paul valery mars 450 - loer.tamug.edustddeviation).pdfpaul valery 2 mars 450 data reduction and...

““The trouble with our times is that theThe trouble with our times is that thefuture is not what it used to befuture is not what it used to be””

““The time for a finite world has startedThe time for a finite world has started””Paul ValeryPaul Valery

2

MARS 450

Data Reduction and

Statistics

33

Analysis as a ProcessAnalysis as a Process

Data Reduction and Method EvaluationData Reduction and Method Evaluation

–– Determine final unknown concentrationDetermine final unknown concentration

–– Determine analytical figures of meritDetermine analytical figures of merit!! Sample meanSample mean

!! Detection limitDetection limit

!! Accuracy estimateAccuracy estimate

!! Precision estimatePrecision estimate

!! Confidence limitsConfidence limits

!! SensitivitySensitivity

!! Working rangeWorking range

4

Analytical Figures of Merit

! Analytical Figures of Merit refers tostatistical information about an analyticaltechnique or analysis.

! They are used to quantitatively comparemethods and provide information aboutthe quality of a data set.

" Justification of data

" Identify limitations of the data

5


! Sample Concentration.

" Best estimate of the concentration of an

unknown

" Determined from mean value

" Corrections made for

! Blank value

! Sample dilution

6


! Precision Estimate. Reported as

" Relative Standard Deviation (RSD) (or

Coefficient of Variation - CV)

! Requires s or !.

" Confidence Limits

7


! Accuracy Estimate

" Reported as % Error

! Sensitivity (of Method)

" Based on slope of calibration curve and

reproducibility

8


! Detection Limit.

" (3 x Std. Dev. of a blank signal; reported as

concentration units)

! May also be determined using a very low standard

! Working Range or Calibration Range

" DL -----> LOL (Detection Limit to Limit of Linearity)

9

Sample Mean

! Mean (or average) – sum of the measured

values divided by the number of measured

values

! Two types based on sample size

"Population Mean (µ) – N > 20

"Sample mean ( x ) - N < 20-

!

x or µ =

xi

i= 0

N

"

N

10

Accuracy and Precision

! Precision" Describes the reproducibility of results

" Describes how well a series of measurements agree

with each other

" Related to random error

! Accuracy" How close a result is to the “true” or accepted value

" Related to systematic error

11

Low accuracy, low precision Low accuracy, high precision

High accuracy, low precision High accuracy, high precision

An Illustration of the difference

between accuracy and precision

12

Data Reduction and Experimental Error

! Data Reduction means to mathematically

process raw signal information into a form that

can be easily understood and communicated

! All measurements have experimental error

! Two types of errors

" Systematic or determinate errors

" Random or indeterminate errors

13

Systematic Errors! Arise from inherent flaws in equipment or experimental

design

! Have a definite value and a known cause

! Reproducible with precision

! Can usually be corrected easily

! Examples" Instrument materials/design

! Active surfaces in GC (sorption)

! Steel instruments used to analyze iron?

" Analyte of interest in reagents (blank)! Grade of reagent

! Reagent cleaning

" Error in making standards! Pipettes

! Balances

! Volumetric flask

" Instrument not calibrated properly14

Methods for Detecting Systematic

Errors

! Analyze samples of known composition

" SRM/CRM

" Develop a calibration curve

! Analyze “blank” samples

" Verify that the instrument and reagents will give a

zero result

! Obtain results using multiple instruments

" Verifies the accuracy of individual instrument/process

! Intercalibration studies (i.e. Black Carbon Ring Trial)

15

Random Errors

! Arise from an unknown source that cannot be

controlled

! Examples

" Variations in how individuals read the measurements

" Instrumentation noise

" Heterogeneity of matrix

! Always present and cannot always be corrected

! Correction for “noise” requires an understanding

of random distributions

16

Characterization of Random

Distributions

! If a continuous random variable is

normally distributed or has a normal

probability distribution, then a relative

frequency histogram of the random

variable has the shape of a Gaussian

curve.

LetLet’’s assume we have a student population (s assume we have a student population (nn = 47) = 47)

Central Tendencies

18

Illustration of the Distribution of Noise -

Replicate Absorbance Measurements

Mean = 0.482

19


! Range = 0.494-0.469 = 0.025

! The range tells you the complete spread ofvalues in the data series.

! However, it does not take into account thenumerical values of each and every observationand thus says nothing about the “internal”variation of a distribution of data.

! A logical measure of variation would be theaverage value of each variation from the mean.

20

Random Error: results in a scatter of results centered on the true value for

repeated measurements on a single sample.

Systematic Error: results in all measurements exhibiting a definite difference

from the true value

Random ErrorSystematic Error

plot of the number of occurrences or population of each measurement

(Gaussian curve)

Comparison of Random and

Systematic Errors

21

How to Describe Accuracy

! Accuracy is determined from the measurement

of a certified reference material (CRM)

! Accuracy is described in terms of Error

" Absolute Error = (X – ")

" Relative Error (%) = 100*(X – ")/"

where: X = The experimental result

" = The true result (i.e. CRM value)

22

Certified Reference Materials

! Certified Reference Materialsare available from nationalstandardizing laboratories" National Institute of Standards

and Technology (US)

" National Research Council(Canada)

! The CRM is analyzed alongwith the samples and itsconcentration is determined asif it were a sample withunknown concentration

! Accuracy is then evaluated bycomparing the determinedvalue with the certified valuefrom the standardizinglaboratory

23 24

25

Calculating the Variance

! By squaring difference between each value (xi) andthe mean (x or µ) and then taking the arithmetic

mean of these sum of squares.

!

