14 - Statistics Lite - Color

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Lecture 14: Statistics •  Accuracy & precision •  Counting statistics •  Detection limit (sensitivity)

GEOL-4700/5700 – Summer 2015

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X-Ray Microanalysis – Precision and Sensitivity Recall…

K-ratio Element = [IElement)in)unknown)/)IElement)in)std.])x)CF CF relates concentration in std to pure element

K x 100 = uncorrected wt.%, and…

K x (ZAF) x 100 = corrected wt.%

Precision, Accuracy and Sensitivity (detection limits) Precision: - Reproducibility

- Analytical scatter due to nature of X-ray measurement process

Accuracy: - Is the result correct?

Sensitivity: - How low a concentration can you expect to see?

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Accuracy and Precision

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Wt.% Fe

20 25 30 35

Correct value Wt.% Fe

20 25 30 35

Correct value

Measured value

Standard deviation

Low precision, but relatively accurate

Avg

Std error

High precision, but low accuracy

Avg

Std error

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Wt.% Fe

20 25 30 35

Correct value

Low precision, but relatively accurate

Wt.% Fe

20 25 30 35

Correct value

Avg

Std error

Avg

Std error

Precise and accurate

Measured value

Standard deviation

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Characterizing Error What are the basic components of error? 1)  Short-term random error (data set)

•  Counting statistics •  Instrument noise •  Surface imperfections •  Deviations from ideal homogeneity

2)  Short-term systematic error (session to session) •  Background estimation •  Calibration •  Variation in coating

3) Long-term systematic error (overall systematic errors that are reproducible session-to-session) •  Standards •  Physical constants •  Matrix correction and Interference algorithms •  Dead time, current measurement, etc.

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Frequency of X-ray counts

Counts

Short-Term Random Error - Basic assessment of counting statistics

X-ray production is random in time, and results in a fixed mean value – follows Poisson statistics

At high count rates, count distribution follows a normal (Gaussian) distribution

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68.3% of area

95.4% of area

99.7% of area

3σ 2σ 1σ N 1σ 2σ 3σ

The standard deviation is:

σ theoretical = N

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0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

1-si

gma

erro

r %

Counts

Variation in percentage of total counts

=100 N / N =100(σ theoretical / N )

So to obtain a result to 1% precision, must collect at least 10,000 counts.

…and for 0.5% precision, must collect around 40,000 counts.

…and for 0.1% precision = 1,000,000!

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Evaluation of count statistics for an analysis must take into account the variation in all acquired intensities

•  Peak (sample and standard)

•  Background (sample and standard)

And errors propagated…

Addition and subtraction

Multiplication and division Relative std. deviation

σC = σ 2A +σ

2B

εC = ε 2A +ε2B εC =

σx

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i Current from the Faraday cup

tp Counting time of the peak (p)

r+, r– Positive and negative offsets for the background measurement, relative to the peak position

tb Total counting time on background

P(n) Peak counts (for n-th measurement)

B(n) Background counts: - For n-th measurement; - Assume LINEAR background.

Cs Element concentration in the standard

Is Intensity (Peak-background) in cps/nA of the element in the standard

Ce Element concentration in the sample

Ie Intensity (Peak-background) in cps/nA of the element in the sample

np, nb Index of measurement on the peak and on the background

np_max, nb_max Total number of measurements on the peak and on the background

B = (B+r− −B−r+ )r− − r+

tb = tb− + tb

+

Abbreviations used in the following cps = counts per second cps/nA = count per second per nanoamp

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For the calibration…

And standard deviation…

Mean =(Is )n

np

∑

np_max

σ measure =

(Pn −Bn −P +B)2

np_max −1np

∑#

$%%

&

'((

1/2

itp

Mean and standard deviation for a series of n measurements on STANDARD

NOTE: Pn, Bn = individual n-th measurement P, B = average of n-th measurements

Sum of the intensities divided by the number of measurement

Values are normalized to time (tp) and current (i)!

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Measured standard deviation

compared to… Theoretical standard deviation…

Theo.Dev(%) = 100 * σtheoretical/Is

σ theoretical =

1np_max

Pn(tp )nnp

∑"

#

$$

%

&

''

2

+1

nb_maxBn(tb )nnb

∑"

#$$

%

&''

2

i

σ measure =

(Pn −Bn −P +B)2

np_max −1np

∑#

$%%

&

'((

1/2

itp

The larger of the two represents the useful error on the standard calibration:

σ2s = max((σmeasure)2,

(σtheoretical)2)

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For the sample, the variance for the intensity can be estimated as…

σ 2Ie=

π e

(tp )n in1

nmax

∑+

βe

(tb )n in1

nmax

∑

The intensity on the sample is…

Or, in the case of a single measurement…

Pk – Bkg [cps/nA] Ie =

Pn(tp )n

−Bn(tb )nn

"

#$$

%

&''1

nmax

∑

innmax

Mean and standard deviation for a series of n measurements on SAMPLE

βe =

Bn(tb )n in1

nmax

∑nb_max

π e =

Pn(tp )n in1

nmax

∑

np_maxWith…

N.b.: standard deviation = square root of variance …or… (standard deviation)2 = variance

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The TOTAL count statistical error is then (3σ)…

σCe= 3 Ce

Ie

!

