AnalChem chapter3.pdf

Chapter 3: Experimental Errors

Chapter 4: Statistics

Steps in a Typical Quantitative Analysis

Data of unknown quality are useless!

All laboratory measurements contain experimental error.

It is necessary to determine the magnitude of the accuracy and reliability in your measurements.

Then you can make a judgment about their usefulness.

Replicates - two or more determinations on the same sample

Example 3-1: One student measures Fe (III) concentrations six times. The results are listed below:

19.4, 19.5, 19. 6, 19.8, 20.1, 20.3 ppm (parts per million)

6 replicates = 6 measurements

The "middle" or "central" value for a group of results:

Mean: average or arithmetic mean

Median: the middle value of replicate data If an odd number of replicates, the middle value of replicate data If an even number of replicates, the middle two values are averaged to obtain the median

N

N

1ii∑

==x

x

Terms & Definitions

Example 3-2: measurements of Fe (III) concentrations: 19.4, 19.5, 19. 6, 19.8, 20.1, 20.3 ppm (parts per million)What are the mean and median of these measurements

Mean =

= 19.78 ppm = 19.8 ppm

6 replicatesAn even number of replicates !!!

Median =

= 19.7 ppm

Calculation: Mean and Median

620.320.119.819.619.519.4 +++++

219.819.6 +

Calculation: Mean and MedianExample 3-3: measurements of Fe (III) concentrations:

19.4, 19.5, 19. 6, 19.8, 20.1 ppm (parts per million)

What are the mean and median of these measurements

Mean =

= 19.68 ppm = 19.7 ppm

5 replicatesAn odd number of replicates !!!

Median = 19.6 ppm

520.119.819.619.519.4 ++++

Any Questions???

Precision - describes the reproducibility of measurements.

How close are results which have been obtained in exactly the same way?

The reproducibility is derived from the deviation from the mean:

Deviation from the mean = di = |xi - |

Standard deviation

Variance

Coefficient of variation

Terms & Definitions

X

Accuracy - the closeness of the measurement to the true or accepted value.

This "closeness" called as the error:

absolute or relative error of a result to its true value.

Terms & Definitions

absolute error

relative error

Outlier - Occasional error that obviously differs significantly from the rest of the results.

Terms & Definitions

Precision & Accuracy

Mean & True Value

Mean : X Xt = true value

Absolute and Relative ErrorsAbsolute Error (E) - the difference between the experimental

value and the true value. Has a sign and experimental units:

Experimental value – true (acceptable) value

Relative Error (Er) - the absolute error corrected for the size of the measurement or expressed as the fraction, %, or parts-per-thousand (ppt) of the true value. Er has a sign, but no units.

parts per hundred (pph) = Er x100 parts per thousand (ppt) = Er x1000

100%x t

tirE x

xx −=

ti E xx −=

Calculation: Absolute and Relative Errors

Example 3-4: measurements of Fe concentrations:

19.4, 19.5, 19. 6, 19.8, 20.1, 20.3 ppm Assumed we already knew the true value of Fe (III) concentration at 20.0-ppm. What are absolute and relative errors of each measurement?E = 19.4 - 20.0 = -0.6 ppm Er =(-0.6/20)x100% = - 3%

E = 19.5 - 20.0 = -0.5 ppm Er =(-0.5/20) x100%= -2.5% ~ -3%

E = 19.6 - 20.0 = -0.4 ppm Er =(-0.4/20) x100% = -2%

E = 19.8 - 20.0 = -0.2 ppm Er =(-0.2/20) x100% = -1%

E = 20.1 - 20.0 = 0.1 ppm Er=(0.1/20)x100% = 0.5%

E = 20.3 - 20.0 = 0.3 ppm Er = (0.3/20)x100% =1.5% ~ 2%

A method of analysis yields weights for gold that are low by 0.3 mg. Calculate the percent relative error caused by this uncertainty if the weight of gold in the sample is

(a) 800 mg; (b) 500 mg; (c) 100 mg; (d) 25 mg

E = -0.3 mg

100%x t

tirE x

xx −=

ti E xx −=

Er = (-0.3 mg/500 mg) x100% = -0.06% = -0.06 pph= -0.6 ppt

Example 3-5

Any Questions???

