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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 =
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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