This article was downloaded by: [University of Chicago Library]On: 15 November 2014, At: 00:40Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Analytical LettersPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lanl20

On The Interpretation Of Errors In CountingExperimentsLloyd A. Currie a ba Physikalisches Institut der Universität Bern , 3000, Bern, Switzerlandb Analytical Chemistry , Division National Bureau of Standards , Washington, D. C., 20234Published online: 05 Dec 2006.

To cite this article: Lloyd A. Currie (1971) On The Interpretation Of Errors In Counting Experiments, Analytical Letters, 4:12,873-882, DOI: 10.1080/00032717108066074

To link to this article: http://dx.doi.org/10.1080/00032717108066074

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

ANALYTICAL LETTERS, 4(12), 873-882 (1971)

ON THE INTERPRETATION O F ERRORS I N

COUNTING EXPERIMENTS

KEY WORDS: r a d i o a c t i v i t y , a c t i v a t i o n a n a l y s i s , Binomial and Poisson d i s t r i b u t i o n s , compound d i s t r i b u t i o n s , chi-squared, d i s p e r s i o n t e s t .

Lloyd A. C u r r i e

Phys ika l i s ches I n s t i t u t d e r U n i v e r s i t a t Bern 3000 Bern, Switzer land

and

A n a l y t i c a l Chemistry Divis ion Na t iona l Bureau of Standards Washington, D. C. 20234

ABSTRACT

D i g i t a l experiments, such as t h e measurement of r ad io -

a c t i v i t y , are u s u a l l y c h a r a c t e r i z e d by random e r r o r s which

fol low t h e Binomial o r Poisson d i s t r i b u t i o n . Three types of

d i f f i c u l t y which commonly a r i s e i n t h e s t a t i s t i c a l t r ea tmen t

of such experiments i nc lude :

a ) f a i l u r e t o recognize t h a t t h e u n c e r t a i n t y i n t h e p h y s i c a l

q u a n t i t y of i n t e r e s t may be governed by a compound proba-

b i l i t y d i s t r i b u t i o n , p a r t i c u l a r l y when d e a l i n g with s h o r t -

l i v e d r a d i o a c t i v i t y ;

b ) t h e improper use of t h e chi-squared s t a t i s t i c when making

s t anda rd error estimates i n weighted l eas t - squa res calcu-

l a t i o n s ;

c) s e l e c t i n g t h e c o r r e c t number of degrees of freedom and

choosing between t h e x2 d i s p e r s i o n t e s t and t h e goodness-

o f - f i t t e s t f o r grouped d a t a .

873

Copyright @ 1971 by Marcel Dekker, Inc. NO PART of this work may be reproduced or utilized in any form or by any means, electronic or mechanical, including xerography, photocopying, microfilm, and re- cording, or by any information storage and retrieval system, without the written permission of the publisher.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:40

15

Nov

embe

r 20

14

L. A. CURRIE

Examination of t h e n a t u r e of each of t h e s e d i f f i c u l t i e s has

been followed by a recommended cour se of a c t i o n .

INTRODUCTION

P h y s i c a l measurements which a r e fundamentally d i g i t a l i n

n a t u r e - i . e . , "count ing" experiments - may f r e q u e n t l y

be i n t e r p r e t e d i n terms of t h e Poisson d i s t r i b u t i o n . The

most common such measurement p rocesses are t h o s e invo lv ing

r a d i o a c t i v i t y or n u c l e a r r e a c t i o n s . However, s e v e r a l o t h e r

t ypes of a n a l y t i c a l measurement which invo lve p u l s e coun t ing ,

such as m a s s spectrometry and e l e c t r o n microprobe a n a l y s i s ,

a l s o f a l l i n t o t h i s class.

Assessment of t h e p r e c i s i o n and ( s t a t i s t i c a l ) f i t of t h e

r e s u l t s of such experiments g e n e r a l l y p r e s e n t s no d i f f i c u l t y .

