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Sampling, Statistics and

Electroanalysis

Dónal Leech

donal.leech@nuigalway.ie 

Ext 3563

Room C205, Physical Chemistry

http://www.nuigalway.ie/chemistry/staff/donal_leech/teaching.html  

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Analytical ChemistryDefinition: A scientific discipline that

develops and applies methods,instruments and strategies to obtain

information on the composition and nature

of matter in space and time.

Importance to Society: qual i tat ive  (what’s

there?) and quanti tat ive  (how much is

there) analysis   of clinical samples  (blood,

tissue and urine), industrial samples  (steel,

mining ores, plastics), pharmacologicalsamples  (drugs and medicines), food

samples  (agriculture) and environmental

samples  (quality of air, water, soil and

biological materials)

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The Analytical Approach

• Statement of Problem

• Definition of Objective

• Selection of Procedure

• Sampling, Sample Transport and Storage

• Sample Preparation

• Measurement/Determination

• Data Evaluation

• Conclusions and Report

Link: http://www.ivstandards.com/tech/reliability /

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Sampling

Definition: a defined procedure whereby a

part of a substance is taken to provide, fortesting, a representative sample of the whole

or as required by the appropriate

specification for which the substance is to be

tested.

Sampling from a shipload of ore for metal content?Sampling for mercury pollution in a stream?

Sampling clothing for propellant residues?

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How to decide?

• Size of bulk to be sampled

 – Shipload or biological cell?

• Physical state of fraction to be analysed – Solid, liquid, gas

• Chemistry of the material to be analysed

 – Searching for a specific species?

Sampling method is linked to the measurement

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Random Sampling

Random: to eliminate questions ofbias in selection. Three types.

• Simple: any sample has an equalchance of being selected 

examples 

stockpiles of cereals: take incrementsfrom surface and interior

compact solids: random drilling to sample manufactured products: divide batch (lot)

into imaginary segments and use arandom number generator to selectincrements to be sampled

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Example• School with a 1000 students, divided equally into boys

and girls. Want to select 100 of them for further study.You might put all their names in a drum and then pull100 names out. Not only does each person have anequal chance of being selected, we can also easilycalculate the probability of a given person being chosen,

since we know the sample size (n) and the population(N) and it becomes a simple matter of division:

• n/N x 100 or 100/1000 x 100 = 10%

• This means that every student in the school has a 10%or 1 in 10 chance of being selected using this method.

• For other populations, can replace names with anidentifier (number)

• Many computer statistical packages, including SPSS,are capable of generating random numbers

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Random Samplnig

• Systematic: first sample selectedrandomly and subsequent samplestaken at arranged intervals 

most commonly used procedure

examples  solid material in motion (conveyor belt):

periodically transfer portion into a samplecontainer

liquids: sample during discharge (from

tanks) at fixed time/volume increments NOTE: manufactured products: sample

more frequently at problematic times(changeover of shift, breaks etc.)

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Example

• Using the same example as before

(school). If the students in our school had

numbers attached to their names ranging

from 0001 to 1000, and we chose arandom starting point, e.g. 533, and then

pick every 10th name thereafter to give us

our sample of 100 (starting over with 0003after reaching 0993). The choice of the

first unit will determine the remainder.

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Random Sampling

• Stratified: the lot is subdivided and a simple randomsample selected from each stratus 

examples  scrap metals: sort into metal type before sampling

material lots delivered at different times: take proportional weightsof material from each lot

sedimented liquids: sample from decanted liquid and sediment byproportional weight, proportion the sample on the basis of volume

or depth

There are a number of potential problems with simple and systematic

random sampling.If the population is widely dispersed, it may be extremely costly to reach

them.

On the other hand, a current list of the whole population we are interested

in (sampling frame) may not be readily available.

Or perhaps, the population itself is not homogeneous and the sub-groups

are very different in size. In such a case, precision can be increased

through stratified sampling 

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Selective Sampling

Selective: screens out or selects materialswith certain characteristics 

Usually attempted following test results onrandom samples

examples 

contaminated foods: attempt to locate theadulterated portion of the lot

toxic gases in factory: total level acceptable buta localised sample may contain lethalconcentrations

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 A Composite Sample

Composite: portions ofmaterial selected inproportion to the amount ofmaterial they represent. The

ratio of the components

taken up to make thecomposite can be in terms

of bulk, time or flow.

