Principles of Analytical Chemistry (F13I11)

1
Lecture Note Statistics for Analytical
Chemistry (MKI 322) Bambang Yudono
Recommended textbook Statistics for Analytical
Chemistry J.C. Miller and J.N. Miller, Second
Edition, 1992, Ellis Horwood Limited Fundamentals
of Analytical Chemistry Skoog, West and Holler,
7th Ed., 1996 (Saunders College Publishing)
2
Applications of Analytical Chemistry
Industrial Processes analysis for quality
control, and reverse engineering (i.e.
finding out what your competitors are doing).

Environmental Analysis familiar to those who
attended the second year Environmental
Chemistry modules. A very wide range of
problems and types of analyte
Regulatory Agencies dealing with many problems
from first two.
Academic and Industrial Synthetic Chemistry of
great interest to many of my colleagues. I will
not be dealing with this type of problem.
3
The General Analytical Problem
Select sample
Extract analyte(s) from matrix
Separate analytes
Detect, identify and quantify analytes
Determine reliability and significance of results
4
Errors in Chemical Analysis
Impossible to eliminate errors. How reliable are
our data? Data of unknown quality are useless!

• Carry out replicate measurements
• Analyse accurately known standards
• Perform statistical tests on data

5
Mean
Defined as follows
Where xi individual values of x and N number
of replicate measurements
Median
The middle result when data are arranged in order
of size (for even numbers the mean of middle
two). Median can be preferred when there is an
outlier - one reading very different from rest.
Median less affected by outlier than is mean.
6
Illustration of Mean and Median
Results of 6 determinations of the Fe(III)
content of a solution, known to contain 20 ppm
Note The mean value is 19.78 ppm (i.e.
19.8ppm) - the median value is 19.7 ppm
7
Precision
Relates to reproducibility of results.. How
similar are values obtained in exactly the same
way?
Useful for measuring this Deviation from the
mean
8
Accuracy
Measurement of agreement between experimental
mean and true value (which may not be
known!). Measures of accuracy
Absolute error E xi - xt (where xt true
or accepted value)
Relative error
(latter is more useful in practice)
9
Illustrating the difference between accuracy
and precision
Low accuracy, low precision
Low accuracy, high precision
High accuracy, high precision
High accuracy, low precision
10
Some analytical data illustrating accuracy and
precision
Benzyl isothiourea hydrochloride
Analyst 4 imprecise, inaccurate Analyst 3
precise, inaccurate Analyst 2 imprecise,
accurate Analyst 1 precise, accurate
Nicotinic acid
11
Types of Error in Experimental Data
Three types
(1) Random (indeterminate) Error Data scattered
approx. symmetrically about a mean
value. Affects precision - dealt with
statistically (see later).
(2) Systematic (determinate) Error Several
possible sources - later. Readings all too high
or too low. Affects accuracy.
(3) Gross Errors Usually obvious - give
outlier readings. Detectable by carrying out
sufficient replicate measurements.
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Sources of Systematic Error
1. Instrument Error Need frequent calibration -
both for apparatus such as volumetric flasks,
burettes etc., but also for electronic devices
such as spectrometers.
2. Method Error Due to inadequacies in physical
or chemical behaviour of reagents or reactions
(e.g. slow or incomplete reactions) Example from
earlier overhead - nicotinic acid does not react
completely under normal Kjeldahl conditions for
nitrogen determination.
3. Personal Error e.g. insensitivity to colour
changes tendency to estimate scale readings to
improve precision preconceived idea of true
value.
13
Systematic errors can be constant (e.g. error
in burette reading - less important for larger
values of reading) or proportional (e.g.
presence of given proportion of interfering
impurity in sample equally significant for all
values of measurement)
Minimise instrument errors by careful
recalibration and good maintenance of equipment.
Minimise personal errors by care and
self-discipline

• Method errors - most difficult. True value may
  not be known.
• Three approaches to minimise
• analysis of certified standards
• use 2 or more independent methods
• analysis of blanks

14
Statistical Treatment of Random Errors
There are always a large number of small, random
errors in making any measurement.
These can be small changes in temperature or
pressure random responses of electronic
detectors (noise) etc.