Three s p e c i a l problems, however, f r e q u e n t l y g e n e r a t e confusion

and erroneous conc lus ions . In t h e f i r s t of t h e s e , s t a n d a r d

e r r o r s are o f t e n underest imated because on ly t h e l a s t (Binomial)

s t a g e of a m u l t i s t a g e (Compound) p rocess i s cons ide red , and t h e

Index of Dispersion (variance/mean) i s consequent ly f a l s e l y

assumed t o be less than u n i t y . The second problem relates

t o t h e improper use of x2 t o " c o r r e c t " t h e s t a n d a r d e r r o r

estimate i n weighted l eas t - squa res c a l c u l a t i o n s . A simple

m u l t i p l i c a t i v e c o r r e c t i o n i s s e e n t o apply on ly when t h e

r e l a t i v e weights ( i n v e r s e v a r i a n c e s ) a r e known i n advance.

F i n a l l y , one may be f aced wi th t h e problem of dec id ing

between t h e goodness-of-f i t test based upon t h e observed

d i s p e r s i o n and t h e ( d i s t r i b u t i o n - f r e e ) t e s t based upon t h e

observed (vs expected) f r e q u e n c i e s f o r grouped d a t a , and - i n e i t h e r case - d e c i d i n g upon t h e c o r r e c t number of degrees

87 4

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:40

15

Nov

embe

r 20

14

INTERPRETATION OF ERRORS IN COUNTING EXPERIMENTS

of freedom. In what fol lows w e s h a l l a t tempt t o expose t h e

sou rces of d i f f i c u l t y and t o s u g g e s t t h e proper t r ea tmen t

f o r each of t h e s e problems.

COMPOUND DISTRIBUTIONS

Random e r r o r s a r i s i n g i n t h e measurement of r a d i o a c t i v i t y

are g e n e r a l l y considered t o fol low t h e Binomial d i s t r i b u t i o n .

This conclusion fol lows provided t h a t one has a l a r g e number

( N ) of e q u i v a l e n t r a d i o a c t i v e n u c l e i having an e q u a l proba-

b i l i t y (p ) t o decay and be d e t e c t e d . For a simple d e r i v a t i o n ,

one may c o n s u l t a s t anda rd t e x t such as t h e one by F r i ed lande r , where t h e l i m i t i n g Poisson ( p < < l ) and Normal ( p c < l , Np>>l)

d i s t r i b u t i o n s are a l s o deduced.

1

A b a s i c d i s t i n c t i o n between t h e Binomial case and i t s

Poisson l i m i t relates t o t h e va r i ance . Although, i n e i t h e r

case t h e mean i s Np, t h e va r i ance e q u a l s Np(1-p) f o r t h e

Binomial d i s t r i b u t i o n , b u t Np ( t h e mean) f o r t h e Poisson

d i s t r i b u t i o n . Therefore , experiments i n which r ad ionuc l ides

are observed with high e f f i c i e n c y f o r a p e r i o d t h a t i s n o t

s h o r t compared t o t h e mean l i f e may l e a d t o s i g n i f i c a n t

dec reases i n t h e va r i ance from t h e Poisson va lue , due t o t h e

f a c t o r , (1-p) . This conclusion -- t h a t t h e va r i ance is

smaller than t h e mean -- has been noted by many au tho r s i n

connection with experiments i nvo lv ing s h o r t - l i v e d radio-

a c t i v i t y l r 2 .

t h e s t anda rd d e v i a t i o n of t h e r e s u l t of an experiment i n

a c t i v a t i o n a n a l y s i s i s d e r i v e d from t h e expres s ion f o r t h e

v a r i a n c e of t h e Binomial d i s t r i b u t i o n , Np(1-p) I with N and

p d e f i n e d as above.

In r e f e r e n c e 2 (Eq ' s . 2 4 and 2 5 ) , f o r example,

875

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:40

15

Nov

embe

r 20

14

L. A . CURRIE

Herein lies an error. Although t h e v a r i a n c e , Np(1-p) ,

prope r ly r e l a t e s t o t h e i n i t i a l number of r a d i o a c t i v e n u c l e i

i n a given, a c t i v a t e d sample, it does n o t re la te t o t h e b a s i c

q u a n t i t y of i n t e r e s t -- t h e o r i g i n a l number of s t a b l e n u c l e i

