1/15/2013

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Lecture Note ;Statistics for Analytical Chemistry

(Chem 222)

GIRMAGIRMAGIRMAGIRMA SELALESELALESELALESELALE

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)1/15/2013 1

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.

1/15/2013 2

The General Analytical Problem

Select sample

Extract analyte(s) from matrix

Detect, identify and

quantify analytes

Determine reliability and

significance of results

Separate analytes

1/15/2013 3

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

1/15/2013 4

Mean Defined as follows:

x

x

N

i

N

= i = 1

∑

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.1/15/2013 5

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

1/15/2013 6

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Precision

Relates to reproducibility of results.

How similar are values obtained in exactly the same way?

Useful for measuring this:

Deviation from the mean:

d x xi i= −

1/15/2013 7

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: E

r

xi

xt

xt

=−

× 100%

(latter is more useful in practice)

1/15/2013 8

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 precision1/15/2013 9

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, accurate1/15/2013 10

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.1/15/2013 11

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.1/15/2013 12

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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 blanks1/15/2013 13

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:

1/15/2013 14

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 form1/15/2013 15

Frequency Distribution for

Measurements Containing Random Errors

4 random uncertainties 10 random uncertainties

A very large number of

random uncertainties

This is a

Gaussian or

normal error

curve.

Symmetrical about

the mean.1/15/2013 16

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 ml1/15/2013 17

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.1/15/2013 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 (µµµµ) : defined as earlier (N → ∞). In absence of systematic error,

µ is the true 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 (σσσσ) - defined on next overhead

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σσσσ : measure of precision of a population of data,

given by:

σ

µ

=

−=

∑ ( )x

N

ii

N

2

1

Where µ = population mean; N is very large.

The equation for a Gaussian curve is defined in terms of µ and σ, as follows:

ye

x

=

− −( ) /µ σ

σ π

2 22

2

1/15/2013 20

Two Gaussian curves with two different

standard deviations, σA and σB (=2σA)

General Gaussian curve plotted in

units of z, where

z = (x - µµµµ)/σσσσ

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.1/15/2013 21

Area under a Gaussian Curve

From equation above, and illustrated by the previous curves,

68.3% of the data lie within ±σ±σ±σ±σ of the mean (µµµµ), i.e. 68.3% of

the area under the curve lies between ±σ±σ±σ±σ of µµµµ.

Similarly, 95.5% of the area lies between ±2σ±2σ±2σ±2σ, and 99.7%

between ±3σ±3σ±3σ±3σ.

There are 68.3 chances in 100 that for a single datum the

random error in the measurement will not exceed ±σ±σ±σ±σ.

The chances are 95.5 in 100 that the error will not exceed ±2σ±2σ±2σ±2σ.

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Sample Standard Deviation, s

The equation for σσσσ must be modified for small samples of data, i.e. small N

s

x x

N

i

i

N

=

−

−

=

∑ ( )2

1

1

Two differences cf. to equation for σσσσ:

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

σσσσ, N and N - 1 are effectively equal. 1/15/2013 23

Alternative Expression for s

(suitable for calculators)

s

x

x

N

N

i

i

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 (µµµµg/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

ΣΣΣΣxi = 0.69 ΣΣΣΣxi2= 0.0533

Mean = Σxi/N= 0.077µg/g (Σxi)2/N = 0.4761/9 = 0.0529

Standard Deviation of a Sample

s =−

−= =

0 0533 0 0529

9 10 00707106 0 007

. .. .

Coefficient of variance = 9.2% Concentration = 0.077 ± 0.007 µg/g

Standard deviation:

1/15/2013 25

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:

s sN

m =

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Pooled Data

To achieve a value of s which is a good approximation to σσσσ, 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:

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) No. of obs. Deviations from mean

1 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 0.097

2 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

1/15/2013 28

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%

1/15/2013 29

Use of Statistics in Data

Evaluation

1/15/2013 30

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How can we relate the observed mean value (x ) to the true mean (µµµµ)?

The latter can never be known exactly.

The range of uncertainty depends how closely s corresponds to σσσσ.

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

with a given degree of probability.

