EXPERIMENTAL ERRORS & STATISTICS

EXPERIMENTAL ERRORS & STATISTICS

SUHAILI ZAINAL ABIDINFACULTY OF APPLIED SCIENCE

UITM NEGERI SEMBILAN

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Scientific NotationThe number of atoms in 12 g of carbon:

602,200,000,000,000,000,000,000

6.022 x 1023

The mass of a single carbon atom in grams:

0.00000000000000000000001991.99 x 10-23

N x 10n

N is a number between 1 and 10

n is a positive or negative integer


568.762

n > 0568.762 = 5.68762 x 102

move decimal left

0.00000772

n < 00.00000772 = 7.72 x 10-6

move decimal right

Addition or Subtraction

1. Write each quantity with the same exponent n2. Combine N1 and N2 3. The exponent, n, remains the same

4.31 x 104 + 3.9 x 103 = 4.31 x 104 + 0.39 x 104 = 4.70 x 104


Multiplication1. Multiply N1 and N2

2. Add exponents n1 and n2

(4.0 x 10-5) x (7.0 x 103) = ?= (4.0 x 7.0) x (10-5+3) = 28 x 10-2 = 2.8 x 10-1

Division1. Divide N1 and N2

2. Subtract exponents n1 and n2

8.5 x 104 ÷ 5.0 x 109 = ?= (8.5 ÷ 5.0) x 104 - 9 = 1.7 x 10-5

(a x 10m) x (b x 10n) = (a x b) x 10m+n

(a x 10m) ÷ (b x 10n) = (a ÷ b) x 10m-n


Significant Figures - The meaningful digits in a measured or calculated quantity.

RULES: Any digit that is not zero is significant

1.234 kg 4 significant figures Zeros between nonzero digits are significant606 m 3 significant figures Zeros to the left of the first nonzero digit are not

significant0.08 L 1 significant figure If a number is greater than 1, then all zeros to the

right of the decimal point are significant2.0 mg 2 significant figures


If a number is less than 1, then only the zeros that are at the end and in the middle of the number are significant

0.00420 g 3 significant figures

Numbers that do not contain decimal points, zeros after the last nonzero digit may or may not be significant.

400 cm 1or 2 or 3 significant figures4 x 102 1 significant figures4.0 x 102 2 significant figures


How many significant figures are in each of the following measurements?

24 mL 2 significant figures

3001 g 4 significant figures

0.0320 m3 3 significant figures

6.4 x 104 molecules 2 significant figures

560 kg 2 significant figures


Significant FiguresAddition or SubtractionThe answer cannot have more digits to the right of the decimal point than any of the original numbers.

89.3321.1+

90.432 round off to 90.4one significant figure after decimal point

3.70-2.91330.7867

two significant figures after decimal point

round off to 0.79


Significant FiguresMultiplication or DivisionThe number of significant figures in the result is set by the original number that has the smallest number of significant figures

4.51 x 3.6666 = 16.536366 = 16.5

3 sig figs round to3 sig figs

6.8 ÷ 112.04 = 0.0606926

2 sig figs round to2 sig figs

= 0.061


QUESTION


LOGARITHMS AND ANTILOGARITHMS

Example

Characteristics

Mantissa

log 957 = 2.981

2

0.981

log 9.57 x 10-4 = -3.019

3

0.019

In converting a number to its logarithm, the number of digits in mantissa of the log of the number (957) should be equal to the number of SF in the number (957).

10 0.072 = 1.18For antilogarithm,


Types of Errors in Chemical Analysis

1. Absolute Error

Definition: The difference between the true value and the measured value

E = xi – xt

Where xi = measured value xt = true or accepted value

Example: If 2.62 g sample of material is analyzed to be 2.52 g, so the absolute error is − 0.10g.


2. Relative Error

Definition: The absolute or mean error expressed as a percentage of the true value.

Er = xi – xt x 100% xt

The above analysis has a relative error of

− 0.10 g x 100% = -3.8% 2.62 g

* We are usually dealing with relative errors of less than 1%. A 1% error is equivalent to 1 part in 100.


2.1 Relative Accuracy

Definition: The measured value or mean expressed as a percentage of the true value.

Er = xi x 100% xt

The above analysis has a relative accuracy of

2.52 g x 100% = 96.2 % 2.62 g


3. Systematic Error or determinate error

Definition: A constant error that originates from a fixed cause, such as flaw in the design of an equipment or experiment.

