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Slide 1

Random Errors in Chemical AnalysisChapter 6

Skoog, West, Holler and Crouch8th Edition

Slide 2 Fig 6-CO, p.105

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Slide 3

A. Nature of Random Errors

• Uncontrollable variables are the source of random errors

• Contributors to random errors are not all– identifiable– individually detectable– quantifiable

• The combined effect of random errors produce the fluctuation of replicate measurements around the mean

• Random errors are the major source of uncertainty.

Slide 4

Distribution of Random Errors

Table 6-1, p.106

Assume four contributors to the random error of equal magnitude.Equal probability of occurrence of negative and positive deviation.Each can cause the final result to be high or low by ±U

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Slide 5

Frequency of Occurrence and Probability

• The frequency of a deviation of a given magnitude is a measure of the probability of occurrence of that deviation

Fig 6-2a, p.107

6.25%

25.0%

37.5%

25.0%

6.25%

Slide 6

For ten equal size uncertainty

Fig 6-2b, p.107

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Slide 7

Gaussian Curve or Normal Error Curve

• For a very large number of individual errors

Fig 6-2c, p.107

Slide 8

Distribution of Experimental Errors

Table 6-2, p.108

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Slide 9

Sources of random fluctuations in the calibration of the pipet

Slide 10

Generating a histogramFrequency within ranges

Table 6-3, p.108

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Slide 11 Table 6-f1, p.110

Distribution of the Experimental Errors Approachesa Gaussian Curve

Slide 12

B. Statistical Treatment of Random Errors

• Distribution of the majority of analytical data displays characteristics of the normal distribution

• Therefore, Gaussian distribution is used to approximate distribution of analytical data– Exceptions exists

• Photon counting: poisson distribution• Isotopes of an elements: binomial distribution

• Available standard statistical methods are used to evaluate analytical data assuming random distribution of errors

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Slide 13 Fig 6-3, p.109

Slide 14

Terminology

• Population: all possible observations/ measurements/a universe of data

• Types of population– Finite and real (lot of steel, a lot of Advil Tablets)– Hypothetical or conceptual (Calcium in blood, lead in

lake Ontario).• A sample of the population is analyzed

Sample: subset of the population• Results from the analysis are used to infer the

characteristics of the population

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Slide 15

Properties of Gaussian Curves

• Gaussian curve Equation

( )

deviationdardspopmeanpopulation

valuedatax

xey

−−−

−

−−

=

tan:::

2

22 2

σµ

µσ

σµ

Slide 16

Parameters in the Gaussian Equation

• The gaussian curve is fully characterized by two parameters– the mean:µ– the standard

deviation:σ

• Population mean (µ) and Standard Deviation(σ)

( )N

x

N

x

N

i i

N

i i

∑

∑

=

=

−=

=

12

1

µσ

µ

***

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Slide 17

• A statistic: estimate of a parameter of the population obtained from a sample of data.

• Examples are:– Sample mean:

– Sample standard deviation:

x

s

Slide 18

Universal Gaussian Curve

2

2

21

2

21

21

z

x

ey

xz

ey

−

⎟⎠⎞

⎜⎝⎛ −

−

=

−=

=

µσ

σµ

µσσµ

Abscissa: deviation from the mean in units of standard deviation

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Slide 19

Properties of a normal error curve

• Mean occurs a the central point of maximum frequency

• Symmetrical distribution of positive and negative deviations

• Exponential decrease in frequency as the magnitude of the deviations increases

Slide 20

Using the Gaussian Curve

• Fraction of the population between two limits is given by the area under the curve between the two limits

• The probability of a single event between two limits is given by the fraction of the area between the two limits

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Slide 21

Calculating the Areas Under the Gaussian Curve

• Fraction of the population between two limits is given by the area under the curve between the two limits

• gives the probable error of a single measurement

( )

683.02

2

2

1

1

2

1

1

2

2

2

2

22

==

=

=

∫

∫

∫

−

−

−

−

−

−−

dzearea

dzearea

dxearea

z

z

x

π

πσ

πσ

σ

σ

σµ

Slide 22 p.114

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Slide 23

The Sample Standard Deviation

x

• Number of degrees of freedom: number of independent results needed to compute the standard deviation

• As N approaches infinity, s approaches σ and approaches µ

( )

iancesamples

N

xxs

N

xx

N

i i

N

i i

var:

1

2

1

2

1

−

−

−=

=

∑

∑

=

=

x

Slide 24

Standard Error of the Mean

• The standard deviation of the mean = standard error of the mean N

ssm =

13

Slide 25

Reliability of s as a Measure of Precision

• As N increases s becomes a better estimator of σ– Typically when N>20, s is considered to be a good

estimator of σ• Pooling data improves the reliability of s.

– Assumptions• same sources of random error in all measurements• random samples of the population are drawn (i.e. same σ).

– s pooled is the weighted average of the individual estimates of σ.

Slide 26

Pooled Standard Deviation

( ) ( ) ( )t

N

i

N

kk

N

jji

pooled NNNN

xxxxxx

s−+++

+−+−+−

=∑ ∑∑= ==

....

...

321

1 1

23

1

22

21

1 32

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Slide 27

C. Standard Deviation of Calculated Results

• Standard Deviation of a Sum or DifferenceThe variance of a sum or difference is equal to the sum of the individual variances

• Standard Deviation of a Product or a QuotientThe square of the relative standard deviation of a product or a quotient is equal to the sum of the squares of the relative standard deviations of individual

Slide 28 Table 6-4, p.128

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Slide 29

D. Significant Figures

• All certain digits plus one uncertain digit• Rules

– All initial zeros are not significant– All final zeros are not significant, unless they

follow a decimal point– Zeros between nonzero digits are significant– All remaining digits are significant

• Use scientific notation to exclude zeros that are not significant

Slide 30

Significant figures in Numerical Computations

• Sums and differences• Products and Quotients• Logarithms and Antilogarithms• Rounding Data