• Reduces the cost ofanalysing large numbers ofsamples. Not a samplingtechnique; it is a preparatorytechnique after the samples

have been taken.

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Subsampling

samples received by analytical laboratory

are usually larger than that required for

analysis. Subsampling  of the laboratory

sample is done following homogenisationto give subsamples that are sufficiently

alike

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Continuous Monitoring

• Real-time measurements to provide detailon temporal variability (variability as afunction of time)

ExamplesIndustrial stack emissions (CO, NO2, SO2)

Workplace monitoring (radiation exposure,

toxic gases etc.)Smoke, heat and CO detectors

Water and air quality monitoring

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Sample QualityThe chain of events from the process of taking asample to the analysis is no stronger than itsweakest link.

Each sample should be registered  (have a uniquebarcode) and all details recorded  including the storageconditions and chain of contact.

details to consider:

• sample properties (e.g. volatility, sensitivity to light)

• appropriate container  (e.g. glass is not suitable for

inorganic trace analyses, low molecular weightpolyethylene is not suitable for hydrocarbon samples)

• length of holding time and conditions (e.g. creamseparates out from milk samples when left standing,sedimentation of particles in liquids occurs)

• amount of sample required to perform the analysis.

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Sample pre-treatment

Solids

• Grinding of solids

• Sample drying

• Leaching and extraction of solublecomponents

• Filtering of mixtures of solids, liquids

and gases to leave particulate (solid)matter

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Decomposition and dissolution of

solidsMost measurement methodologies depend upon

presentation of samples in liquid solutions

Preparation method will depend upon materialcomposition and analyte(s) targeted.

• Simple dissolution (appropriate

solvent/T/ultrasound)•  Acid treatment (strong and/or oxidising acids and

heat, see next slide).

• Fusion techniques –  Adding a flux (solid sodium carbonate, for example) and

heating, to aid dissolution – Expensive and last resort

http://www.informaworld.com/smpp/ftinterface~content=a741470469~fulltext=713240928 

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Nitric Acid treatment

• Nitric acid is acting:

as a strong acid where inorganic oxides are brought intosolution...

(1) CaO + 2H3O+ Ca+2 + 3H2O

as an oxidizing agent / acid combo where zero valenceinorganic metals and nonmetals are oxidized and broughtinto solution...

(2) Fe + 3H3O+ + 3HNO3 (conc.)Fe+3 + 3NO2 (brown) +

6H2O

or

(3) 3Cu + 6H3O+ + 2HNO3 (dilute) 2NO (clear) + 3Cu+2 +

10H2O

• In addition, nitric acid does not form any insolublecompounds with the metals and non-metals listed. Thesame cannot be said for sulfuric, hydrochloric,hydrofluoric, phosphoric, or perchloric acids.

Link: http://www.ivstandards.com/tech/reliability /

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Statistics

 An introduction to statistics is necessary inorder to explain the uncertainty associatedwith measurements and sampling.

One cannot go far in AnalyticalChemistry without encountering

statistics!

No quantitative results are of any valueunless they are accompanied by someestimate of the errors inherent in them

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Definitions

•  Arithmetic mean: average of all observations

n

 x

 x

n

i

i

 1

 

If the sample is random then the

arithmetic mean is the best

estimate of the population (true)mean, m 

1  1

2

21

2

2

n

 x x

 sn

 x xn

i

i

n

i

i

  

Variance: measures the extent to which the data differs in relation to

itself. Variance of population is the mean squared deviation from the

population mean, denoted σ2, while the variance of the sample data isdenoted s2.

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More Definitions• Standard deviation: the positive square root of the variance,

used also to indicate the extent to which data differs inrelation to itself.

• Probability distribution: It is possible to make an infinitenumber of measurements to determine the concentration ofan analyte. Normally a small number of test samples istaken…a statistical sample from the population. If there areno systematic errors, then the mean of the population (µ) isthe true value of the measurand. The mean of the sample

gives an estimate of µ.When repeat measurements are made they can take on, intheory, any value…….a Normal (Gaussian) distribution isthe mathematical model used to describe the continuousdistribution of values for repeat measurements, giving a

bell-shaped curve.

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Normal Distribution

 

2

2/exp 22  x y

0 20 40 60 80 100

     y

x

m is 50

is 5 (black dots)

 is 10 (red line)

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Normal Distribution

• Curve is symmetrical and centred at m.

