Basic Statistics for the Estimation of Uncertainty Of

8/19/2019 Basic Statistics for the Estimation of Uncertainty Of

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QUALITY ANALYSISCONSULTANTS 1

BASIC STATISTICS FOR THEESTIMATION OF

UNCERTAINTY OFMEASUREMENT

By Wesley Johnson & Jane Wei !el


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QUALITY ANALYSISCONSULTANTS "

O# line$ No% al 'is %i(# ion an' eas#%e en

s a is i)s$ Si*ni+i)an)e es in*$ Re*%ession e ho's$ Analy i)al ,a a Q#ali y$ -%o)ess Ca.a(ili y


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QUALITY ANALYSISCONSULTANTS /

0oals o+ -%esen a ion

1"/ ..

1 2 o 13 ..


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No% al ,is %i(# ion an'Meas#%e en S a is i)s


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Mean5 S an'a%' ,e6ia ion an'

7a%ian)e$ Mean

$ S an'a%' ,e6ia ion

$ 7a%ian)e

n x xi 8∑=

91:9:

"

−−= ∑

n x x s

i

91:

9: ""−

−= ∑

n

x x s

i


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7a%ian)ex i (x i- x) (x i- x) 2

15.2 0.143 0.0204

14.7 -0.357 0.125715.1 0.043 0.0018

15.0 -0.057 0.0033

15.3 0.243 0.0590

15.2 0.143 0.020414.9 -0.157 0.0247

x = 15.057 = 0.2572


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QUALITY ANALYSISCONSULTANTS ;


7a%ian)ex i (x i- x) (x i- x) 2

15.2 0.143 0.0204

14.7 -0.357 0.125715.1 16.0 0.043 0.0018

15.0 -0.057 0.0033

15.3 16.5 0.243 0.0590

15.2 0.143 0.020414.9 -0.157 0.0247

x = 15.05715.3571

= 0.2572 3.1671


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QUALITY ANALYSISCONSULTANTS <


7a%ian)e$ Mean = 1 4>38; = 14> 4;

$ S an'a%' ,e6ia ion = : >"4;"82918" = > " ;

$ 7a%ian)e = >"4;"82 = > 3"


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QUALITY ANALYSISCONSULTANTS ?

No% ally ,is %i(# e' ,a a$ The os +%e@#en ly o(se%6e' 6al#e is he ea

a6e%a*e$

The 6al#es a%e 'is %i(# e' sy e %i)ally a(o# ean$ The%e a%e o%e 6al#es )lose% o he ean han

a ay

$ I+ he%e is no sys e a i) e%%o%5 he ean a..he %#e 6al#e$ Ran'o e%%o% %es#l s )an (e %ea e' s a is i)a

he la s o+ .%o(a(ili y


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Si #la e' Sa .lin* E .e%i enINTERVAL TALLY COUNT FRE UENCY

!7.00 11 2

7.00 " 7.99 11111 1111 98.00 " 8.99 11111 11111 11111 11111 11111 1 26

9.00 " 9.99 11111 11111 11111 11111 11111 11111 1 31

10.00 " 10.99 11111 11111 11111 1111 19

11.00 " 11.99 11111 11111 1 11

# 11.99 11 2


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HISTOGRAM OF SAMPLING EXPERIMENT

0

5

10

15

20

25

30

35

6.5 7.5 8.5 9.5 10.5 11.5 12.5

RANGE

Series2


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QUALITY ANALYSISCONSULTANTS 1"

Normal Distribution Plot

010

203040

2 9 26 31 19 11 2

6.5 7.5 8.5 9.5 10.5 11.5 12.5

Interval

F r e q

u e n c y


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QUALITY ANALYSISCONSULTANTS 1/

In e%la(o%a o%y ,a a $%& 1 $%& 2 $%& 3

T'i N*. ++, ++, ++, 1 10 12 122 14 8 143 6 4 94 11 15 115 13 6 106 8 12 157 9 5 138 7 17 89 12 6 16

10 10 15 12

,% / = 10.0 10.0 12.0 = 2.6 4.76 2.6


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No% al E%%o% C#%6e

-4 -3 -2 -1 0 1 2 3 4

0


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QUALITY ANALYSISCONSULTANTS 1;

Fa) o%s

is in #ni s o+ s an'a%' 'e6ia ions an' is aeas#%e o+ ho +a% a 6al#e is +%o a e

o% a li i o% so e o he% 6al#e>

s

x x Z

9: −=


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QUALITY ANALYSISCONSULTANTS 1<

- 7al#es

- is he .%o.o% ion o+ o(se%6a ions e .eo lie o# si'e a s.e)i+ie' %an*e>

- )an also %e+e% o he .%o.o% ion o+ .o# .# (eyon' a s.e)i+ie' li i ha is s an'a%' 'e6ia ion #ni s a ay +%o he .%o)ess a6e%a*e>


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QUALITY ANALYSISCONSULTANTS 1?