It caused the mean of a set data to differ from the accepted value. This error tends to cause the results to either high every time or low every time compared to the true value. There are 3 types of systematic error:

Instrumental Error

Method Error

Personal

Error

Oct 2008


3.1 Instrumental Errors

All measuring devices contribute to systematic errors. Glassware such as pipets, burets, and volumetric flasks may hold volume slightly different from those indicated by their graduations.Occur due to significant difference in temperature from the calibration temperatere. Sources of uncertainties:

Decreased power supply voltage

Increases resistance in circuits due to temperature change


3.2 Method Errors

Non-ideal analytical methods are often sources of systematic errors. These errors are difficult to detect. The most serious of the 3 types of systematic errors.

Slow or incomplete

reaction

Instability of reacting species

Occurrence of side

reaction

Interference


3.3 Personal Errors

Involve measurements that require personal judgment. For example :

i) estimation of a pointer between tow scale divisions.

ii) color of solution.iii) level of liquids with respect to a graduation in

a burette.iv) prejudice.


3.4 Effect of Systematic Errors

•Does not change with size of the quantity measured.•Become more obvious as the size of the quantity decreases.•Approach to minimize the effect is use as large a sample as possible.

Constant Error

•Increase and decrease in proportion to the size of the sample for analysis.•Source of error : Interference due to contaminants in the sample.

Proportional Error


3.5 Detection and Control of Systematic Errors

Standard reference materials (SRM)

• There are certified samples containing a known concentration or quantities of particular analytes.

• Can be purchased from a number of governmental or industrial; sources such as U. S. National Institute of Standards and Technology (NIST).

Independent analysis

• If SRM are not available, an independent and largely different analysis can be used in parallel with the method evaluated.

• A statistical test must be used to determine whether the difference is due to random errors in the 2 methods.

Analysis of blank sample

• Blank contains the reagents and solvents used in analysis but no analyte.

• Reveals errors due to interfering contaminants from the reagents and vessels used in analysis.


4 Random Error or Indeterminate error

Cause data to be scattered more or less symmetrically around a mean value.

It reflects the precision of the measurement.

This error is caused by the many uncontrollable variables in physical or chemical measurements.


5 Gross Error

Differ from indeterminate and determinate errors.

They usually occur only occasionally, may cause a result to be either high or low.

For example:i) part of precipitate is lost before weighing,

analytical results will be low.ii) touching a weighing bottle with your fingers

after empty mass will cause a high mass reading for a solid weighed.

Lead to outliers, results in replicate measurements that differs significantly from the rest of the 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

Mean Defined as follows: • The average of the numbers• Add up all the numbers, then divide by how

many numbers there are.

xx

N

i

N

= i = 1Where 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 whenthere 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• Precision of a measurement system, also called reproducibility

or repeatability, is the degree to which repeated measurements under unchanged conditions show the same results

• 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

http://en.wikipedia.org/wiki/Reproducibility

http://en.wikipedia.org/wiki/Repeatability

http://en.wikipedia.org/wiki/Result

Accuracy

• Measurement of agreement between experimental mean andtrue value (which may not be known!).

• Measures of accuracy:Absolute error: E = xi - xt (where xt = true or accepted value)

Relative error: Er

xi xtxt

100%

(latter is more useful in practice)

• Accuracy of a measurement system is the degree of closeness of measurements of a quantity to its actual (true) value

http://en.wikipedia.org/wiki/Measurement

http://en.wikipedia.org/wiki/Quantity

http://en.wikipedia.org/wiki/Value_(mathematics)

Illustrating the difference between “accuracy” and “precision”

Low accuracy, low precision Low accuracy, high precision

High accuracy, low precision High accuracy, high precision

Some analytical data illustrating “accuracy” and “precision”

HHS

NH3+Cl-NH

N

OH

O

Benzyl isothioureahydrochloride

Nicotinic acid

Analyst 4: imprecise, inaccurateAnalyst 3: precise, inaccurateAnalyst 2: imprecise, accurateAnalyst 1: precise, accurate

Sample Standard Deviation, s

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

sx x

N

ii

N

( )2

1

1Two 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 areknown, the last value is defined. Thus only N - 1 degreesof freedom. For large values of N, used in calculatings, N and N - 1 are effectively equal.

• Standard deviation is a widely used measurement of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the "average" (mean, or expected / budgeted value).

http://en.wikipedia.org/wiki/Statistics

http://en.wikipedia.org/wiki/Probability_theory

http://en.wikipedia.org/wiki/Statistical_dispersion

http://en.wikipedia.org/wiki/Mean

Alternative Expression for s(suitable for calculators)

sx

x

NN

ii

N ii

N

( )( )

2

1

1

2

1

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.