Systematic or determinate errors

affect accuracy!

Random or indeterminate errors

affect precision!

Gross errors or blunders

Lead to outlier’s and require statistical techniques to be rejected.

Types of Errors

1. Instrument errors - failure to calibrate, degradation of parts in the instrument, power fluctuations, etc.

2. Method errors - errors due to no ideal physical or chemical behavior - completeness and speed of reaction, interfering side reactions, sampling problems

3. Personal errors - occur where measurements require judgment, result from prejudice, color acuity problems

Systematic or Determinate Errors

Systematic or determinate errorsPotential Instrument Errors

Variation in temperature

Contamination of the equipment

Power fluctuations

Component failureAll of these can be corrected by calibration or proper instrumentation maintenance.

Systematic or determinate errors

Method Errors

Slow or incomplete reactions

Unstable species

Nonspecific reagents

Side reactions

These can be corrected with proper method development.

Systematic or determinate errorsPersonal Errors

Misreading of data

Improper calibration

Poor technique/sample preparation

Personal bias

Improper calculation of resultsThese are blunders that can be minimized or eliminated with proper training and experience.

The Effect of Systematic Error - normally "biased" and often very "reproducible".

1. Constant errors - Es is of the same magnitude, regardless of the size of the measurement.

This error can be minimized when larger samples are used. In other words, the relative error decreases with increasing amount of analyte.

Er = (Es/Xt )x100%

Constanteg. Solubility loss in gravimetric analysiseg. Reading a buret

2. Proportional errors - Es increases or decreases with increasing or decreasing sample size, respectively. In other words, the relative error remains constant.

ProportionalTypically a contaminant or interference in the sample

Detection of Systematic Method Errors

1. Analysis of standard samples

2. Independent Analysis: Analysis using a "Reference Method" or "Reference Lab"

3. Blank determinations

4. Variation in sample size: detects constant error only

Gross ErrorGross errors cause an experimental value to be discarded.

Lead to outlier’s and require statistical techniques to be rejected.

Examples of gross error are an obviously "overrun end point" (titration), instrument breakdown, loss of a crucial sample, anddiscovery that a "pure" reagent was actually contaminated.

We do NOT use data obtained when gross error has occurred during collection.

Random Errorscaused by uncontrollable variables which normally cannot

be defined

The accumulated effect causes replicate measurements to fluctuate randomly around the mean.

Random errors give rise to a normal or gaussian curve.

Results can be evaluated using statistics

Usually statistical analysis assumes a normal distribution

Term & DefinitionThe Nature of Random Errors –

also called "indeterminate" and follow a predictable pattern.

Error is the deviation from the "true value"

Random error results in values that are higher or lower than the "true value".

The Statistical Treatment of Random Error

A. The Population and the Sample Data

The population data is an infinite number of observations (all the possible results in the universe!).

The sample data is a finite number of observations that are, hopefully, representative of the population.

A Normal or Gaussian CurveFr

eque

ncy

X - μ

The Statistical Treatment of Random Error

B. Properties of a Gaussian Curve - has a population mean, µ, and a population standard deviation.

1. Population mean. In the absence of systematic error, µ, is the true value for the measurement. The sample mean, x, approaches µ when the number of observations approach infinity.

2. Population standard deviation.

Standard Deviation

The Population of Standard Deviation (σ)

1-N)(

s Deviation Standard 2

1ixxi

N

−∑== =

N)(

2

1iμ

σ−∑

= = i

N

x

Sample of Standard Deviation (s)

Sample Standard Deviation as a measure of precision

Reliability of the sample standard deviation (s) increases with the number of replicates (N).