Si*ni+i)an)e Tes in*


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QUALITY ANALYSISCONSULTANTS "

OneD aile' an' T oD aile'

,is %i(# ions$ I+ he%e is a %e@#i%e en o e .%ess .%o(a(ili y ha a 6al#e ill lie o# si'e o+ %an*e :ei he% a(o6e o% (elo i 9 oD .%o(a(ili ies a%e #se'>

$ I+ he%e is a %e@#i%e en o e .%ess a

o% a lo e% li i (elo hi)h o% a(o6ehi)h s.e)i+ie' .%o.o% ions o+ 'a a .oin sill lie5 oneD aile' .%o(a(ili ies a%e #se'>


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QUALITY ANALYSISCONSULTANTS "1


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QUALITY ANALYSISCONSULTANTS ""

Con+i'en)e In e%6al +o% a Me$ This is a si*ni+i)an)e es +o% )o .a%in*

ean o a %#e 6al#e>

$ I ells5 i h a *i6en le6el o+ )on+i'en)e5he%e he .o.#la ion ean lies i h %es.eo he sa .le ean>

CI +o% G = nts x±


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QUALITY ANALYSISCONSULTANTS "/

T oD aile' Ta(le F &*' *' C*/ i %/ % I/&%'

(/-1) 90 95 99

1 6.31 12.7 63.7

2 2.92 4.30 9.92

4 2.13 2.78 4.60

9 1.81 2.26 3.25

14 1.76 2.14 2.98

∞ 1.64 1.96 2.58


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7AMSTAT Re6ie


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Con+i'en)e In e%6al +o% h

Es i a e o+ S an'a%' ,e6ia io


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S a is i)al Tole%an)e In e%6$ This 'es)%i(es he li i s i hin hi)h a

s.e)i+ie' .e%)en a*e o+ he .o.#la ion ise .e) e' o lie i h a *i6en .%o(a(ili y>:+%o J> > Taylo%9

$ Tole%an)e in e%6al = ± s he%e is a en+%o a a(le o+ +a) o%s : oD aile'9


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QUALITY ANALYSISCONSULTANTS ";

A Si*ni+i)an)e Tes o see i+ T

Means a%e ,i++e%en :19$ Cal)#la e s .oole' an' o(s +%o

" 99:1:99:1: "1

"

""

"

11−+

−+−=

nn

sn sn s pooled

"1

"1 119:

nn s

x xt

pooled obs +

−=


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QUALITY ANALYSISCONSULTANTS "<

A Si*ni+i)an)e Tes o see i+ T

Means a%e ,i++e%en :"9$ Co .a%e o(s i h )%i +%o a oD aile' a(l

o+ 6al#es +o% he a..%o.%ia e 'e*%ees+%ee'o :n1 K n " "9 an' )on+i'en)e le6el

$ I+ o(s )%i yo# )an say he eans a%e'i++e%en a ha )on+i'en)e le6el


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QUALITY ANALYSISCONSULTANTS "?

-ai%e' D es :19$ This is #se' o es +o% a (ias (e een o e

:o% analys s5 o% la(o%a o%ies9 hen %#nnin*

sa .les o+ 6a%yin* )o .osi ion$ Cal)#la e he ean 'i++e%en)e : 'i++ 9 an' hes an'a%' 'e6ia ion o+ he 'i++e%en)es :s'i++ 9 (e een

he o se s o+ %es#l s +o% ea)h o+ n sa .les )al)#la e o(s +%o

diff

diff obs sn

xt =


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QUALITY ANALYSISCONSULTANTS /

-ai%e' D es :"9$ I+ o(s )%i a en +%o a oD aile' a(l

hen he%e is no e6i'en)e ha he%e is a (e een he o a ha )on+i'en)e le6el nD1 'e*%ees o+ +%ee'o >


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QUALITY ANALYSISCONSULTANTS /1

A si*ni+i)an)e Tes +o% a Co .a%i

o+ 7a%ian)es D he FD es $ s1 = s an'a%' 'e6ia ion +%o n1 eas#%e en s$ s" = s an'a%' 'e6ia ion +%o n" eas#%e en s