1/15/2013 31

Define some terms:

CONFIDENCE LIMITS

interval around the mean that probably contains µ.

CONFIDENCE INTERVAL

the 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 σ.

1/15/2013 32

Percentages of area under Gaussian curves between certain limits of z (= x - µ/σµ/σµ/σµ/σ)

50% of area lies between ±0.67σ

80% “ ±1.29σ

90% “ ±1.64σ

95% “ ±1.96σ

99% “ ±2.58σ

What this means, for example, is that 80 times out of 100 the true mean will lie

between ±1.29σ of any measurement we make.

Thus, at a confidence level of 80%, the confidence limits are ±1.29σ.

For a single measurement: CL for µ = x ± zσ (values of z on next overhead)

For the sample mean of N measurements ( x ), the equivalent expression is:

CL for µ σ= ±x zN

1/15/2013 33

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.1/15/2013 34

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

value of 8.53 µµµµg Cu/ml. Pooled results of many analyses showed s →→→→ σσσσ = 0.32 µµµµg

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

1853 052

85 05

CL g / ml

i.e. g / ml

= ± = ±

±

.( . )( . )

. .

. .

µ

µ

(a)

99% 8 532 58 0 32

1853 083

85 08

CL g / ml

i.e. g / ml

= ± = ±

±

.( . )( . )

. .

. .

µ

µ

(b)

90% 8 53164 0 32

4853 0 26

85 0 3

CL g / ml

i.e. g / ml

= ± = ±

±

.( . )( . )

. .

. .

µ

µ

99% 8 532 58 0 32

48 53 0 41

8 5 0 4

CL g / ml

i.e. g / ml

= ± = ±

±

.( . )( . )

. .

. .

µ

µ

(c)

90% 8 53164 0 32

168 53 013

8 5 01

CL g / ml

i.e. g / ml

= ± = ±

±

.( . )( . )

. .

. .

µ

µ

99% 8532 58 0 32

16853 0 21

8 5 0 2

CL g / ml

i.e. g / ml

= ± = ±

±

.( . )( . )

. .

. .

µ

µ

Confidence Limits when σσσσ is known

1/15/2013 35

If we have no information on σσσσ, 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 - µ)/s

i.e. just like z, but using s instead of σ.

By analogy we have: CL for

(where = sample mean for measurements)

µ = ±x tsN

x N

The calculated values of t are given on the next overhead

1/15/2013 36

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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

1/15/2013 37

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

Σxi = 21.72 Σxi2 = 157.3742

xx

N

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 → σ = 0.28%: 90% CL

= ± = ±

= ±

x zN

σ 7 24164 0 28

3

7 24 0 27%

.( . )( . )

. .

Confidence Limits where σσσσ is not known

1/15/2013 38

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 (µµµµA = µµµµt) and

(B) the experimental curve (µµµµB)

1/15/2013 39

Bias = µB- µA = µB - xt.

Test for bias by comparing with the

difference caused by random error

x xt−

Remember confidence limit for µ (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).

µ = ±

∴

− = ±

∴ − >

xts

N

x xts

N

x xts

N

t

t

1/15/2013 40

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

= ∴ − = −

= =

∴ =−

=

∑ ∑

378% 11%

1134 4208 30

4208 30 113 4 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.

1/15/2013 42

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8

Test for significant difference between two sets of data

Two different methods for the analysis of boron in plant samples

gave the following results (µg/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 N1 2 1 2 3 36 0 267 10 25 - i . e . ( . )( . )i.e. ± 0.5674, or ±0.57 µg/g.

B u t g / gx x1 2

2 8 0 2 6 2 5 1 7 5− = − =. . . µ

i . e . x x t s N N N Np o o l e d1 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.1/15/2013 43

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

1/15/2013 44

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 0.634

9 0.437 0.493 0.598

10 0.412 0.466 0.568

1/15/2013 45

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 e x p . . ( . . ) .= − − =0 3 8 0 0 4 0 1 0 4 1 0 0 3 8 0 0 7

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?

Q e x p . . ( . . ) .= − − =0 3 8 0 0 4 0 0 0 4 1 3 0 3 8 0 0 6 0 6

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

1/15/2013 46