• The greater the value of σ, the greater the

spread of the curve.

• Whatever values of µ and σ,

• 68.27% of observations are within µ  σ 

• 95.45% of observations are within µ  2 σ • 99.97% of observations are within µ  3 σ 

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Confidence Limits

Confidence limits: extreme values of the confidenceinterval which defines the range in which the true value ofa measurand is expected to be found. For small (n<30)samples the confidence limits can be given by:

where t  is the value determined from the Student’s t  distribution tables for a given confidence level and with(n-1) degrees of freedom (ν).

n st  x /

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Confidence Limits

n 90%

 95%

 99%

 99.9%

 2

3

4

5

10

20

30 

2.920

2.353

2.132

2.015

1.812

1.725

1.697 

4.303

3.182

2.776

2.571

2.228

2.086

2.042 

9.925

5.841

4.604

4.032

3.169

2.845

2.750 

31.596

12.941

8.610

6.869

4.587

3.850

3.646 

Worked example:

Fluoride content of a sample determined potentiometrically in water is (mg/l)

4.50, 3.80, 3.90, 4.20, 5.00 and 4.80 for separate analyses.

Mean = 4.37 Standard deviation = 0.48

90% confidence limits are:

µ = 4.37  2.015 x (0.48/6) = 4.37  0.39

99% confidence limits are:

µ = 4.37  4.032 x (0.48/6) = 4.37  0.79

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More useful definitions• Uncertainty: a parameter characterising the range of values within which the

value of the quantity being measured is expected to lie.use the confidence limits as estimates  of uncertainty

• Error: the difference between an individual result and the true value of thequantity being measured.

 Accuracy Precision

nearness of the result nearness of a series ofto the true value of the replicate measurements

quantity being measured to each other

determine by comparing determine by evaluating

result to those obtained the standard deviation or

using other methods and the confidence limitsother laboratories.

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Linear Calibration Curves

Straight-line plot takes the form:y = bx + a

correlation co-efficient, r:

thus +1  r  -1, the closer to 1 the value, the betterthe correlation.

 

2/1

22

 

 

i

i

i

i

iii

 y y x x

 y y x xr 

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Linear Regression

Linear regression of y on x:We seek a line that minimises the deviations in the y-

direction between the experimental points and the

calculated line (using the sum of the square of these

deviations)-method of “least squares”. 

 

 xb ya

 x x

 y y x x

b

i

i

i

ii

 

2

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Worked Example

0 2 4 6 8 10

0

5

10

15

20

25

       S       i     g     n     a       l

Concentration

Conc Signal

1 2.1

2 4.23 5.8

4 7

5 9.5

6 11.8

7 14

8 16.19 18.2

10 21

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Worked Example (Microcal Origin)

Conc Signal

1 2.1

2 4.23 5.8

4 7

5 9.5

6 11.8

7 14

8 16.19 18.2

10 21

0 2 4 6 8 10

0

5

10

15

20

25

       S       i     g     n     a       l

Concentration

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Results Log

Linear Regression for DATA1_B:

 Y = A + B * X

Parameter Value Error

------------------------------------------------------------A -0.46 0.33438

B 2.07818 0.05389

------------------------------------------------------------

R SD N P

------------------------------------------------------------0.99732 0.48948 10 <0.0001

------------------------------------------------------------

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How to do it in Excel!• Start EXCEL

• Input “Concentration” in cell A3  Input “Signal” in cell B3 

• Input Concentration data Input Signal data• Select Cells and use Chart Wizard to produce a chart: Use XY (Scatter)

and Chart Type1 (Scatter, Compare pairs of values, top chart)

• Input Chart title and input legends for the x and y-axes. Click onNext/Finish.

• To superscript the –1 on the x-axis, left click on the legend and then use

the cursor to select the –1 part of the legend. Click on Format/Selected Axis Title on the Menu. Check Superscript. Click OK. 

• To add the least squares line to the plot. 

• Left Click on the chart area (this will select the chart).

• Left Click on Chart on the Menu.

• Left Click on Add Trendline.

• Left Click on Linear.

• Left Click on Options and Check Display Equation on Chart and DisplayR-squared value on Chart.

• Click on OK.

• Move the text to the margins by dragging and dropping it.