Chapter 3: Experimental Errors doc

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: ¾ Mean : average or arithmetic mean ¾ Median : the middle

Chapter 3: Experimental Errors

Chapter 4: Statistics

Steps in a Typical Quantitative Analysis

¾Data of unknown quality are useless !

¾All laboratory measurements contain

experimental error

¾It is necessary to determine the magnitude of

the accuracy and reliability in your measurements

¾Then 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:

¾ Mean : average or arithmetic mean

¾ Median : the middle value of replicate data

¾ If an odd number of replicates, the middle value of replicate data

¾ If an even number of replicates, the middle two values are averaged to obtain the median

N

N

1 i

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

19.8 19.6

19.5

2 19.8 19.6 +

Calculation: Mean and Median

Example 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

19.6 19.5

Any Questions???

¾ 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

¾ 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

¾absolute error

¾relative error

¾ 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 Errors

¾ Absolute Error (E) - the difference between the experimental value and the true value Has a sign and experimental units:

Experimental value – true (acceptable) value

¾Relative 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

t

i r

E = x x − x

t i

E = x − x

Calculation: Absolute and Relative Errors

Example 3-4: measurements of Fe concentrations:

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

t

i r

E = x x − x

t i

E = x − x

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

= - 0.6 ppt

Example 3-5

Any Questions???

¾ Systematic or determinate errors

affect accuracy!

¾ Random or indeterminate errors

affect precision!

¾ 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

¾ Variation in temperature

¾ Contamination of the equipment

¾ Power fluctuations

¾ Component failure

All of these can be corrected by calibration

or proper instrumentation maintenance.

Systematic or determinate errors

Systematic or determinate errors

¾ Improper calculation of results

These are blunders that can be minimized or

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%

Constant

eg Solubility loss in gravimetric analysis

eg 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.

Proportional

Typically 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 Error

¾Gross errors cause an experimental value to be discarded

¾ Lead to outlier’s and require statistical techniques to be

rejected

¾Examples 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

¾We do NOT use data obtained when gross error has

occurred during collection

¾ Random errors give rise to a normal or gaussian curve.

¾ Results can be evaluated using statistics

¾ Usually statistical analysis assumes a normal distribution

Term & Definition

also called "indeterminate" and follow a

predictable pattern

¾ Error is the deviation from the "true value"

¾ 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

¾ The population data is an infinite number of

observations (all the possible results in the

universe!)

¾ The sample data is a finite number of

observations that are, hopefully, representative of the population.

A Normal or Gaussian Curve

The Statistical Treatment of Random Error

mean, µ, and a population standard deviation

1 Population mean

¾ In the absence of systematic error, µ, is the true

value for the measurement

¾ The sample mean, x, approaches µ when the

number of observations approach infinity

2 Population standard deviation

Standard Deviation

1 - N

)

( s

Deviation Standard

2 1

Sample Standard Deviation as a

measure of precision

¾ Reliability of the sample standard deviation (s)

increases with the number of replicates (N).

¾ For N greater than 20, s ⇒ σ

¾ Measuring 20 replicates is usually not practical!

Standard Deviation

1 - N

)

( s

Deviation Standard

2 1

iN xi − x

∑

=

Other measures of precision

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)

¾The difference between the largest and the smallest

values in the set of data

¾ Another term that is occasionally used to described the precision

of 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?

Calculation: S, Variance, RSD, CV, Range

6

20.3 20.1

19.8 19.4

19.6

)

( s

Deviation Standard

2 1

) 78 19 3 20 ( ) 78 19 1 20 ( ) 78 19 8 19 ( ) 78 19 4 19 ( ) 78 19 6 19 ( ) 78 19 5

2 2

2 2

− +

− +

− +

− +

− +

−

=

= 0.396 = 0.40 ppm

Standard Error of a Mean

or

¾It shows that the standard error of the mean is inversely

proportional to the square root of the number of data (replicates), N

N

s s

mean a

of Deviation

¾The 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?

1 - 6

) 78 19 3 20 ( ) 78 19 1 20 ( ) 78 19 8 19 ( ) 78 19 4 19 ( ) 78 19 6 19 ( ) 78 19 5 19

(

s

2 2

2 2

2 2

− +

− +

− +

− +

− +

mean a

of Deviation

16

0 6

0.40

=

Calculation of Sm (the standard error of the mean)

8

00080

0 50

1

0012

0 25

Pooled standard deviation

• Combine standard deviation from different

experiments to obtain a reliable estimate of the precision of a method

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

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

Error (Uncertainty) Propagation

Error (Uncertainty)

Propagation

Error Propagation:Addition and Subtraction

Example 3-7: The volume delivered by a buret is the difference between

the final and the initial reading If the uncertainty in each reading

is ± 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 reading

is 17.88 (± 0.02) mL

17.88 (± 0.02)0.05 (± 0.02)

2 b

S y

S y = 2

c

2 b

b 2

a

c

S(

)b

S(

)a

S(y

S

c

aby

++

=

=

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

1 Determining the number of significant figures in a number.

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

Price ($)

Precision (s) (mg) Type of balance

~ $16,000

± 0.0003 mg

Analytical balance

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

Methods for Reporting Data: Significant Figures

¾ Disregard all initial zeros

¾ All remaining digits including zeros between nonzero integers are significant

¾ Addition and Subtraction – the smallest number of digits to the right of the decimal sets the significance

¾ Products and Quotients - the smallest number of significant

digits determines significance

¾ Logarithms - for logs, keep as many digits to the right of the decimal as there are significant figures in the original number

¾ 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

¾ Disregard all initial zeros

¾ All 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

¾ Addition and Subtraction – the smallest number of digits to the right of the decimal sets the significance

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?

47 5

67

47 5

9.5423 9.5421

9.5 9.542

9.54

=

=

+ +

+ +

= 9.5 3 g

Rounding Data

¾ Round up for digits > or = 5, and round down for digits

< 5

¾ Use common sense when rounding Remember that

even though 3 significant figures may be permissible for a S value, S is ± term so that, 2.1 0 ± 0.0111

becomes 2.10 ± 011.

¾ Remember not to round off calculations until the final result is obtained!

Example 3-13:

Any Questions???

Absolute and Relative Errors

™ Systematic or determinate errors

™ Random or indeterminate errors

™ Gross errors or blunders

9Relative standard deviation

9Standard deviation of the mean (sm)

9Pooled standard deviation

9Coefficient of variation

9Variance (s 2 )

9 Significant Figures

The End!