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# Evaluation of Analytical Data.pdf

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University of Toronto Mississauga

Chemistry

CHM211H5

Steven M Short

Fall

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Evaluation of Analytical Data
Some Important Definitions
Mean ( x:
• also known as the average
• obtained by dividing the sum of replicate measurements by the number of
measurements in the set
• given mathematically as
N
x
∑ i
x = i=1
N
where xirepresents the individual valxemaking up the set Nfreplicate
measurements . Evaluation of Analytical Data
Some Important Definitions
Median:
• middle result of replicate data when arranged in ascending or descending order.
• There should exist an equal number of results above and below the median value
• If working with an even-numbered data set, the median can be evaluated as the
mean of the middle pair of results.
Comparison of the Mean and Median
• Ideally, the mean and the median should be identical. Note: This may not always
be the case for very small data sets.
• Large discrepancy between the mean and median occurs when the data set
contains an outlier.
• The outlier will have a significant effect on the mean, however, no effect on the
median. Evaluation of Analytical Data
Some Important Concepts
Precision:
• describes the reproducibility of measurements.
• Often described in more quantitative terms as standard deviation, variance or coefficient of variation.
• the above metrics are all functions of how much any individual result xidiffers from the mean.
• the deviation from the mean, dp, is the measure of how any one result differs from the mean, as given
by:
d = x − x
p i
Accuracy:
• indicates the closeness of the measurement to the true or accepted value.
• the level of accuracy of a measurement is expressed by the error.
• the absolute error, E, in the measurement xi is given by: E = x i x t, where xtis the true value.
• Most often, the error in a measurement is expressed in relative terms. The relative error, E r in a
measurement is given as:
x i x t
E r ×100%
xt Evaluation of Analytical Data
Some Important Concepts
Precision versus Accuracy
Low Accuracy, Low Precision Low Accuracy, High Precision
High Accuracy, Low Precision High Accuracy, High Precision
Figure adapted from: Fundamentals of Analytical Chemistry, 8 Edition, Skoog, West, Holler and Crouch, Eds., Nelson (Toronto) 2004, p.93. Evaluation of Analytical Data
Types of Errors
Random Error (Indeterminant Error)
• causes data to scatter symmetrically about the mean value.
• affects precision
Systematic Error (Determinate Error)
• causes the mean of the data to differ from the accepted/true value
• affects accuracy by biasing the data (negative bias if results are low w.r.t. the true value,
positive bias if results are greater than the true value)
Gross Error
• often large in magnitude and with rare occurrence.
• often a result of human errors. (e.g. spillage of a sample prior to analysis)
• lead to outliers, which are results that are markedly different all other data in the set of
replicate measurements.
• Statistical tests are available for determination of outliers (e.g. Q Test) Evaluation of Analytical Data
Types of Systematic Error:
• Instrumental Error
– e.g. faulty calibration, component failure, use of instrument outside of normal operating parameters
• Method Error
– non-ideal chemical or physical behaviour of the reagents or reactions on which the analysis is
based.
– e.g. slow or incomplete reactions, instability of reactants or products, side product formation, etc.
• Personal Error
– e.g. carelessness, inattention, conducting lab work while under the influence of drugs/alcohol, etc.
Systematic Errors can be Constant or Proportional
– constant errors are independent of the size of the sample being analysed and hence become of
greater consequence as the sample size decreases.
– proportional errors decrease or increase in proportion to the size of the sample. Evaluation of Analytical Data