" 2 =(x

i#µ)2

i= 0

N

$N

!

s2

=(x

i" x )

2

i= 0

N

#N "1

26

Calculating the Standard Deviation

Since the variance is in units of measurement that are squared,

it is convenient to take the square root of the variance and

define the quantity known as Standard Deviation:

!

" 2 =(x

i#µ)2

i= 0

N

$N

!

s2

=(x

i" x )

2

i= 0

N

#N "1

Standard Deviation =

!

Variance

27

Standard Deviation

! Standard Deviation – measures how closely the

data are clustered about the mean.

" The smaller the deviation, the more precise the

measurements

! We distinguish two types of standard deviations

based on the number of samples involved

" Population Standard Deviation (!) – (N > 20)

" Sample Standard Deviation (s) - (N < 20)

28


Distributions – The Normal Distribution

! A normal distribution isbell-shaped andsymmetric.

! The distribution ischaracterized by themean, (x or µ) and thestandard deviation (s or!, sigma).

! The mean defines thecenter value andstandard deviationdefines the spread.

-

29

Measures of Variability! Range: the high to low values measured in a

repeat series of experiments.

! Standard Deviation: describes the distribution ofthe measured results about the mean or averagevalue.

" Relative Standard Deviation

(or Coefficient of Variation)

!=

""=n

i

i nXXSD

1

2 )1/()(

where: n = total number of measurements

Xi = measurement made for the nth trial

= mean result for the data sample!

RSD(%) = (SD /X ) "100

X

30


Distributions – The Standard Deviation

! The standard deviation is thedistance from the mean tothe inflection point of thenormal curve; the placewhere the curve changesfrom concave down toconcave up.

! A smaller standard deviationmeans that your results aremore reproducible (they don’tvary as much frommeasurement tomeasurement).

31

Standard Deviations and Areas

Under the Normal Curve

! For any normal curve withmean mu (µ) and standard

deviation sigma (!):

" 68 percent of the observations

fall within ±1 standard deviation

of the mean.

" 95 percent of observation fall

within ± 2 standard deviations.

" 99.7 percent of observations fall

within ± 3 standard deviations of

the mean.

32

Illustration of the Distribution of Noise -


MEAN = 0.482

Range = 0.025

Std Dev. = 0.0056

33


Mean = 0.482 ; Range = 0.025 ; Std Dev. = 0.0056

68% of variation: 0.4864-0.4876

95% of variation: 0.4708-0.4932

99% of variation: 0.4652-0.4988

34

Curve Symmetry and Number of

Observations - Range of a Projectile

N = 100

N = 40 N = 200

N = 500

N = 1000

N = 5000

35

Curve Symmetry and Number of

Observations - Range of a Projectile

Symmetry increases as the number of observations

increases

N = 5000

36

Standard Normal Distribution

The Z Distribution! The standard normal distribution has mean = 0 and

standard deviation sigma = 1.

37

Z-Score Formula

! Any normal distribution with mean = mu and

standard deviation = sigma, can be converted

into a standard normal Z distribution by the

following transformation:

!

µ"=X

Z

38

What does a Z-Score tell us?

! Z-score describes the location of the raw

score in terms of distance from the mean,

measured in standard deviations

! Gives us information about the location of

that score relative to the “average”

deviation of all scores

39

Confidence Levels for

Various Values of z

40

Illustration of Normalization

! Zeke got 680 on the SAT math exam. The

mean on this exam is 500 and the std. dev. is

100.

! Gerald got 27 on the math ACT. The mean

score for the ACT was 18 and the std. dev.

was 6.

! Who received the better score?

41


! Zeke got 680 on the SAT math exam. The

mean on this exam is 500 and the std. dev. is

100.

! Zeke’s standardized score is:

80.1100

180

100

500680==

!=Z

42


(Continued)

! Gerald got 27 on the math ACT. The meanscore for the ACT was 18 and the std. dev.was 6.

! Gerald’s Z-Score is:

! Zeke, did better !

50.16

9

6

1827==

!=Z

43

Determining the concentration of an unknown:

Standard Addition

! Standard Addition: Useful method for analyzing complex sample in

which matrix effect can be substantial

! Common form: Adding one or more increments of a standard solution

(solid) to sample aliquots of the same size # “spiking” the sample.

44


Standard Addition

! Let “C” = concentrations and “V” = volume

! “unk” = unknown; “std” = standard

!

Cunk =CunkVunk + CstdVstd

Vtot

=CunkVunk

Vtot

+CstdVstd

Vflask

!

S = kCunkVunk

Vtot

+ kCstdVstd

Vtot

45


Standard Addition

! Let “S” = instrument signal and “k” = proportionality constant

!

S = kCstdVstd

Vtot

+ kCunkVunk

Vtot

!

S = mVstd

+ b

!

m = kCstd

Vtot

!

b = kVunkCunk

Vtot

46


Standard Addition

! Let “S” = instrument signal and “k” = proportionality constant

!

b

m=

kVunkCunk

Vtot

kCstd

Vtot

!

b

m=VunkCunk

Cstd

!

Cunk

=bC

std

mVunk

47

Standard Addition! Alternatively, a linear portion of the plot may be extrapolated to the left

of the origin (red line below)

!

Cunk

= "(V

std)0Cstd

Vunk

!

S = kCstdVstd

Vtot

+ kCunkVunk

Vtot

= 0

48

Standard Addition! Standard Additions can also be applied to solids:

!

C[ ]tot

=C[ ]

unkM

unk( ) + C[ ]stdM

std( )M

unk+ M

std

!

C[ ]unk

=C[ ]

tot" M

unk+ M

std( )( ) # C[ ]stdM

std( )M

unk

paul valery mars 450 - loer.tamug.edustddeviation).pdfpaul valery 2 mars 450 data reduction and...

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