"#

$

%& σ Ie

2 +IeIs

!

"#

$

%&

2

σ Is2

Standard deviation from SAMPLE measurement

(counts)

Standard deviation from STANDARD

measurement (counts)

Standard deviation from SAMPLE concentration

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An example of calibration

X-Ray Th Ma Pk-Bg Mean (cps/nA) 155.2335 Std.Dev (%) 0.372 Theo.Dev (%) 0.136 3 Sigma (Wt%) 0.563 Pk Mean (cps) 3119.686 Bg Mean (cps) 34.455 Raw cts Mean (cts) 61657 Beam (nA) 19.87 SD measure 0.5775

Point Th Ma

(cps/nA) 1 154.6281 2 155.3082 3 154.8897 4 154.8656 5 156.4651 6 155.6509 7 156.8881 8 155.5401 9 154.8923

10 154.8614 Average,

omitting #7 155.2335 SD 0.5772

SD% 0.3718 154$

155$

156$

157$

0$ 2$ 4$ 6$ 8$ 10$

Complete statistics given by software (e.g. Cameca “PeakSight”)

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This is a very precise number

An example of sample analysis Sample Th data @ ~200 nA

Wt% Current Peak cps Peak t (sec) Bkg cps Pk-bkg cps 6.4992 200.35 nA 4098.57 800 285.0897 3813.483

Ie (net intensity for sample) 19.0337 πe (peak int) 20.4567 βe (bkg int) 1.4229 σ2

e (sample variance) 0.000137 Is (net intensity for std) 155.2335 σ2

s (std variance) 0.3335

σe

0.0735 (~ 0.39%)

σCe= 3 6.50

19.03!

"#

$

%& (0.0735)22 + 19.03

155.23!

"#

$

%&2

(0.578)2 = 0.105Error on the

concentration (including std)

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Sensitivity and Detection Limits Ability to distinguish two concentrations that are nearly equal (C and C�);95% confidence approximated by:

N Average counts NB Average

background counts n Number of analysis

points

Actual standard deviation ~ 2σC, so ΔC about 2x above equation

If N)>>)NB, then…

ΔC =C −C ' ≥ 2.33n

σC

N − NB

$

%&

'

()

ΔC =C −C ' ≥ 2.33nN

Sensitivity in % is then… ΔC /C(%) = 2.33

nNx100

Ziebold (1967)

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Example gradient:

•  If take 1 micron steps o  gradient = 0.04 wt.% / step o  Sensitivity at 95% confidence must be ≤ 0.04 wt.% o  Must accumulate ≥ 85,000 counts / step

•  If take 2.5 micron steps o  Gradient = 0.1 wt.% / step o  Must accumulate ≥ 13,600 counts / step ! So can cut count time by 6x

To achieve 1% sensitivity, must accumulate at least 54,290 counts.

As concentration decreases, must increase count time to maintain precision.

0 distance (microns) 25

Wt%

Ni

1%

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Detection Limits •  At low concentration N is no longer much larger than NB (rather N ≈ NB) •  What value of N-NB can be measured with statistical significance?

Liebhafsky limit: Element is present if counts exceed 3x precision of background: N > 3(NB)1/2

Ziebold approximation:

CDL =3.29a

ntpPPB

tp = measurement time n = # of repetitions of each measurement P = pure element count rate B = background count rate on pure element a = relates composition to intensity

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Or…

Ave Z = 79

Ave Z = 14

Ave Z = 14, 4X counts as b

CDL =3.29(wt%)IP (tpi) / IB

IP = peak intensity IB = background intensity tp = acquisition time i = current

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Detection limit & counts You can decrease the detection limit by increasing the total number of counts!

Several options to increase the number of counts:

•  Change the acceleration voltage to reach an optimum, both in term of spatial resolution (volume increase with increasing voltage!) and overvoltage (optimum usually around 2-3x);

•  Increase the current;

•  Increase the counting time;

HOWEVER, keep in mind that your sample might suffer from beam exposure, in this case, you need to lower the energy density:

•  Increase the analytical volume:

o  Increase the acceleration voltage (loss of depth resolution);

o  Increase the beam diameter (loss of lateral resolution);

•  Decrease the current and/or the counting time.