Sample Selenium content (mg/g) (xI) xi2

1 0.07 0.00492 0.07 0.00493 0.08 0.00644 0.07 0.00495 0.07 0.00496 0.08 0.00647 0.08 0.00648 0.09 0.00819 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 of a Sample

s

0 0533 0 0529

9 10 00707106 0 007

. .. .

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

Standard deviation:

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 sNm

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:

sx x x x x x

N N N tpooled

i i ii

N

i

N

i

N

( ) ( ) ( ) ....

......

12

22

32

111

1 2 3

321

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

Analysis of 6 bottles of wine for residual sugar.

Bottle Sugar % (w/v) No. of obs. Deviations from mean1 0.94 3 0.05, 0.10, 0.082 1.08 4 0.06, 0.05, 0.09, 0.063 1.20 5 0.05, 0.12, 0.07, 0.00, 0.084 0.67 4 0.05, 0.10, 0.06, 0.095 0.83 3 0.07, 0.09, 0.106 0.76 4 0.06, 0.12, 0.04, 0.03

s

sn

1

2 2 20 05 010 0 082

0 01892

0 0972 0 097

( . ) ( . ) ( . ) .

. .

and similarly for all .

Set n sn

1 0.0189 0.0972 0.0178 0.0773 0.0282 0.0844 0.0242 0.0905 0.0230 0.1076 0.0205 0.083

Total 0.1326

( )x xi 2

spooled

0132623 6

0 088%.

.

Pooled Standard Deviation

Two alternative methods for measuring the precision of a set of results:

VARIANCE: This is the square of the standard deviation:

sx x

N

ii

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:

CVsx

( ) 100%

Use of Statistics in

Data Evaluation

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

Examples later. First assume that the known s is a goodapproximation to s.

Percentages of area under Gaussian curves between certain limits of z (= x - m/s)

50% of area lies between 0.67s80% “ 1.29s90% “ 1.64s95% “ 1.96s99% “ 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

Values of z for determining Confidence Limits

Confidence level, % z

50 0.6768 1.080 1.2990 1.6495 1.9696 2.0099 2.5899.7 3.0099.9 3.29

Note: these figures assume that an excellent approximationto 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.

90% 853164 0 32

1853 052

8 5 0 5

CL g / ml

i.e. g / ml

.( . )( . )

. .

. .

m

m

(a)

99% 8532 58 0 32

18 53 083

8 5 08

CL g / ml

i.e. g / ml

.( . )( . )

. .

. .

m

m

(b)

90% 853164 0 32

4853 0 26

8 5 0 3

CL g / ml

i.e. g / ml

.( . )( . )

. .

. .

m

m

99% 8532 58 0 32

48 53 0 41

8 5 0 4

CL g / ml

i.e. g / ml

.( . )( . )

. .

. .

m

m

(c) 90% 853164 0 32

168 53 013

85 01

CL g / ml

i.e. g / ml

.( . )( . )

. .

. .

m

m

99% 8532 58 0 32

168 53 0 21

8 5 0 2

CL g / ml

i.e. g / ml

.( . )( . )

. .

. .

m

m

Confidence Limits when s is known

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: CL for

(where = sample mean for measurements)

m x tsN

x N

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 63.72 1.89 2.92 4.30 9.923 1.64 2.35 3.18 5.844 1.53 2.13 2.78 4.605 1.48 2.02 2.57 4.036 1.44 1.94 2.45 3.717 1.42 1.90 2.36 3.508 1.40 1.86 2.31 3.369 1.38 1.83 2.26 3.2519 1.33 1.73 2.10 2.8859 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.

xi% xi2

7.47 55.80096.98 48.72047.27 52.8529

Sxi = 21.72 Sxi2 =

157.3742 xx

Ni

21723

7 24.

.

sx

xN

Ni

i

22

2

1157 3742 2172

32

0 246 0 25%

( ). ( . )

. .

90% CL

x tsN 7 24

2 92 0 253

7 24 0 42%

.( . )( . )

. .

If repeated analyses showed that s ® s = 0.28%:

90% CL

x zN

s 7 24164 0 28

37 24 0 27%

.( . )( . )

. .

Confidence Limits where s is not known

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

The parameter Qexp is defined as follows:

Detection of Gross Errors

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 0.9944 0.765 0.829 0.9265 0.642 0.710 0.8216 0.560 0.625 0.7407 0.507 0.568 0.6808 0.468 0.526 0.6349 0.437 0.493 0.59810 0.412 0.466 0.568

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 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?

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


ANY QUESTIONS ??


GOOD LUCK !!

~SUHAILI ZAINAL ABIDIN~

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EXPERIMENTAL ERRORS & STATISTICS