$ s1 s"5 'e*%ees o+ +%ee'o ν = nD1$ Cal)#la e F = s1"8s"" $ Loo #. he )%i i)al 6al#e F) +o% he a..%o.%ia e

'e*%ees o+ +%ee'o : ν9 +o% ea)h se >$ I+ F F) hen s1 s" is si*ni+i)an a he )on+i'en)e

le6el$ I+ F F) hen s" is no si*ni+i)an ly less han s1>


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QUALITY ANALYSISCONSULTANTS /"

Si*ni+i)an)e Tes in* +o% O# li

The%e a%e any es s a6aila(leP ,i on Tes 0%#((s Tes H#*e E%%o% Tes ol o*o%o6DS i%no6 Tes

I is i .o% an o es s#s.e) 'a a .oin s o%'e% o a..ly he s a is i)s 'es)%i(e' he%


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QUALITY ANALYSISCONSULTANTS //

The H#*e E%%o% Tes$ M = sD& 8s he%e s is he s#s.e) 'a a

.oin an' & an' s a%e )al)#la e' i h s e )l#'e'>

$ I+ M 35 he 'a a .oin is %e e) e'> I+ n≥ <hen %e e) ion is )onsi'e%e' )o%%e) i

.%o(a(ili y o+ ?4


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Re*%ession Me ho's


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Re*%ession Me ho's$ Re*%ession e ho's a%e )o only

#n'e%s oo' as a eans o+ +i in* he (ess %ai*h line h%o#*h a )ali(%a ion 'a a The%e a%e o he% non linea% %ela ionshi.s We ill 'eal i h leas s@#a%es e)hni@#e as

is os )o ony =α Kβ


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Resi'#als$ The leas s@#a%es %e*%ession .%o'#)es a

'es)%i(in* he %ela ionshi. (e een an' ys#)h ha he s# o+ he s@#a%es o+ h'i++e%en)es (e een he a) #al e .e%i en6al#es o+ y :yi9 an' hose ha )an (e

)al)#la e' +%o he +i e' lineŷ i is aini # > The 6al#es :yi D ŷ i ) a%e )alle''% i >


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QUALITY ANALYSISCONSULTANTS /;

Resi'#als an' 0oo'ness o+ Fi$ The %esi'#als sho#l' (e %an'o ly

'is %i(# e' a(o# a ean o+ i h as an'a%' 'e6ia ion sy8B

$ I+ a .lo o+ sy8B 6s is a'e5 he%e sho#l' (ea %an'o 'is %i(# ion o+ sy8B a(o# a line a

i h "8/ o+ he .oin s lyin* i hin a %e*iha is±" sy8B o+ he ean line a


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QUALITY ANALYSISCONSULTANTS /<

Resi'#als an' 0oo'ness o+ Fi$ A )o on +i*#%e ha is *ene%a e' in

)o .# e% %e*%ession )al)#la ions is he " 6al#e he .%o.o% ion o+ he 6a%ian)e o he %e*%ession> The (e e% he 'a as %ai*h line5 he )lose% R " ill (e o 1> This

)an (e islea'in* ho e6e%5 an' he%esho#l' (e a 6is#al ins.e) ion o+ he .lo e''a a o )on+i% he %ela ionshi. is linea%


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QUALITY ANALYSISCONSULTANTS /?

An R " 7al#e 0%ea e% Than >?? Can BO( aine' F%o a NonDlinea% Cali(%a

y = 1.0114x + 0.0381

R 2 = 0.9938

y = -0.0393x 2 + 1.2079x - 0.0929

R 2 = 0.997

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6

Concentration

R e a d i n g


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Cali(%a ion as a So#%)e o+ M

$ Fo% %e*%ession he #n)e% ain y in o is *i6en (y he e@#a ionP

∑ −−++= ""

"8

9:9:11 x xb

y ynmb

s s

i

o x y xo


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Cali(%a ion as a So#%)e o+ M$ This es i a e o+ he s an'a%' 'e6ia ion o+ o )an

(e #se' as a )on %i(# ion +%o he )ali(%a ion

an' he sho% e% ins %# en 6a%ia(ili y es i a in* he o6e%all eas#%e en #n)e% aihe eas#%e en .%o)ess> I+ i is 'e e% ine'

less han 18/ o+ he la%*es so#%)e o+ #n)e%

:s#)h as sa .lin* #n)e% ain y +o% e a .le9 in o6e%all eas#%e en .%o)e'#%e hen his )an ('e ons %a e' an' i s )on %i(# ion i*no%e'>


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QUALITY ANALYSISCONSULTANTS 3"

Analy i)al ,a a Q#ali y


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QUALITY ANALYSISCONSULTANTS 3/

Me ho' 7ali'a ion as a

So#%)e o+ MU In+o% a ion$ We ill 'is)#ss he e ho' 6ali'a ion%e.o% as a so#%)e o+ in+o% a ion +o%'e e% inin* MU


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Re+e%en)e$ The Fi ness +o% -#%.ose o+ Analy i)al

Me ho's

A La(o%a o%y 0#i'e o Me ho' 7ali'a ionan' Rela e' To.i)s

$ E#%o)he Wo% in* 0%o#.