(No) . r e l e v a n t s t a t i s t i c a l t h e o r y has been developed by Feller . Following F e l l e r ' s development, w e may c o n s i d e r t h e o v e r a l l

act ivat ion-decay-count ing p rocess as a compound p rocess , i n

which t h e f i r s t s t e p ( a c t i v a t i o n ) is a Poisson process and

succeeding s t e p s a r e simply b i n a r y ( B e r n o u l l i ) t r i a l s (decay/no

decay, count/no coun t , e t c . ) Feller t r ea t s t h e problem by

d e r i v i n g t h e d i s t r i b u t i o n of t h e sum,

3 This d i s t i n c t i o n has been noted by Stevenson4 and t h e 5

s = x 1 + x 2 + x 3 + + %

where N as w e l l as each of t h e x ' s are independent, random

v a r i a b l e s . The mean (us) and va r i ance (Vs) of S are given by,

- ' s - 'N'x

2 - vs - PNVx + VNIJX

Taking t h e case of n u c l e a r a c t i v a t i o n , where S r e p r e s e n t s

t h e number of counts ob ta ined and x = 0,l f o r each of t h e N

r a d i o a c t i v e atoms -- a c t i v a t e d w i t h p r o b a b i l i t y T , one f i n d s

VN = NOIT ( 1 - T ) -N IT 0

PN = NOT

u, = P vx = p ( l - p )

Thus ,

!Js = NoTP

Vs = No'Tp(l-p) + NoTp2 = Norp

As a r e s u l t , t h e d i s t r i b u t i o n f o r t h e o v e r - a l l (compound)

process i s e q u i v a l e n t t o a s imple Poisson d i s t r i b u t i o n with

mean and va r i ance both e q u a l t o Nonp.

p r o b a b i l i t y of producing a count is high ( p z l ) , t h e sma l l

Thus, even when t h e

876

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:40

15

Nov

embe

r 20

14

INTERPRETATION OF ERRORS IN COUNTING EXPERIMENTS

r e a c t i o n p r o b a b i l i t y ( n < < l ) p r e se rves t h e Poisson c h a r a c t e r

of t h e p rocess . 7

S i m i l a r conclusions apply t o a l a r g e number of phys i ca l

experiments -- i n which one i s b a s i c a l l y i n t e r e s t e d i n t h e

i n i t i a l number of atoms, a r e a c t i o n c r o s s s e c t i o n , o r an

( a c t i v a t i n g ) p a r t i c l e f l u x . I t should be noted, i n pas s ing ,

t h a t o t h e r compound d i s t r i b u t i o n s do n o t l ead t o such simple

s o l u t i o n s as t h e above Poisson-Binomial combination. The

more gene ra l compound Poisson d i s t r i b u t i o n -- of concern i n

t h e fol lowing s e c t i o n -- has a va r i ance which exceeds t h e

mean, because t h e Poisson parameter i t s e l f i s randomly d i s t r i b

uted5 ' 6.

C H I - SQUARED AND WEIGHTED LEAST SQUARES

Weighted least squa res c a l c u l a t i o n s , which are commonly

used f o r f i t t i n g nuc lea r s p e c t r a o r decay curves, i nvo lve t h e

a p p l i c a t i o n of weights (wi) t o t h e obse rva t ions ( y . ) t o d e r i v e

a t e s t s t a t i s t i c , 1

2 - pi) / d . f .

which i s d i s t r i b u t e d as X2/d.f., provided t h a t t h e weights

G = C W . (y i i

are t h e i n v e r s e va r i ances . (d . f . r e p r e s e n t s t h e number of

degrees of freedom.) Comparison of G w i th i t s expected

va lue , u n i t y , p rov ides a d i s p e r s i o n t e s t of t h e f i t . On

t h e o t h e r hand, i f t h e weights are only p r o p o r t i o n a l t o t h e

i n v e r s e va r i ances , t hen G s e r v e s t o estimate t h e cons t an t of

p r o p o r t i o n a l i t y , and t h e s t a n d a r d e r r o r estimate must t hen

include t h e f a c t o r c. Confusion between t h e s e two uses f o r G r e s u l t s i n t h e

common mistake of mul t ip ly ing t h e s t a n d a r d e r r o r e s t i m a t e by

fi whenever G i s n o t w i t h i n accep tab le bounds. Except f o r

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:40

15

Nov

embe

r 20

14

L. A . CURRIE

t h e case where t h e var iances a re a l l underestimated by the

same f a c t o r , t h i s i s not a co r rec t procedure; it may prevent

t h e recogni t ion of i nco r rec t models or var iances and it may

lead t o wrong s tandard e r r o r es t imates .

