statistics for analytical chemistry (girma selale)

7/29/2019 Statistics for Analytical Chemistry (Girma Selale)

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Lecture Note ;

Statistics for Analytical Chemistry(Chem 222)

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)2/4/2013 1


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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 yearEnvironmental 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.

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The General Analytical Problem

Select sample

Extract analyte(s) from matrix

Detect, identify and

quantify analytes

Determine reliability and

significance of results

Separate analytes

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Errors in Chemical Analysis

Impossible to eliminate errors.

How reliable are our data?

Data ofunknown quality are useless!

Carry out replicate measurementsAnalyse accurately known standards

Perform statistical tests on data

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Mean Defined as follows:

x

x

N

i

N

= i = 1

Wherexi = individual values ofx 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.2/4/2013 5


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Illustration of Mean and Median

Results of 6 determinations of the Fe(III) content of a solution, known to

contain 20 ppm(a standard solutions ):

Note: The mean value is 19.78 ppm (i.e. 19.8ppm) - the median value is 19.7 ppm

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Precision

Relates toreproducibilityof results.

How similar are values obtained in exactly the same way?

Useful for measuring this:

Deviation from the mean:

d x xi i

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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:

Er

xi

xt

xt

100%

(latter is more useful in practice)

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Illustrating the difference between accuracy and precision

Using a pattern of darts on a dartboards.

Low accuracy, low precision Low accuracy, high precision

High accuracy, low precision High accuracy, high precision

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Some analytical data illustrating accuracy and precision

This figure summarize the result for determining nitrogen in

two pure compound

HHS

NH3+Cl-NH

N

OH

O

Benzyl isothiourea

hydrochloride

Nicotinic acid

Analyst 4: imprecise, inaccurate

Analyst 3: precise, inaccurate

Analyst 2: imprecise, accurate

Analyst 1: precise, accurate2/4/2013 10


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Types of Error in Experimental

DataThree types:

(1) Random (indeterminate) Error

Data scattered approx. symmetrically about a mean valu

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.2/4/2013 11


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Sources of Systematic Error

1. Instrument Error

Need frequent calibration - both for apparatus such asvolumetric flasks, burettes etc., but also for electronic

devices such as spectrometers.

2. Method Error

Due to inadequacies in physical or chemical behaviourof 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.2/4/2013 12


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Systematic errors can beconstant(e.g. error in burette reading -

less important for larger values of reading) orproportional (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 goodmaintenance 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 blanks2/4/2013 13


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Statistical Treatment of

Random ErrorsThere are always a large number ofsmall, 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:

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Combination of Random Errors

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 form2/4/2013 15


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Frequency Distribution for

Measurements Containing Random Errors

4 random uncertainties 10 random uncertainties

A very large number of

random uncertainties

This is aGaussian or

normal error

curve.

Symmetrical about

the mean.

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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 9.9868 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 9.98015 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 ml2/4/2013 17


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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.2/4/2013 18


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SAMPLE = finite number of observations

POPULATION = 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 ( ): defined as earlier (N ). In absence of systematic error,

m is thetrue value (maximum on Gaussian curve).

Remember, sample mean ( x ) defined for small values of N.

(Sample mean population mean when N 20)

Population Standard Deviation ( )- defined on next overhead

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: measure ofprecision of a population of data,

given by:

s

m

( )x

N

ii

N

2

1

Where m = population mean;Nis very large.

The equation for a Gaussian curve is defined in terms ofm and s, as follows:

y ex

( ) /m s

s

2 2

2

2

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Two Gaussian curves with two different

standard deviations, sA and sB(=2sA)

General Gaussian curve plotted in

units of z, where

z = (x - )/ 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,

andany standard deviation.

