULTS

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When a sufficiently large number of measurements, s frequency distribution like that shown in Figure 7 (a).

Theoretical distribution for ten equal-sized uncertainties is shown in Figure 7 (b).

When a same procedure is applied to a very large number of individual errors, a bell-shaped curve like that shown in Figure 7(c).

Such a plot is called a Gaussian curve or a normal error curve.

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DISTRIBUTION OF EXPERIMENTAL RESULTS

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Figure 7: Frequency distribution for measurements containing (a) four random

uncertainties (4U).

(b) ten random uncertainties (10 U)

(c) a very large number of random uncertainties.

(a)

(b)

(c)

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DISTRIBUTION OF EXPERIMENTAL RESULTS

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A Gaussian or normal error curve: A curve that shows the symmetrical distribution of data around the mean of an infinite set of data.

A random uncertainties (± U): An assumed fixed amount (high or low) of

error where each error has an equal probability of occurring and that each can cause the final result.

The spread in a set of replicate measurements is the difference between the highest and lowest result.

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DISTRIBUTION OF EXPERIMENTAL RESULTS

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A histogram is a bar graph.

Figure 8: A histogram showing distribution of the results and a Gaussian curve for data having the same mean & standard deviation

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STATISTICAL TREATMENT OF RANDOM ERRORS

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Statistical analysis: Only reveals information that is already present in a data set.

No new information is created by statistical treatments.

Statistical methods do not allow us to categorize and characterize data in different ways and to make objective and intelligent decisions about data quality and interpretation.

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STATISTICAL TREATMENT OF RANDOM ERRORS

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As a rule of thumb, if we have more than 30 results and the data are not heavily skewed, we can safely use a Gaussian distribution.

(1) Samples and Populations We gather information about a

population or universe from observation made on a subset or sample.

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STATISTICAL TREATMENT OF RANDOM ERRORS

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A population: the collection of all measurements of

interest to the experiment, while a sample is a subset of measurement selected from the population.

(2) Properties of Gaussian curves Figure 9 shows two Gaussian curves.

A normalized Gaussian curve can be described by:

πσ

ey

σμx

2

22 2

Population

mean (μ)

Population

standard deviation

(σ)

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STATISTICAL TREATMENT OF RANDOM ERRORS

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Figure 9: Normal error curves. The standard deviation for curve B is twice that for curve A, σB= 2σA.(a)x-axis = deviation from mean (x-μ), (b)X-axis = deviation from mean (in σ).

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STATISTICAL TREATMENT OF RANDOM ERRORS

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Sample mean: The arithmetic average of

a limited sample drawn from a population of data.

Population mean: is the true mean for the population. In the absence of systematic error, μ is the true value for the measured quantity.

N

xx

N

ii

1

Number of measuremen

t in the sample set.

N

xμ

N

ii

1

Total number of measurement in the population.

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STATISTICAL TREATMENT OF RANDOM ERRORS

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Population standard deviation: A measure of the precision of the

population. Precision of the data of curve A is twice as good as that of curve B.

N

μxσ

N

ii

1

2 Number of data points making up

the population.

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STATISTICAL TREATMENT OF RANDOM ERRORS

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The quantity z represents the deviation of a result from the population mean relative to the standard deviation.

It is commonly given as a variable in statistical tables since it is a dimensionless quantity.

The equation for Gaussian error curve:

σ

μxz

πσ

e

πσ

ey

zσμx

22

22 222

σ2 =

variance

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STATISTICAL TREATMENT OF RANDOM ERRORS

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(3) Areas under a Gaussian Curve In reference to Figure 7, regardless of its

width, 68.3% if the area beneath a Gaussian curve for a population lies within one standard deviation (± 1σ) of the mean (μ).

Approximately 95.4% of all data points are within ± 2σ of the mean.

Approximately 99.7% data are within ± 3σ.