Random Error Distribution:
• Consider four independent unknown random error sources (U) are imposed onto an
analytical measurement
• Assume the magnitude of the error is identical for all four sources and that the magnitude
of the error can either be added or subtracted from the analytical results.
Combinations of Four Equal Uncertainties (U)
Uncertainty Number of Relative !#("
Error Magnitude
Combination Combinations Frequency !#'$"
+U + U + U + U ! + 4U 1 1/16 = 0.0625
1 2 3 4 . !#'"
,
+U +1U + U2- U 3 4 +!#&$"
)
+U +1U - U2+ U 3 4 + 2U 4 4/16 = 0.250 " !#&"
+U -1U + 2 + U 3 4 %
-U + U + U + U "!#%$"
1 2 3 4 ! !#%"
-U - U + U + U !#!$"
1 2 3 4
+U +1U - U2- U 3 4 !"
+U -1U + 2 - U 3 4
0 6 6/16 = 0.375 )*" )(" )&" !" &" (" *"
-U 1 U - 2 + U3 4
-U 1 U + 2 - U 3 4 /"&0$%1,'2)13'4"$,'
+U - U - U + U
1 2 3 4
+U -1U - 2 - U3 4
-U 1 U - 2 - U3 4 - 2U 4 4/16 = 0.250
-U 1 U +2U - U3 4
-U 1 U -2U + 3 4
-U - U - U - U - 4U 1 1/16 = 0.0625
1 2 3 4
!
Table and Figure adapted from: Fundamentals of Analytical Chemistry, 8 Edition, Skoog, West, Holler and Crouch, Eds., Nelson (Toronto) 2004, p.106-107 Evaluation of Analytical Data
Random Error Distribution:
• Consider the case of ten independent unknown random error sources (U) are imposed onto
an analytical measurement
• Assume the magnitude of the error is identical for all ten sources and that the magnitude of
the error can either be added or subtracted from the analytical results.
!#'"
!#&$"
.
,
+ !#&"
)
" !#%$"
%
" !#%"
!
!#!$"
!"
(%&" (%!" ()" (*" (+" (&" !" &" +" *" )" %!" %&"
/"&0$%1,'2)13'4"$,'
• This is distribution about the mean caused by random error produces a bell-shaped curve
which is known as a Gaussian Curve or Standard Error Curve. Evaluation of Analytical Data
Properties of Gaussian Curves:
• Gaussian curves can be described as a function two parameters, the population mµ,n,
and the standard deviationσ , where: −( )µ 2
2
e 2σ
y =
σ 2π
• the population mean, µ, is calculated in the same manner as the mean calculation shown
on the first slide, however, is based on the total number of measurements in the population
and not a limited sample set. As such, in the absence of systematic error, the population
mean should equal the true value of the measurement.
• the population standard deviationσ , is a measure of the precision of a population of
data and is given by:
N
2
∑ (xi− µ )
σ = i=1
N
where N is the number of data points making up the population.
€ Evaluation of Analytical Data
Properties of Gaussian Curves:
• A more conventional approach taken in the analysis of Gaussian distributions is to define
the abscissa ( x -axis) in terms of a parameter, z, which is the deviation of a data point from
the mean relative to one standard deviation:
(x−µ)
z =
σ
• This serves to normalise error curves so that common statistical analysis of the data is
facilitated. The resultant curve describes all populations of data, regardless of standard
deviation. When z = 1 ,x - µ = σ . When z = 2 , x - µ = 2σ. When z = 3 , x - µ = 3σ .
5
" !5 0▯5 " !▯ !▯
$ % ! #
$%&'%(
!8 0▯8
8 "▯ !#▯
675 075 !"#$%&'"()*"+,"-./0
!"#$%&'"()*"+,"-./0
6▯8 07▯8
"-▯ ")▯ "#▯ "+▯ ,▯ +▯ #▯ )▯ -▯
3"4%&$#")!"#$%&$'( / 1'2$&$#")!"#$%&$'( $%*%"&"▯"
−z 2 ▯"
!"#$%&$'()*+',)&-"),"%! (".#)"▯" 2
e
• The equation for the Gaussian curve is then written as: y =
σ 2π
Figures adapted from: Fundamentals of Analytical Chemistry, 8 Edition, Skoog, West, Holler and Crouch, Eds., Nelson (Toronto) 2004, p.111. Evaluation of Analytical Data
Properties of Gaussian Curves:
• Regardless of width:
– 68.3% of the area beneath a Gaussian curve exists within one standard deviation (±1σ) of the
mean.
– 95.4% of the area beneath a Gaussian curve exists within two standard deviations (±2σ) of the
mean.
– 99.7% of the area beneath a Gaussian curve exists within three standard deviation (±3σ) of the
mean. Evaluation of Analytical Data
Error Analysis of Small Sample Sets:
• The sample standard deviation, s, should be used in place of the population standard
deviation, σ, when analyzing small sample sets.
N 2
$ '
N &∑ xi N 2 N
∑ xi− %i=1 ( ∑ (x i x ) ∑ di
i=1 N i=1 i=1
s = = =
N −1 N −1 N −1
• the sample standard deviation is said to be an unbiased estimator of the population
st€ndard deviation aN (population size) is replaced in the denominator of the standard
deviation equation wi(N-1) , which is known as the number of degrees of freedom.
• As a rule of thumb, the sample set is considered smallNw< 30. Evaluation of Analytical Data
Pooling Data:
• if we have several sets of data for the same experiment, these data may be pooled (or
combined) so as to provide a better estimate of the population mean and population
standard deviation.
• The pooled estimate of σ is tspooledand is a weighted averagsfor each data set,
which is calculated as:
N1 N2 N 3
2 2 2
∑ (xi− x 1) + ∑ ( xj− x 2) + ∑ (xk− x 3) +...
i=1 j=1 k=1
spooled
N 1 N +2N +..3− N t
where N is the number of results in Neis the number of results inNseis the
number of results in set 3, and so the series Nonis the total number of data sets
€ t
pooled. Evaluation of Analytical Data
Reporting errors in experimental data:
• Most commonly, the standard deviation is used to report the precision of analytical data.
• The standard deviation can also be put into a relative context. The two most common
relative precision measures used are the relative standard deviation (RSD) and the
coefficient of variation (CV).
• Relative standard deviation (RSD) is calculated by dividing the standard deviation by the
mean value of the data set as follows:
s
RSD =
x
• The Coefficient of Variation (CV) is simply the RSD multiplied by 100%:
s
CV = ×100%
x Evaluation of Analytical Data
Propagation of Errors Through Arithmetic Data Processing:
Type of Calculation Sample Equation Standard Deviation of y
Addition or Subtraction y = a+b+c sy= s +as +cs c
2 2 2
#b& sy "sa% " sb% "sc
Multiplication or Division y = a "$c' = $ ' +$ ' + $ '
y # a& # b & #c&
! x ! sy "sa%
Exponentiation y = a = x$ '
y #a &
! ! " sa%
Logarithm y = log10 ) sy= 0.434$ '
# a &
! ! sy
Antilogarithm y = antilo10( ) y = 2.303 " a
! ! !
Note: a ,b and c represent experimental variables with standard deviations of s a
sb and sc, respectiv!ly. Values of sy/y should be!taken as absolute values.
Table adapted from: Fundamentals of Analytical Chemistry, 8 Edition, Skoog, West, Holler and Crouch, Eds., Nelson (Toronto) 2004, p.128. Evaluation of Analytical Data
Confidence Intervals:
• True values (i.e. real population means) are never known in the real world.
• However, if numerous measurements are made, the mean value ( ) of the xet containing
many experimental outcomes may closely approach the true value ( µ ).
• It is often the case that resources available to us (quantity of sample, time and $$$) only
permit for a limited set of data to be collected for any experiment.
• We can establish a confidence interval ( CI ) surrounding an experimentally determined
mean within which the population mean should exist
• Vice versa, a suitable number of experiments to conduct that will provide us with the
confidence required in the experimental outcome can be established a priori.
• NOTE: This assumes there is an absence of determinate error!!! Evaluation of Analytical Data
Confidence Intervals:
!"#$%&'#('
If a good estimate of σ exists, )'*'+,
zσ &"-./▯ %"# 0"-./▯
%# &
CI for µ = x ±
N
&+-""▯ .!-'# 0+-""▯
Confidence Levels as a Functioz of &+-*$▯ !"# 0+-*$▯
Confidence Level (%)!
!
50.0 0.67 -'+./%*',01'23'#(45 $"# 0+-.,▯
68.3 1.00 &+-$.▯ 0+-$.▯
80.0 1.28 $%#
90.0 1.64 &*-%!▯ $$# 0*-%!▯
95.0 1.96
95.4 2.00 &,▯ &'▯ &*▯ &+▯ "▯ +▯ *▯ '▯ ,▯
99.0 2.58 "#$#▯#
99.7 3.00 !()(
99.9 3.29 ▯#
!
Table adapted from: Fundamentals of Analytical Chemistry, 8 Edition, Skoog, West, Holler and Crouch, Eds., Nelson (Toronto) 2004, p.144. Evaluation of Analytical Data
Confidence Intervals for Limited Sets of Data:
• It is often th

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