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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 must be modified for small samples of data, i.e. smallN

s

x x

N

i

i

N

( )2

1

1Two differences cf. to equation for :

1. Use sample mean instead of population mean.

2. Usedegrees of freedom,N- 1, instead ofN.Reason is that in working out the mean, the sum of the

differences from the mean must be zero. IfN- 1 values are

known, the last value is defined. Thus onlyN- 1 degrees

of freedom. For large values ofN, used in calculating

,NandN- 1 are effectively equal.2/4/2013 23


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Alternative Expression for s

(suitable for calculators)

sx

x

N

N

ii

N i

i

N

( )

( )2

1

1

2

1

Note: NEVER round off figures before the end of the calculation

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Reproducibility of a method for determining

the % of selenium in foods. 9 measurements

were made on a single batch of brown rice.

Sample Selenium content ( g/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 0.00647 0.08 0.0064

8 0.09 0.0081

9 0.08 0.0064

xi = 0.69 xi2= 0.0533

Mean = Sxi/N= 0.077mg/g (Sxi)2/N = 0.4761/9 = 0.0529

Standard Deviation of a Sample

s

00533 00529

9 1000707106 0 007

. .. .

Coefficient of variance = 9.2% Concentration = 0.077 0.007 mg/g

Standard deviation:

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Standard Error of a Mean

The standard deviation relates to the probable error in asingle measurement.

If we take a series ofNmeasurements, 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:

s sN

m

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Pooled Data

To achieve a value of s which is a good approximation to , i.e. N 20,it is sometimes necessary topooldata from a number of sets of measurements

(all taken in the same way).

Suppose that there aret small sets of data, comprisingN1,N2,.Nt measurements.

The equation for the resultant sample standard deviation is:

s

x x x x x x

N N N tpooled

i i ii

N

i

N

i

N

( ) ( ) ( ) ....

......

1

2

2

2

3

2

111

1 2 3

321

(Note: one degree of freedom is lost for each set of data)

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Analysis of 6 bottles of wine

for residual sugar.

Bottle Sugar % (w/v) o. of obs. Deviations from mean1 0.94 3 0.05, 0.10, 0.08

2 1.08 4 0.06, 0.05, 0.09, 0.06

3 1.20 5 0.05, 0.12, 0.07, 0.00, 0.08

4 0.67 4 0.05, 0.10, 0.06, 0.09

5 0.83 3 0.07, 0.09, 0.10

6 0.76 4 0.06, 0.12, 0.04, 0.03

s

sn

1

2 2 20 05 010 0 08

2

0 0189

20 0972 0 097

( . ) ( . ) ( . ) .. .

and similarly for all .

Set n sn

1 0.0189 0.0972 0.0178 0.077

3 0.0282 0.084

4 0.0242 0.090

5 0.0230 0.107

6 0.0205 0.083

Total 0.1326

( )x xi

2

spooled

01326

23 60 088%

..

Pooled Standard Deviation

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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:

s

x x

N

i

i

N

2

2 2

1

1

( )

COEFFICIENT OF VARIANCE (CV)

(or RELATIVE STANDARD DEVIATION):

Divide the standard deviation by the mean value and express as a percentage:

CVs

x ( ) 100%

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Use of Statistics in Data

Evaluation

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How can we relate the observed mean value (x

) to the true mean ( )?

The latter can never be known exactly.

The range of uncertainty depends how closely s corresponds to .

We can calculate the limits (above and below) aroundx that must lie,

with a given degree of probability.

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Define some terms:

CONFIDENCE LIMITS

interval around the mean that probably contains m.

CONFIDENCE INTERVALthe magnitude of the confidence limits

CONFIDENCE LEVEL

fixes the level of probability that the mean is within the confidence limits

Examples later. First assume that the known s is a good

approximation to s.

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Percentages of area under Gaussian curves between certain limits ofz (=x - )

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 ( x ), the equivalent expression is:

CL for m s x zN

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Values ofz 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.

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Atomic absorption analysis for copper concentration in aircraft engine oil gave a

value of 8.53 g Cu/ml. Pooled results of many analyses showed s = 0.32 g

Cu/ml.Calculate 90% and 99% confidence limits if the above result were based on(a) 1, (b) 4, (c) 16 measurements.

90% 853164 0 32

18 53 0 52

8 5 05

CL g / ml

i.e. g / ml

.( . )( . )

. .

. .

m

m

(a)

99% 8 532 58 0 32

18 53 0 83

8 5 0 8

CL g / ml

i.e. g / ml

.( . )( . )

. .