For N greater than 20, s ⇒ σ

Measuring 20 replicates is usually not practical!

Standard Deviation

1-N)(

s Deviation Standard 2

1ixxi

N

−∑== =

Other measures of precisionStandard deviation

Variance (s2)

Relative standard deviation

Coefficient of variation

Spread or range

Other measures of precision

• Variance (s2)

•The advantage of working with variance is that variances from independent sources of variation may be summed to obtain a total variance for a measurement

Other measures of precisionRelative standard deviation (RSD)

• Coefficient of variation (CV)

100 pph

Spread or Range (w)

The difference between the largest and the smallest values in the set of data.

Another term that is occasionally used to described the precisionof a set of replicate data.

Example 3-6: measurements of Fe (III) concentrations: 19.5, 19. 6, 19.4, 19.8, 20.1, 20.3 ppm (parts per million)What are the standard deviation, variance, RSD, coefficient of variation (CV) and range (w) of the data set?

Mean = = 19.78 ppm

Calculation: S, Variance, RSD, CV, Range

620.320.119.819.419.619.5 +++++

1-N)(

s Deviation Standard 2

1ixxi

N

−∑== =

N = replicates = 6

N-1 = 6-1 = 5 (number of degrees of freedom)

1-6)78.193.20()78.191.20()78.198.19()78.194.19()78.196.19()78.195.19( s

222222 −+−+−+−+−+−=

= 0.396 = 0.40 ppm

Standard Error of a Mean or

Standard Deviation of a Mean (sm)

It shows that the standard error of the mean is inverselyproportional to the square root of the number of data (replicates), N.

Ns s mean a ofDeviation Standard m ==

The standard deviation of each mean is known as the standard error of the mean or Standard Deviation of a Mean

Example 3-7: measurements of Fe (III) concentrations: 19.5, 19. 6, 19.4, 19.8, 20.1, 20.3 ppm (parts per million)What are the standard deviation, Sm, variance, RSD, coefficient of variation (CV) and range (w) of the data set?

Calculation: S, Sm, Variance, RSD, CV, Range

1-6)78.193.20()78.191.20()78.198.19()78.194.19()78.196.19()78.195.19( s

222222 −+−+−+−+−+−=

= 0.396 = 0.40 ppm

Ns s mean a ofDeviation Standard m ==

16.06

0.40 == ppm

Calculation of Sm (the standard error of the mean)

a) For 50 trails

b) For 1-25 trails

Ns sm =

4-x100.8

00080.050

0.0056

=

==

mL

3-x102.1

0012.025

0.0058

=

==

mL

S1-25 =0.0058 mL

S26-50 =0.0054 mL

Pooled standard deviation

• Combine standard deviation from different experiments to obtain a reliable estimate of the precision of a method

Example 3-8

Example 3-8

Pooled standard deviation

• Combine standard deviation from different experiments to obtain a reliable estimate of the precision of a method

1. Systematic errors affect _______(a) accuracy (b) precision (c) none of these

(a) accuracy

2. Random errors affect _________(a) accuracy (b) precision (c) none of these

(b) precision

Quiz 3-1: Multiple choices: please circle the best answer.

1. What is used to describe precision of measurements?(a) relative error (b) standard deviation (c) mean (d) medium (e) none of these

(b) standard deviation

2. What is used to describe accuracy of measurements?(a) relative error (b) standard deviation (c) mean (d) medium (e) none of these

(a) relative error

Quiz 3-2: Multiple choices: please circle the best answer.

Error (Uncertainty) Propagation

Error (Uncertainty) Propagation

Error Propagation:Addition and SubtractionExample 3-7: The volume delivered by a buret is the difference betweenthe final and the initial reading. If the uncertainty in each readingis ± 0.02 mL, what is the uncertainty in the volume delivered?

