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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
Statistics for 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.
Extract analyte(s) from matrix
Detect, identify and
Determine reliability and
significance of results
Impossible to eliminate errors.
How reliable are our data?
Data of unknown quality are useless!
Defined as follows:
Where xi = individual values of x and N = number of replicate measurements
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.
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
Relates to reproducibilityof results..
How similar are values obtained in exactly the same way?
Useful for measuring this:
Deviation from the mean:
Measurement of agreement between experimental mean and
true value (which may not be known!).
Measures of accuracy:
Absolute error: E = xi - xt (wherext = true or accepted value)
(latter is more useful in practice)
Low accuracy, low precision
Low accuracy, high precision
High accuracy, high precision
High accuracy, low precision
Some analytical data illustrating “accuracy” and “precision”
Analyst 4: imprecise, inaccurate
Analyst 3: precise, inaccurate
Analyst 2: imprecise, accurate
Analyst 1: precise, accurate
(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
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
3. Personal Error
e.g. insensitivity to colour changes; tendency to estimate
scale readings to improve precision; preconceived idea of
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
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:
Total Error No. Relative Frequency
+U+U+U+U +4U 1 1/16 = 0.0625
-U+U+U+U +2U 4 4/16 = 0.250
-U-U+U+U 0 6 6/16 = 0.375
+U-U-U-U -2U 4 4/16 = 0.250
-U-U-U-U -4U 1 1/16 = 0.01625
The next overhead shows this in graphical form
Measurements Containing Random Errors
4 random uncertainties
10 random uncertainties
This is a
A very large number of
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 9.982 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 9.978 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
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.
= finite number of observations
= total (infinite) number of observations
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
Where m = population mean; N is very large.
The equation for a Gaussian curve is defined in terms of m and s, as follows:
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.
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%
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 .
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.
(suitable for calculators)
Note:NEVER round off figures before the end of the calculation
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 0.0081
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
Coefficient of variance = 9.2% Concentration = 0.077 ± 0.007 mg/g
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:
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)
Analysis of 6 bottles of wine
for residual sugar.
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:
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.
interval around the mean that probably contains m.
the magnitude of the confidence limits
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.
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:
Confidence level, % z
Note: these figures assume that an excellent approximation
to the real standard deviation 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.
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
Degrees of freedom 80% 90% 95% 99%
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 2.02 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) < , t > z, I.e. greater uncertainty
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%:
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
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)
Remember confidence limit for m (assumed to be xt, i.e. assume no bias)
is given by:
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.
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.
Two different methods for the analysis of boron in plant samples
gave the following results (mg/g):
Each based on 5 replicate measurements.
At the 99% confidence level, are the mean values significantly
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.
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:
Qcrit (reject if Qexpt > 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 > Qcrit for the desired confidence level.
Note:1. 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.
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
But Qcrit = 0.829 (at 95% level) for 4 values
Therefore, Qexp < 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 > 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.
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.
(named after developer).
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