. .

m

m

(b)

90% 853164 0 32

4853 0 26

85 0 3

CL g / ml

i.e. g / ml

.( . )( . )

. .

. .

m

m

99% 8532 58 0 32

4853 0 41

85 0 4

CL g / ml

i.e. g / ml

.( . )( . )

. .

. .

m

m

(c)

90% 853 164 0 3216

853 013

85 01

CL g / ml

i.e. g / ml

. ( . )( . ) . .

. .

m

m

99% 8 532 58 0 32

16853 0 21

85 0 2

CL g / ml

i.e. g / ml

.( . )( . )

. .

. .

m

m

Confidence Limits when is known

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If we have no information on , and only have a value for s -

the confidence interval is larger,

i.e. there is a greater uncertainty.

Instead ofz, it is necessary to use the parametert, defined as follows:

t= (x - m)/s

i.e. just likez, but using s instead ofs.

By analogy we have: CL for

(where = sample mean for measurements)

m x tsN

x N

The calculated values oftare given on the next overhead

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Values oft 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 4.036 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 2.661.29 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

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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.

xi% xi2

7.47 55.8009

6.98 48.7204

7.27 52.8529

Sxi = 21.72 Sxi2 = 157.3742 x xN

i 2172

37 24. .

s

xx

N

N

i

i

2

22

1

157 37422172

3

2

0 246 0 25%

( ).

( . )

. .

90% CL

x tsN

7 242 92 0 25

3

7 24 0 42%

.( . )( . )

. .

If repeated analyses showed that s s = 0.28%: 90% CL

x zN

s 7 24164 0 28

3

7 24 0 27%

.( . )( . )

. .

Confidence Limits where is not known

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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 forthe true value (mA = mt) and

(B) the experimental curve (mB)

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Bias = mB- mA = mB - xt.

Test for bias by comparing with the

difference caused by random error

x xt

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

is given by:

CL for

at desired confidence level, random

errors can lead to:

if , then at the desired

confidence level bias (systematic error)

is likely (and vice versa).

m

x

ts

N

x xts

N

x xts

N

t

t

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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?

x x x

x x

s

t

i i

37 8% 11%

1134 4208 30

4208 30 1134 3

20 943%

2

2

. .

. .

. ( . ).

Assume null hypothesis (no bias). Only reject this if

x x ts Nt

But t (from Table) = 4.30, s (calc. above) = 0.943% and N = 3

ts N

x x ts Nt

4 30 0 943 3 2 342%. . .

Therefore the null hypothesis is maintained, and there is no

evidence for systematic error at the 95% confidence level.

Detection of Systematic Error (Bias)

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Are two sets of measurements significantly different?

Suppose two samples are analysed under identical conditions.

Sample 1 from replicate analyses

Sample 2 from replicate analyses

x N

x N

1 1

2 2

Are these significantly different?

Using definition of pooled standard deviation, the equation on the last

overhead can be re-arranged:

x x tsN N

N Npooled1 2

1 2

1 2

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.

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T f i ifi diff b f d


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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

ts N N N N 1 2 1 2 336 0267 10 25- i.e. ( . )( . )i.e. 0.5674, or 0.57 mg/g.

But g / gx x1 2 28 0 26 25 175

. . . m

i.e. x x ts N N N Npooled1 2 1 2 1 2

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.2/4/2013 43


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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.

Qexp xq xn /w

where xq = questionable result

xn = nearest neighbour w = spread of entire set

Consider a set of results

The parameter Qexp is defined as follows:

Detection of Gross Errors

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Q i th d t t f l Q


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Qexp is then compared to a set of values Qcrit:

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.

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 0.6349 0.437 0.493 0.598

10 0.412 0.466 0.568

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Th f ll i l bt i d f


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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?

Qexp . . ( . . ) . 0 380 0 401 0 410 0 380 0 7But 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?

Qexp . . ( . . ) . 0 380 0 400 0 413 0 380 0 606

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.

Q Test for Rejection

of Outliers

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statistics for analytical chemistry (girma selale)

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