--

A d e t a i l e d examination of t h i s problem has been given

elsewhere f o r an ac t iva t ion ana lys i s experiment involving

t h e more general compound Poisson d i s t r i b u t i o n , i n which

the index of d i spers ion exceeds uni ty . In t h i s experiment,

t h e variances d i f f e r e d from t h e simple Poisson values by an

add i t iona l (cons tan t ) var iance, a r a t h e r common s i t u a t i o n i n

8

r ad ioac t iv i ty measurements. Ignoring t h a t f a c t l ead t o a

"bad" f i t , but mu l t ip l i ca t ion of t h e s tandard e r r o r estimate

by fi did not y i e l d a co r rec t answer. The r e s u l t s of t h e

experiment -- t h e determination of oxygen by f a s t neutron

ac t iva t ion analysis ' -- appear i n Table 1. The f i r s t row

gives t h e (weighted) mean r a t i o of unknown t o s tandard (ne t )

c o u t s , based upon 10 observat ions, and t h e following rows

r e l a t e t o the respec t ive s tandard e r r o r s .

TABLE 1

Oxygen Act ivat ion Analysis -- Weighted

Mean Ratios and Standard Errors

Poisson Weights*

Mean Ratio 2 .006

Standard Error

(1) from weights 0 .o1g2

( 2 ) x a 0.0884

( 3 ) cor rec t value 0 . 1 4 9

True Weights

2 . 1 4 6

0 .119

-- --

2 -1 *wi =

878

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:40

15

Nov

embe

r 20

14

I N T E R P R E T A T I O N OF ERRORS I N COUNTING EXPERIMENTS

The r e s u l t of u s ing count ing s t a t i s t i c s on ly ("Poisson

Weights") -- 2 .006 - + 0.019 -- w a s accompanied by a value of

4 .60 f o r a, i n d i c a t i v e of a very poor f i t . Erroneously

mul t ip ly ing by fi produced t h e r e v i s e d s t anda rd error

e s t i m a t e 0.0884.

e r r o r of t h e r e s u l t 2 .006 i s 0 . 1 4 9 and it may be de r ived , as

shown i n r e f e r e n c e 8 , from t h e r a t i o s of t h e t r u e t o assumed

(Poisson) va r i ances .

However, t h e c o r r e c t value f o r t h e s t anda rd

A b e t t e r procedure i n t h i s case is t o inc lude t h e 2 a d d i t i o n a l va r i ance component (axs) i n t h e weights -- i . e .

w . = ( a + (I ) -' Then G serves a s a t es t s t a t i s t i c i f t h e 1 xs p i '

a d d i t i o n a l va r i ance component i s known, o r G may be se t e q u a l

t o i t s expected value ( u n i t y ) i n o r d e r t o estimate t h e

a d d i t i o n a l va r i ance . The c a l c u l a t i o n i s d i scussed i n some

d e t a i l i n r e f e r e n c e 8 , where it i s shown t h a t t h e best estimate

and i t s s t anda rd e r r o r i s 2 . 1 4 6 f 0.119. The r e s u l t of u s ing

j u s t Poisson weights , 2.006, i s accep tab le i n t h a t it i s

unbiased, b u t i t s va r i ance i s l a r g e r t han t h a t of t h e 2 . 1 4 6

estimate by a f a c t o r of about 1 . 6 . On t h e o t h e r hand, "cor-

r e c t i o n " by t h e f a c t o r fi l eads t o an e s t ima ted va r i ance ( f o r

2.006) which i s t o o small by a f a c t o r of about 2.8.

