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Lecture Note ; Statistics for Analytical Chemistry (MKI 322) Bambang Yudono

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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)

Applicationsof 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.

The General Analytical Problem

Select sample

Extract analyte(s) from matrix

Separate analytes

Detect, identify and

quantify analytes

Determine reliability and

significance of results

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

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.

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

Precision

Relates to reproducibilityof results..

How similar are values obtained in exactly the same way?

Useful for measuring this:

Deviation from the mean:

Accuracy

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)

Relative error:

(latter is more useful in practice)

Illustrating the difference between “accuracy” and “precision”

Low accuracy, low precision

Low accuracy, high precision

High accuracy, high precision

High accuracy, low precision

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

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.

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.

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

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:

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

+U+U-U+U

+U+U+U-U

-U-U+U+U 0 6 6/16 = 0.375

-U+U-U+U

-U+U+U-U

+U-U-U+U

+U-U+U-U

+U+U-U-U

+U-U-U-U -2U 4 4/16 = 0.250

-U+U-U-U

-U-U+U-U

-U-U-U+U

-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

Gaussian or

normal error

curve.

Symmetrical about

the mean.

A very large number of

random uncertainties

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 9.982 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

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.

= 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

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:

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.

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 .

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

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 0.0081

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

Standard deviation:

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)

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:

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.

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.

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:

Values of z for determining Confidence Limits

Confidence level, % z

50 0.67

68 1.0

80 1.29

90 1.64

95 1.96

96 2.00

99 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.

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)

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

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 2.02 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

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%:

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)

Remember confidence limit for m (assumed to be xt, i.e. assume no bias)

is given by:

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.

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.

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.

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:

Qexp is then compared to a set of values Qcrit:

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 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 < 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.

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.

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.

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.

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