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STATISTICAL TREATMENT OF RANDOM ERRORS

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Area under the Gaussian curve:

68302

1

1

22

.dzπ

eArea

z

95402

2

2

22

.dzπ

eArea

z

99702

3

3

22

.dzπ

eArea

z

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THE SAMPLE STANDARD DEVIATION

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It is a measure of precision. The sample standard deviation:

The sample variance (s2) is also of importance in statistical calculations.

111

2

1

2

N

d

N

xxs

N

ii

N

ii

Deviation (di) of value xi from the

mean Ẋ.Number of degree of freedom

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STANDARD ERROR OF THE MEAN

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If a series of replicate results, each containing N measurements, are taken randomly from a population of results, each containing N measurements, are taken randomly from a population of results, the mean of each set will show less and less scatter as N increases.

Standard deviation of each mean is known as the standard error of the mean. N

ssm

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VARIANCE AND OTHER MEASURES OF PRECISION

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1) Variance It is just the square of the standard

deviation. It is an estimate of the population

variance σ2.

111

2

1

2

2

N

d

N

xxs

N

ii

N

ii

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VARIANCE AND OTHER MEASURES OF PRECISION

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2) Relative Standard Deviation (RSD) and Coefficient of Variation (CV)

Frequently standard deviations are given in relative rather than absolute terms.

The relative standard deviation is calculated by dividing the standard deviation by the mean value of the data set. x

ssRSD r

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VARIANCE & OTHER MEASURES OF PRECISION

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RSD is often expressed in parts per thousand (ppt):

RSD multiplied by 100% is called the coefficient of variation (CV).

3) Spread or Range (w) It is used to describe the precision of a

set of replicate results.

pptxx

spptinRSD 1000

%xx

spresentinRSDCV 100

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VARIANCE AND OTHER MEASURES OF PRECISION

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Example 1: From a set of data: 0.752, 0.756, 0.752, 0.751, 0.760 it was found that: , s = 0.0038 ppm Pb

Calculate:(a) the variance(b) The relative standard deviation (in ppt)(c) The coefficient of variation (d) The spread

ppm.x 7540

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REPORTING COMPUTED DATA

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1) Significant figures The significant figures in a number are

all of the certain digits plus the first uncertain digit.

Example: The liquid level in a buret: 30.24 ml

Rules of determining the number of significant figures:i. Disregard all initial zeros.ii. Disregard all final zeros unless they

follow a decimal point.

Certain

Uncertain

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REPORTING COMPUTED DATA

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iii. All remaining digits including zeros between nonzero digits are significant.

2) Significant figures in numerical computations

For addition & subtraction, the result should contain the same number of decimal places as the number with the smallest number of decimal places.

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REPORTING COMPUTED DATA

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For multiplication and division is the number with the smallest number of significant figures.

For logarithms and antilogarithms:(i) In a logarithm of a number, keep as many

digits to the right of the decimal point as there are significant figures in the original number.

(ii)In an antilogarithm of a number, keep as many digits as there are digits to the right of the decimal point in the original number.

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REPORTING COMPUTED DATA

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Example 2 Round the following calculated number

so that only significant digits are retained:(a) Log 4.000 x 10-5 = - 4.3979400(b) Antilog 12.5 = 3.162277 x 1012

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REPORTING COMPUTED DATA

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3) Rounding Data In rounding a number ending in 5,

always round so that the result ends with an even number. For example, 0.635 rounds to 0.64

0.625 rounds to 0.62

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EXAMPLE 1(a) s2 = (0.0038)2 = 1.4 x 10-5

(b) RSD = 0.0038 / 0.754 x 1000 ppt = 5.0 ppt

(c)CV = 0.0038 / 0.754 x 100% = 0.50%

(d) W = 0.760 – 0.751 = 0.009 ppm Pb.

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EXAMPLE 2(a) Log 4.000 x 10-5 = - 4.3979

[Rule 1]

(b) Antilog 12.5 = 3 x 1012

[Rule 2]