Suppose there are 4 small random errors
possible. Assume all are equally likely, and that
each causes an error of ?U in the
reading. Possible combinations of errors are
shown on the next slide
15
Combination of Random Errors
Total Error No. Relative Frequency UUUU 4
U 1 1/16 0.0625 -UUUU 2U 4 4/16
0.250 U-UUU UU-UU UUU-U -U-UUU 0 6
6/16 0.375 -UU-UU -UUU-U U-U-UU U-UU-U
UU-U-U U-U-U-U -2U 4 4/16
0.250 -UU-U-U -U-UU-U -U-U-UU -U-U-U-U -4U 1
1/16 0.01625
The next overhead shows this in graphical form
16
Frequency Distribution for Measurements
Containing Random Errors
4 random uncertainties
10 random uncertainties
This is a Gaussian or normal error curve. Symmetri
cal about the mean.
A very large number of random uncertainties
17
Replicate Data on the Calibration of a 10ml
Pipette
No. Vol, ml. No. Vol, ml. No. Vol,
ml 1 9.988 18 9.975 35 9.976 2 9.973 19 9.980
36 9.990 3 9.986 20 9.994 37 9.988 4 9.980 21
9.992 38 9.971 5 9.975 22 9.984 39 9.986 6 9.98
2 23 9.981 40 9.978 7 9.986 24 9.987 41 9.986
8 9.982 25 9.978 42 9.982 9 9.981 26 9.983 43
9.977 10 9.990 27 9.982 44 9.977 11 9.980 28 9.
991 45 9.986 12 9.989 29 9.981 46 9.978 13 9.97
8 30 9.969 47 9.983 14 9.971 31 9.985 48 9.980
15 9.982 32 9.977 49 9.983 16 9.983 33 9.976
50 9.979 17 9.988 34 9.983
Mean volume 9.982 ml Median volume 9.982
ml Spread 0.025 ml Standard deviation 0.0056 ml
18
Calibration data in graphical form
A histogram of experimental results
B Gaussian curve with the same mean value, the
same precision (see later) and the same area
under the curve as for the histogram.
19
SAMPLE
finite number of observations
total (infinite) number of observations
POPULATION
Properties of Gaussian curve defined in terms of
population. Then see where modifications needed
for small samples of data
Main properties of Gaussian curve
Population mean (m) defined as earlier (N ?
?). In absence of systematic error, m is the
true value (maximum on Gaussian curve).
Remember, sample mean (
) defined for small values of N.
(Sample mean ? population mean when N ? 20)
Population Standard Deviation (s) - defined on
next overhead
20
s measure of precision of a population of
data, given by
Where m population mean N is very large.
The equation for a Gaussian curve is defined in
terms of m and s, as follows
21
Two Gaussian curves with two different standard
deviations, sA and sB (2sA)
General Gaussian curve plotted in units of z,
where z (x - m)/s i.e. deviation from the
mean of a datum in units of standard deviation.
Plot can be used for data with given value of
mean, and any standard deviation.
22
Area under a Gaussian Curve
From equation above, and illustrated by the
previous curves, 68.3 of the data lie within ??
of the mean (?), i.e. 68.3 of the area under
the curve lies between ?? of ?.
Similarly, 95.5 of the area lies between ???,
and 99.7 between ???.
There are 68.3 chances in 100 that for a single
datum the random error in the measurement will
not exceed ??. The chances are 95.5 in 100
that the error will not exceed ???.
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Sample Standard Deviation, s
The equation for s must be modified for small
samples of data, i.e. small N
Two differences cf. to equation for s
1. Use sample mean instead of population mean.
2. Use degrees of freedom, N - 1, instead of
N. Reason is that in working out the mean, the
sum of the differences from the mean must be
zero. If N - 1 values are known, the last value
is defined. Thus only N - 1 degrees of freedom.
For large values of N, used in calculating s, N
and N - 1 are effectively equal.
24
Alternative Expression for s (suitable for
calculators)
Note NEVER round off figures before the end of
the calculation
25
Reproducibility of a method for determining the
of selenium in foods. 9 measurements were
made on a single batch of brown rice.
Standard Deviation of a Sample
Sample Selenium content (mg/g)
(xI) xi2 1 0.07 0.0049 2 0.07 0.0049 3
0.08 0.0064 4 0.07 0.0049 5 0.07 0.0049
6 0.08 0.0064 7 0.08 0.0064 8 0.09 0.0
081 9 0.08 0.0064 Sxi 0.69 Sxi2 0.0533
Mean Sxi/N 0.077mg/g (Sxi)2/N 0.4761/9
0.0529
Standard deviation
Coefficient of variance 9.2 Concentration
0.077 0.007 mg/g
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Standard Error of a Mean
The standard deviation relates to the probable
error in a single measurement. If we take a
series of N measurements, the probable error of
the mean is less than the probable error of any
one measurement.