By way of summary, s t anda rd e r r o r m u l t i p l i c a t i o n by fi

should be used only i f r e l a t i v e weights (and va r i ances ) are

c o r r e c t l y known. The more common p h y s i c a l s i t u a t i o n i n which

t h e (assumed) va r i ances are underestimated because of a missing

e r r o r component r a t h e r t han by a common f a c t o r must be t r e a t e d

as d i s c u s s e d i n t h e preceding paragraph. Otherwise model

e r r o r s may go unrecognized, b e s t (minimum va r i ance ) estimates

879

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:40

15

Nov

embe

r 20

14

L. A. CURRIE

w i l l n o t r e s u l t , and s t a n d a r d e r r o r estimates w i l l be inco r - 10 rect .

POISSON DISPERSION TEST

A b r i e f remark about L e x i s ' d i s p e r s i o n t e s t f o r Poisson

v a r i a b l e s may prove h e l p f u l i n exposing ano the r p o t e n t i a l

source of confusion i n coun t ing experiments . I n t h i s t e s t ,

t h e sum of t h e squa res of t h e d e v i a t i o n s of n-repeated counts

(xi) from t h e mean va lue (x) is d i v i d e d by t h e mean value:

- X

1 2 2 2

X / d . f . a sympto t i ca l ly approaches t h e v a r i a n c e r a t i o s /a ,

and t h e r e f o r e it has approximately t h e X2/d.f. d i s t r i b u t i o n .

Th i s r e s u l t , i n which d . f . = n-1, i s t r e a t e d i n some d e t a i l by

Cox and Lewis".

D i f f i c u l t i e s may a r i s e , however, when one compares

t h i s d i s p e r s i o n t e s t with Pea r son ' s x 2 t e s t f o r goodness-

o f - f i t u s ing c - c l a s s e s of d a t a . I n t h i s case where one

examines t h e frequency of occurrence ( f . ) f o r d i f f e r e n t

c l a s s e s (magnitudes) of counts r a t h e r t han t h e counts them-

s e l v e s , t h e number of degrees of freedom i s c-1-PI where P

e q u a l s t h e number of parameters e s t i m a t e d from t h e d a t a . (As

t h e c-classes cover t h e e n t i r e sample space , t h e number of

independent f r equenc ie s cannot exceed c-1) . Thus, count ing

(Poisson) experiments , i n which t h e mean i s e s t i m a t e d from

t h e d a t a , y i e l d c-2 degrees of freedom.

3

Confusion d i sappea r s immediately i f one recognizes t h e

f a c t t h a t t h e r e are two d i f f e r e n t tests invo lved , one f o r

d i s p e r s i o n and one f o r goodness -o f - f i t f o r grouped observa-

t i o n s . B o t h tests a r e c a r r i e d o u t with t h e x2 s t a t i s t i c ,

000

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:40

15

Nov

embe

r 20

14

INTERPRETATION OF ERRORS IN COUNTING EXPERIMENTS

and, f o r Poisson v a r i a b l e s , t h e two tes t s formally look

very much a l i k e . Th i s formal s i m i l a r i t y , i n f a c t , goes

deeper , €or i n t h e goodness-of-f i t t e s t ( f o r any assumed

d i s t r i b u t i o n ) t h e obse rva t ions i n each class may be considered

t o fo l low independent Poisson d i s t r i b u t i o n s 1 2 -- a f a c t

which i n c i d e n t a l l y i s h e l p f u l i n dec id ing upon t h e minimum

accep tab le frequency p e r class i n t e r v a l .

In g e n e r a l it would seem d e s i r a b l e t o use t h e d i s p e r s i o n

t e s t ( d . f . = n-1) r a t h e r t han t h e goodness-of-€it (class)

t e s t ( d . f . = c-2) f o r Poisson va r i ab le s13 .

tes t , which g e n e r a l l y has many more degrees of freedom, f r e -

quen t ly proves t h e more s e n s i t i v e . A d e t a i l e d and very il-

luminat ing d i s c u s s i o n of t h e e n t i r e problem has been given

by Cochran .