Supposed that the initial reading is 0.05 (± 0.02) mL and the final readingis 17.88 (± 0.02) mL

17.88 (± 0.02)

0.05 (± 0.02)-17.83(± S) mL

S Volume delivered = 222

f

2

i

2

b

2

a (0.02)(0.02)SSSS +=+=+= 0.028 ≈ 0.03 mL

17.83 (± 0.03) mLVolume delivered =

aS

yS

ay x=

y = ax

Example 3-9:

Sy =2

c

2

b

2

a SSS ++

Y = a + b -c

2c2b2ay )cS()

bS()

aS(

yS

caby

++=

=

Example 3-10:

Any Questions???

Significant Figures• The number of digits reported in a measurement reflect the accuracy of the measurement and the precision of the measuring device.

• All the figures known with certainty plus one extra figure are called significant figures.

• In any calculation, the results are reported to the fewest significant figures (for multiplication and division) or fewest decimal places (addition and subtraction).

Significant Figures

30.2? mL

30.24 mL

30.25 mL

30.26 mL

Significant Figures

1. Determining the number of significant figures in a number.

Rule 1. Significant figures are: all the certain figures and the first uncertain figure!

Significant Figures

~ $13,000± 0.001 mgAnalytical balance

~ $7,000± 0.01 mgSemimicroAnalyticalbalance

~$3,000± 0.1 mgMacrobalance

Price($)

Precision(s) (mg)

Type of balance

~ $16,000± 0.0003 mgAnalytical balance

a) 1.23 g b) 1.230 g c) 1.2300 g

Methods for Reporting Data: Significant Figures

Disregard all initial zerosAll remaining digits including zeros between nonzero integers are significant.Addition and Subtraction – the smallest number of digits to the right of the decimal sets the significanceProducts and Quotients - the smallest number of significant digits determines significance Logarithms - for logs, keep as many digits to the right of the decimal as there are significant figures in the original number Antilogarithms - keep as many digits as there are digits to the right of the decimal point in the original number

Methods for Reporting Data: Significant Figures

Disregard all initial zerosAll remaining digits including zeros between nonzero integers are significant.

Example 3-11: (A)0.002 has _______ significant figure. (B) 0.0202 has _______ significant figure. (C) 0.0020 has _______ significant figure (D) 24.00 has _______ significant figure.

Answers

(A) 1(B) 3(C) 2(D) 4

Methods for Reporting Data: Significant Figures

Addition and Subtraction – the smallest number of digits to the right of the decimal sets the significance

Products and Quotients - the smallest number of significant digits determines significance

Example 3-12: measurements of sample weight using different types of balances: 9.54, 9.542, 9.5, 9.5421, 9.5423 g.What are the sum and mean of these measurements?

Sum = 9.54 + 9.542 + 9.5 + 9.5421 + 9.5423 = 47.6664 g

= 47.7 g9.549.5429.59.54219.5423Mean =

Calculation: Sum and Mean

57.47

567.47

59.54239.54219.59.5429.54 ==++++

= 9.53 g

Rounding Data

Round up for digits > or = 5, and round down for digits < 5

Use common sense when rounding. Remember that even though 3 significant figures may be permissible for a S value, S is ± term so that, 2.10 ± 0.0111 becomes 2.10 ± .011.

Remember not to round off calculations until the final result is obtained!

Example 3-13:

Any Questions???

SummaryMean:

Median:

Accuracy:

Precision:

Errors:Absolute and Relative Errors

Systematic or determinate errors

Random or indeterminate errors

Gross errors or blunders

SummaryAbsolute standard deviation or standard deviation

Relative standard deviation

Standard deviation of the mean (sm)

Pooled standard deviation

Coefficient of variation

Variance (s2)

Spread or range

Significant Figures

Homework

Chapter-3: 3-1, 2, 5, 10, 12, 13, 15, 16, 22

Practice with all examples that we discussed in the classand related examples in textbook !!

Before working on Homework,

The End!