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
Analytical Chemistry
Definition: 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: qualitative (what’s
there?) and quantitative (how much is
there) analysis of clinical samples (blood,
tissue and urine), industrial samples (steel,
mining ores, plastics), pharmacological
samples (drugs and medicines), food
samples (agriculture) and environmental
samples (quality of air, water, soil and
biological materials)
The Analytical Approach
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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/
Sampling
Definition: a defined procedure whereby a
part of a substance is taken to provide, for
testing, 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?
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
Random Sampling
Random: to eliminate questions of
bias in selection. Three types.
• Simple: any sample has an equal
chance of being selected
examples
 stockpiles of cereals: take increments
from surface and interior
 compact solids: random drilling to sample
 manufactured products: divide batch (lot)
into imaginary segments and use a
random number generator to select
increments to be sampled
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 pull
100 names out. Not only does each person have an
equal chance of being selected, we can also easily
calculate 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 an
identifier (number)
• Many computer statistical packages, including SPSS,
are capable of generating random numbers
Random Samplnig
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Systematic: first sample selected
randomly and subsequent samples
taken at arranged intervals
most commonly used procedure
examples
 solid material in motion (conveyor belt):
periodically transfer portion into a sample
container
 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.)
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 a
random starting point, e.g. 533, and then
pick every 10th name thereafter to give us
our sample of 100 (starting over with 0003
after reaching 0993). The choice of the
first unit will determine the remainder.
Random Sampling
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
•
Stratified: the lot is subdivided and a simple random
sample selected from each stratus
examples

scrap metals: sort into metal type before sampling

material lots delivered at different times: take proportional weights
of material from each lot

sedimented liquids: sample from decanted liquid and sediment by
proportional weight, proportion the sample on the basis of volume
or depth
Selective Sampling
Selective: screens out or selects materials
with certain characteristics
Usually attempted following test results on
random samples
examples
 contaminated foods: attempt to locate the
adulterated portion of the lot
 toxic gases in factory: total level acceptable but
a localised sample may contain lethal
concentrations
A Composite Sample
Composite: portions of
material selected in
proportion to the amount of
material they represent. The
ratio of the components
taken up to make the
composite can be in terms
of bulk, time or flow.
• Reduces the cost of
analysing large numbers of
samples. Not a sampling
technique; it is a preparatory
technique after the samples
have been taken.
Subsampling
samples received by analytical laboratory
are usually larger than that required for
analysis. Subsampling of the laboratory
sample is done following homogenisation
to give subsamples that are sufficiently
alike
Continuous Monitoring
• Real-time measurements to provide detail
on temporal variability (variability as a
function of time)
Examples
Industrial stack emissions (CO, NO2, SO2)
Workplace monitoring (radiation exposure,
toxic gases etc.)
Smoke, heat and CO detectors
Water and air quality monitoring
Sample Quality
The chain of events from the process of taking a
sample to the analysis is no stronger than its
weakest link.
Each sample should be registered (have a unique
barcode) and all details recorded including the storage
conditions and chain of contact.
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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 weight
polyethylene is not suitable for hydrocarbon samples)
length of holding time and conditions (e.g. cream
separates out from milk samples when left standing,
sedimentation of particles in liquids occurs)
amount of sample required to perform the analysis.
Sample pre-treatment
Solids
• Grinding of solids
• Sample drying
• Leaching and extraction of soluble
components
• Filtering of mixtures of solids, liquids
and gases to leave particulate (solid)
matter
Decomposition and dissolution of
solids
Most measurement methodologies depend upon
presentation of samples in liquid solutions
Preparation method will depend upon material
composition 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=a7414
70469~fulltext=713240928
Nitric Acid treatment
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Nitric acid is acting:
 as a strong acid where inorganic oxides are brought into
solution...
(1) CaO + 2H3O+ Ca+2 + 3H2O
 as an oxidizing agent / acid combo where zero valence
inorganic metals and nonmetals are oxidized and brought
into solution...
(2) Fe + 3H3O+ + 3HNO3 (conc.) Fe+3 + 3NO2 (brown) +
6H2O
or
(3) 3Cu + 6H3O+ + 2HNO3 (dilute) 2NO (clear) + 3Cu+2 +
10H2O
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In addition, nitric acid does not form any insoluble
compounds with the metals and non-metals listed. The
same cannot be said for sulfuric, hydrochloric,
hydrofluoric, phosphoric, or perchloric acids.
Link: http://www.ivstandards.com/tech/reliability/
Statistics
An introduction to statistics is necessary in
order to explain the uncertainty associated
with measurements and sampling.
One cannot go far in Analytical
Chemistry without encountering
statistics!
No quantitative results are of any value
unless they are accompanied by some
estimate of the errors inherent in them
Definitions
• Arithmetic mean: average of all observations
n
x
x
i 1
If the sample is random then the
arithmetic mean is the best
estimate of the population (true)
mean, m
i
n
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 is
denoted s2.
n
n
2
2
i
i
2
2
i 1
i 1
 
 x  x 
n
s 
 x  x 
n 1
More Definitions
• Standard deviation: the positive square root of the variance,
used also to indicate the extent to which data differs in
relation to itself.
• Probability distribution: It is possible to make an infinite
number of measurements to determine the concentration of
an analyte. Normally a small number of test samples is
taken…a statistical sample from the population. If there are
no systematic errors, then the mean of the population (µ) is
the true value of the measurand. The mean of the sample
gives an estimate of µ.
When repeat measurements are made they can take on, in
theory, any value…….a Normal (Gaussian) distribution is
the mathematical model used to describe the continuous
distribution of values for repeat measurements, giving a
bell-shaped curve.
Normal Distribution

exp  x  m  / 2 2
y
 2
2
y
m is 50
is 5 (black dots)
 is 10 (red line)
0
20
40
60
x
80
100

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
Confidence Limits
Confidence limits: extreme values of the confidence
interval which defines the range in which the true value of
a measurand is expected to be found. For small (n<30)
samples the confidence limits can be given by:

m  x t s/ n

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).
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
More useful definitions
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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 the
quantity being measured.
Accuracy
nearness of the result
to the true value of the
quantity being measured
Precision
nearness of a series of
replicate measurements
to each other
determine by comparing
result to those obtained
using other methods and
other laboratories.
determine by evaluating
the standard deviation or
the confidence limits
Linear Calibration Curves
Straight-line plot takes the form:
y = bx + a
correlation co-efficient, r:
r
 x  x  y  y 
i
i
i
1/ 2

2 
2 
   xi  x     yi  y   
 i

 i
thus +1  r  -1, the closer to 1 the value, the better
the correlation.
Linear Regression
Linear regression of y on x:
We seek a line that minimises the deviations in the ydirection between the experimental points and the
calculated line (using the sum of the square of these
deviations)-method of “least squares”.
 x  x  y  y 
b
 x  x 
i
i
i
2
i
i
a  y  bx
Worked Example
25
20
Conc
1
2
3
4
5
6
7
8
9
10
Signal
15
10
5
0
0
2
4
6
Concentration
8
10
Signal
2.1
4.2
5.8
7
9.5
11.8
14
16.1
18.2
21
Worked Example (Microcal Origin)
25
20
Conc
1
2
3
4
5
6
7
8
9
10
Signal
15
10
5
0
0
2
4
6
Concentration
8
10
Signal
2.1
4.2
5.8
7
9.5
11.8
14
16.1
18.2
21
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 on
Next/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 Display
R-squared value on Chart.
Click on OK.
Move the text to the margins by dragging and dropping it.