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Microprobe dating of monazite •  Monazite is a U-Th-bearing REE phosphate: can be dated by

measuring the quantity (NOT the isotopes!) of U, Th and Pb. This assume no common lead (Pb present when the crystal form), which is a reasonable assumption (typically ~2-3 ppm common Pb):

•  Pb in monazite is often less than a couple thousands ppm of Pb •  Age accuracy will depend on how accurate the Pb-content (and

actinide) can be measured ! TRACE element analysis…

Pb = Th232

(eλ232*t −1)

"

#$%

&'* 208

+U

2380.9928(eλ

238*t −1)"

#$%

&'* 206

+U

2350.0072(eλ

235*t −1)"

#$%

&'* 207

+ 204Pb

208Pb from 232Th

206Pb from 238U

207Pb from 235U

Common lead (assumed = 0) ~ a few ppm in Mnz

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0

10

20

30

40

50

60

70

80

90

100

0 100000 200000 300000 400000 500000

ppm

current * time (nA * sec)

Detection limit for Pb Pb Mα measured on VLPET

200nA, 800 sec

Can increase current and / or count time to come up with low detection limits and relatively high precision

But is it right?

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Accuracy All results are approximations. Many factors… Level 1 •  Quality and characterization of standards •  Precision •  Matrix corrections

o  Mass absorption coefficients o  Ionization potentials o  Backscatter coefficients o  Ionization cross sections

•  Dead time estimation and implementation Level 2 – inaccuracy due to sample •  Inhomogeneous excitation volume •  Background estimation •  Change in peak position (peak shift) or change in peak shape •  Polarization in anisotropic crystalline solids •  Changes in Φ(ρZ) shape with time

! Evaluate by cross checking standards of known composition (secondary standards)

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ALL measurements include errors! •  Measurement of time •  Time-integral effects (*) •  Measurement of current, including linearity, is a nanoamp a

nanoamp? Depends on measurement…

(*) Time-integral acquisition effects

•  Drift in electron optics, measurement circuitry •  Dynamic X-ray production •  Non-steady state absorbed current / charge response in

insulating materials o  Beam damage o  Compositional and charge distribution changes o  Surface contamination

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Overall accuracy Combined effect of all sources of variance….

σT2)=)σC2+σI2+σO2+σS2+σM2 σT = total error

σC = counting error

σI = instrumental error

σO = operational error

σS)= specimen error

σM = miscellaneous error

Each of which can consist of a number of other summed terms

Becomes more critical for more sensitive analyses ! trace element analysis

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Sources of measurement error: Time-integral measurements and sample effects

Sample damage (especially with focus beam) can alter the count rate! Change is apparently correlated

with abnormal behavior of the absorbed current

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Sources of measurement error: Extracting accurate intensities: peak and background measurements Background shape depends on… •  Bremsstrahlung emission (background continuum) •  Spectrometer efficiency

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0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

54000 55000 56000 57000 58000 59000 60000 61000 62000 63000 64000

Wavelength (sinθ)

Sources of measurement error: Background curvature, example in the Pb Mα region (PET crystal)

Cps

/ nA

How (non-)linear is a background? 2σ counting statistics

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Pb Mα

Exponential regression y = 74.4951 e-8.465 E-05 x

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.54 0.56 0.58 0.60 0.62 0.64

Inte

nsity

(cps

/nA

)

Wavelength (sinθ)

Pb-M region on GdPO4

Gd Lβ2 3rd order 59847 7.1 KeV

Background: linear or not?

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y = 0.679x-1

0

2

4

6

8

10

12

14

0.00 0.10 0.20 0.30 0.40 0.50 0.60

% e

rror

of n

et in

tens

ity

Net intensity (cps/nA)

Measured Theoretical based on run5

At ~1000ppm Pb, 10% error can easily produce an age error of 35-40Ma (5 wt.% Th, 4000ppm U)

Becomes 50% error at ~ 0.015 net intensity

Bkg Net intensity (Pk – Bkg)

Actual bkg 0.2354 0.0592 Linear fit 1 0.2422 0.0524

Difference 0.0068 -0.0068

% error 2.884 11.478 Actual bkg 0.2327 0.0956 Linear fit 2 0.2426 0.0857








% error 3.110 3.861

Background & Net Intensity: linear vs. exponential bkg…

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Challenge of trace element analysis: Background & peak interferences

We will come back later on the peak interference issue…

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Concentration (ppm) Y Th Pb U Age (Ma)

fitted bkg.

Ave 19196 48043 1094 3984 400.2 S Std 3518 3043 119 1343 8.0

Std Err 1573 1361 53 601 3.6

linear bkg.

Ave 19194 48007 996 3958 365.4 S Std 3519 3039 123 1340 10.1

Std Err 1574 1359 55 599 4.5

Choosing the wrong background…

…can results in an error of 10’s to 100’s ppm! For U-Th-Pb dating of monazite, this can represents 10’s millions of years!

✔ ✗

Isotopic age: 394 Ma

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