The standard error of the mean, is defined as
follows
27
Pooled Data
To achieve a value of s which is a good
approximation to s, i.e. N ? 20, it is sometimes
necessary to pool data from a number of sets of
measurements (all taken in the same way).
Suppose that there are t small sets of data,
comprising N1, N2,.Nt measurements. The equation
for the resultant sample standard deviation is
(Note one degree of freedom is lost for each set
of data)
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Pooled Standard Deviation
Analysis of 6 bottles of wine for residual sugar.
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Two alternative methods for measuring the
precision of a set of results
VARIANCE This is the square of the standard
deviation
COEFFICIENT OF VARIANCE (CV) (or RELATIVE
STANDARD DEVIATION) Divide the standard
deviation by the mean value and express as a
percentage
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Use of Statistics in Data Evaluation
31
) to the true mean (m)?
How can we relate the observed mean value (
The latter can never be known exactly.
The range of uncertainty depends how closely s
corresponds to s.
that m must lie,
We can calculate the limits (above and below)
around
with a given degree of probability.
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Define some terms
CONFIDENCE LIMITS interval around the mean
that probably contains m.
CONFIDENCE INTERVAL the magnitude of the
confidence limits
CONFIDENCE LEVEL fixes the level of probability
that the mean is within the confidence limits
First assume that the known s is a
good approximation to s.
Examples later.
33
Percentages of area under Gaussian curves between
certain limits of z ( x - m/s)
50 of area lies between ?0.67s 80
?1.29s 90 ?1.64s 95
?1.96s 99 ?2.58s
What this means, for example, is that 80 times
out of 100 the true mean will lie between ?1.29s
of any measurement we make.
Thus, at a confidence level of 80, the
confidence limits are ?1.29s.
For a single measurement CL for m x ? zs
(values of z on next overhead)
For the sample mean of N measurements (
), the equivalent expression is
34
Values of z for determining Confidence Limits
Confidence level, z 50 0.67 68 1.0 8
0 1.29 90 1.64 95 1.96 96 2.00 9
9 2.58 99.7 3.00 99.9 3.29
Note these figures assume that an excellent
approximation to the real standard deviation is
known.
35
Confidence Limits when s is known
Atomic absorption analysis for copper
concentration in aircraft engine oil gave a value
of 8.53 mg Cu/ml. Pooled results of many
analyses showed s s 0.32 mg Cu/ml. Calculate
90 and 99 confidence limits if the above
result were based on (a) 1, (b) 4, (c) 16
measurements.
(b)
(a)
(c)
36
If we have no information on s, and only have a
value for s - the confidence interval is
larger, i.e. there is a greater uncertainty.
Instead of z, it is necessary to use the
parameter t, defined as follows
t (x - m)/s
i.e. just like z, but using s instead of s.
By analogy we have
The calculated values of t are given on the next
overhead
37
Values of t for various levels of probability
Degrees of freedom 80 90 95 99 (N-1) 1 3.08
6.31 12.7 63.7 2 1.89 2.92 4.30 9.92 3 1.64 2
.35 3.18 5.84 4 1.53 2.13 2.78 4.60 5 1.48 2.0
2 2.57 4.03 6 1.44 1.94 2.45 3.71 7 1.42 1.90
2.36 3.50 8 1.40 1.86 2.31 3.36 9 1.38 1.83 2.
26 3.25 19 1.33 1.73 2.10 2.88 59 1.30 1.67 2.
00 2.66 ? 1.29 1.64 1.96 2.58
Note (1) As (N-1) ? ?, so t ? z (2) For all
values of (N-1) lt ?, t gt z, I.e. greater
uncertainty
38
Confidence Limits where s is not known
Analysis of an insecticide gave the following
values for of the chemical lindane 7.47,
6.98, 7.27. Calculate the CL for the mean value
at the 90 confidence level.
Sxi 21.72
Sxi2 157.3742
If repeated analyses showed that s s 0.28
39
Testing a Hypothesis
Carry out measurements on an accurately known
standard.
Experimental value is different from the true
value.
Is the difference due to a systematic error
(bias) in the method - or simply to random error?
Assume that there is no bias (NULL
HYPOTHESIS), and calculate the probability that
the experimental error is due to random errors.
Figure shows (A) the curve for the true value
(mA mt) and (B) the experimental curve (mB)
40
Bias mB- mA mB - xt.
Remember confidence limit for m (assumed to be
xt, i.e. assume no bias) is given by
41
Detection of Systematic Error (Bias)
A standard material known to contain 38.9 Hg
was analysed by atomic absorption spectroscopy.