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7ali'a ion O. ions

$ S an'a%' e ho's :AOAC e )>9 7ali'a ion ISO 1; "4 4>3>"

$ InDho#se e ho' 6ali'a ion ISO 1; "4 4>3>3


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QUALITY ANALYSISCONSULTANTS

32

S an'a%' Me ho's

I+ a e ho' is 6ali'a e' (y a s an'a%'so%*ani!a ion hen he #se% nee' onles a(lish .e%+o% an)e 'a a +o% hei%

#seISO 1; "4 4>3>"


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3;

S an'a%' Me ho's$ -e%+o% an)e o+ he e ho' in yo#% la(

in)l#'es 'e e% inin* A))#%a)y -%e)ision

$ This 'a a )an (e #se' o 'e e% ine MU


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3<

S an'a%' Me ho's$ In e%la(o%a o%y .%e)isions )an (e #se' +o% MU

:AOAC9 See he CAEAL .oli)y

$ In he a(sen)e o+ any o he% so#%)e o+ %e.ea e' 'a a Will o6e%es i a e .%e)ision +o% one la( Tha is5 MU≈ k SR

$ Wi h k = )o6e%a*e +a) o% :(e een " an' /5 'e.en'in* on he

n# (e% o+ la(s9$ SR = Re.%o'#)i(ili y S, = :S% "KSL"9$ S%= Re.ea a(ili y S,$ SL = Be eenDla(o%a o%y S,


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3?

In Ho#se Me ho's$ A))#%a)y

Un)e% ain y o+ CRM

Bias5 Re)o6e%y$ 1 %e.e i ions o% / )on) n / %e.e i ions

$ -%e)ision In %a assay5 in e% assay5 %e.ea a(ili y5in e% e'ia e .%e)ision5 %e.%o'#)i(ili y

$ 1 %e.e i ions o% / )on) n / %e.e i ions


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4

In Ho#se Me ho's$ LO,5 LOQ

,e e% ine i+ .%e)ision 6a%ies i h

)on)en %a ion$ R#**e'ness 0oo' so#%)e o+ .%e)ision in+o% a ion

$ Cali(%a ion The )on %i(# ion o 6a%ia ion +%o )ali(%ay (e in)l#'e' in a))#%a)y an' .%e)ision es s


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4"

Fi ness +o% -#%.ose$ The en i%e .#%.ose o+ an analysis is o

'e e% ine i+ so e hin* is +i +o% #se Is he a e% +i +o% .eo.le o '%in Is he soil +i +o% a*%i)#l #%al #se Is he e++l#en +i +o% a''in* o he lo)al se

sys e$ To assess +i ness +o% #se he MU #sno n


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4/

7a%ia ion F%o Sa .lin*$ Sa .lin* o))#%s hen o%i*inal sa .le is

a en an' hen s#(Dsa .les a%e a en in

he la( La( is %es.onsi(le +o% 6a%ia ion in %o'#)e' (hei% s#(Dsa .lin*

Use o+ %e.li)a es :'#.li)a es9 is an e++e) i6ee)hni@#e o 'e e% ine 6a%ia ion +%o s#(sa .lin*


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44

Fi ness +o% -#%.ose$ The en' #se% o+ he 'a a )an #se he MU

'e e% ine i+ he so e hin* is +i +o% They )an no no ho #)h 6a%ia ion is)a#se' (y he analy i)al 'e e% ina ion

$ Co .a%e 6al#e o a s.e)i+i)a ion

A )o only #se' .a%a e e% is .%o)ess)a.a(ili y


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42

-%o)ess Ca.a(ili y$ So e +o% #las

$ C .l = : V LSL98/σ$ C#l = :USL D V98/σ$ C .J = in C .l 5 C .# X

LSL = Lo e% S.e)i+i)a ion Li iUSL = U..e% S.e)i+i)a ion Li i


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QUALITY ANALYSIS 4

Con)l#sion

1"/ ..

1 2 o 13 ..

Basic Statistics for the Estimation of Uncertainty Of

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