The d i s p e r s i o n

1 2

ACKNOWLEDGMENT

The au tho r i s most g r a t e f u l f o r t h e s t i m u l a t i n g d i scus -

s i o n s and t h e h o s p i t a l i t y of P ro f . H. Oeschger and h i s co l -

leagues of t h e Phys ika l i s ches I n s t i t u t d e r U n i v e r s i t a t Bern,

where t h i s work was c a r r i e d o u t . Helpful d i s c u s s i o n s a l s o

took p l a c e with D r . H. Riedwyl, D r . S t r e i t and P ro f . A. N i r .

P a r t i a l suppor t from t h e “Schweizer ischer Nat ionalfonds” i s

g r a t e f u l l y acknowledged.

1.

2.

3.

4 .

EFEENCES

G. F r i ed lande r , J. W. sennedy, and J. M. M i l l e r , Nuclear and Radiochemistry, 2n E d i t . , Wiley ( N e w York) 1 9 6 4 .

P. C. J u r s and T. L. Isenhour , Anal. Chem. - 39 1388 (1967) .

The d i f f i c u l t y becomes obvious if p+l. Then t h e assumed variance e q u a l s ze ro , even though N i s a random v a r i a b l e .

P. C. Stevenson, ‘Processing of Counting Data ,” NAS-NS- 3109 ( 1 9 6 6 ) .

881

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:40

15

Nov

embe

r 20

14

L. A. CURRIE

5.

6 .

7.

8 .

9 .

10.

11.

1 2 .

13.

W. F e l l e r , An I n t r o d u c t i o n t o P r o b a b i l i t y Theory and Its A p p l i c a t i o n s , 2nd E d i t . , V o l . I , Chapt . X I I , Wiley ( N e w York) 1957.

J. Byrne, P h y s i c a - 41 575 (1969) .

The Binomial e x p r e s s i o n f o r t h e v a r i a n c e , Np(1-p) , may be of use i n o p t i m i z i n g t h e l a s t s t e p ( c o u n t i n g ) of t h e compound p r o c e s s , b u t one must be aware of t h e p o s s i b i l i t y o f c o r r e l a t i o n i f s e q u e n t i a l c o u n t i n g t i m e s are invo lved .

L. A. C u r r i e , "The L i m i t of P r e c i s i o n i n Nuc lea r and A n a l y t i c a l Chemis t ry" , s u b m i t t e d t o Nucl. I n s t . Meth.

F. A. Lundgren and S. S. Nargo lwa l l a , Anal. Chem. - 40 672 (1968) .

One o f t h e few r e p o r t e d cases where t h e l i m i t a t i o n s o f u s i n g fi as a scale f a c t o r h a s been r ecogn ized is i n t h e c o n t i n u i n g review of Fundamental P a r t i c l e P r o p e r t i e s by t h e P a r t i c l e D a t a Group (A. Barbaro-Galtieri , S. E. Derenzo, L. R. P r i c e , A. R i t t e n b e r t , A. H. Rosenfe ld , N. Barash- Schmidt , C. Bricman, M. ROOS, P. Soding and C. G. Wohl, Rev. Mod. Phys. 42 87 ( 1 9 7 0 ) ) . Even h e r e , however, it would seem l i k e l y t h a t t h e a d d i t i o n a l v a r i a n c e component approach might l e a d t o a more reliable and more c o n s e r v a t i v e estimate f o r t h e s t a n d a r d error as w e l l as a more p r e c i s e estimate f o r t h e mean of t h e p a r t i c l e p r o p e r t y i n ques- t i o n .

D. R. Cox and P. A. W. L e w i s , The S t a t i s t i c a l Ana lys i s of Series of Even t s , Chapt . 6 , Wiley (1966) .

W. G. Cochran, Ann. Math. S t a t . 23 315 (1952) .

d . f . is h e r e g i v e n f o r t h e case where x is de te rmined from t h e o b s e r v a t i o n s . If a known mean i s assumed, d . f . f o r t h e t w o tests become n , and c-1, r e s p e c t i v e l y . The "miss ing" d . f . i s due t o t h e s i n g l e l i n e a r c o n s t r a i n t , Cf. = n, whereas t h e r e are no c o n s t r a i n t s among t h e

-

Xi. I

Rece ived October 1 3 , 1971

Accepted October 18 , 1971

882

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

0:40

15

Nov

embe

r 20

14