The results were 38.9, 37.4 and 37.1. At the
95 confidence level, is there any evidence for
a systematic error in the method?
Assume null hypothesis (no bias). Only reject
this if
But t (from Table) 4.30, s (calc. above)
0.943 and N 3
Therefore the null hypothesis is maintained, and
there is no evidence for systematic error at the
95 confidence level.
42
Are two sets of measurements significantly
different?
Suppose two samples are analysed under identical
conditions.
Are these significantly different?
Using definition of pooled standard deviation,
the equation on the last overhead can be
re-arranged
Only if the difference between the two samples is
greater than the term on the right-hand side can
we assume a real difference between the samples.
43
Test for significant difference between two sets
of data
Two different methods for the analysis of boron
in plant samples gave the following results
(mg/g) (spectrophotometry) (fluorimetry)
Each based on 5 replicate measurements. At the
99 confidence level, are the mean values
significantly different? Calculate spooled
0.267. There are 8 degrees of freedom, therefore
(Table) t 3.36 (99 level). Level for rejecting
null hypothesis is
i.e. 0.5674, or 0.57 mg/g.
Therefore, at this confidence level, there is a
significant difference, and there must be a
systematic error in at least one of the methods
of analysis.
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Detection of Gross Errors
A set of results may contain an outlying result
- out of line with the others. Should it be
retained or rejected? There is no universal
criterion for deciding this. One rule that can
give guidance is the Q test.
Consider a set of results
The parameter Qexp is defined as follows
45
Qexp is then compared to a set of values Qcrit
Qcrit (reject if Qexpt gt Qcrit) No. of
observations 90 95 99 confidencelevel 3 0
.941 0.970 0.994 4 0.765 0.829 0.926 5 0.642 0
.710 0.821 6 0.560 0.625 0.740 7 0.507 0.568 0
.680 8 0.468 0.526 0.634 9 0.437 0.493 0.598
10 0.412 0.466 0.568
Rejection of outlier recommended if Qexp gt Qcrit
for the desired confidence level.
Note1. The higher the confidence level, the less
likely is rejection to be recommended.
2. Rejection of outliers can have a marked
effect on mean and standard deviation,
esp. when there are only a few data
points. Always try to obtain more data.
3. If outliers are to be retained, it is often
better to report the median value rather
than the mean.
46
The following values were obtained for the
concentration of nitrite ions in a sample of
river water 0.403, 0.410, 0.401, 0.380
mg/l. Should the last reading be rejected?
Q Test for Rejection of Outliers
But Qcrit 0.829 (at 95 level) for 4
values Therefore, Qexp lt Qcrit, and we cannot
reject the suspect value. Suppose 3 further
measurements taken, giving total values
of 0.403, 0.410, 0.401, 0.380, 0.400, 0.413,
0.411 mg/l. Should 0.380 still be retained?
But Qcrit 0.568 (at 95 level) for 7
values Therefore, Qexp gt Qcrit, and rejection of
0.380 is recommended.
But note that 5 times in 100 it will be wrong to
reject this suspect value! Also note that if
0.380 is retained, s 0.011 mg/l, but if it is
rejected, s 0.0056 mg/l, i.e. precision appears
to be twice as good, just by rejecting one value.
47
Obtaining a representative sample
Homogeneous gaseous or liquid sample No problem
any sample representative.
Solid sample - no gross heterogeneity Take a
number of small samples at random from
throughout the bulk - this will give a suitable
representative sample.
Solid sample - obvious heterogeneity Take small
samples from each homogeneous region and mix
these in the same proportions as between each
region and the whole.
If it is suspected, but not certain, that a bulk
material is heterogeneous, then it is necessary
to grind the sample to a fine powder, and mix
this very thoroughly before taking random samples
from the bulk.
For a very large sample - a train-load of metal
ore, or soil in a field - it is always necessary
to take a large number of random samples from
throughout the whole.
48
Sample Preparation and Extraction
May be many analytes present - separation - see
later.
May be small amounts of analyte(s) in bulk
material. Need to concentrate these before
analysis.e.g. heavy metals in animal tissue,
additives in polymers, herbicide residues in
flour etc. etc.
May be helpful to concentrate complex mixtures
selectively.
Most general type of pre-treatment EXTRACTION.
49
Classical extraction method is
SOXHLET EXTRACTION
(named after developer). Apparatus
Sample in porous thimble. Exhaustive reflux for
up to 1 - 2 days. Solution of analyte(s) in
volatile solvent (e.g. CH2Cl2, CHCl3 etc.)
Evaporate to dryness or suitable concentration,
for separation/analysis.