NATIONAL BUREAU OF STANDARDS REPORT

NBS Project NBS Report

21 January

U03-40-4U40 3913

ammfszs of mmm ms& mmsn mm b, r, a&seb

Hajpy 6c Xatrolla

Statistical &8gi®®®zrlng Lateoroioffg

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

ation, reprinting, or re* bited unless permission in

>nal Bureau of Standards,

This report is issu Approved for public release by the

production in any f Director of the National Institute of

writing is obtained „

Washington 25. D. c

Standards and Technology (NIST) on October 9, 2015
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21 ^January 1955

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Ana lysis ©f Pebble Bata Beceived from H. F. A3Lbe@

L Introduction and §maar^ «kkj3^.a'

. ?3--Tr^ngrt. 'i' ,j^,-aTi,. -jj-.-m-.:; e-y\..r:L^-rx.:^i:,j-l sry^acs-y.-.r.’.^-s-ri —n—

j

y The data consisted of 2 samples ©f ISO pebbles each*

classified by lithology* from each of 6 sites (E&-129*

EA-132* BA-12®.* M-134* 1& -87-T* and HA-91-T)c At te© of

the sitee* IiA-87-T and BA~81~T* as additional sample of

30© pebbles had bee® talsen. A X^ test was applied to the

2 samples at each ®i&@* to test whether or not the samples

differed sore than might be expected on the basis @f ran- d©a sampling fluctuations. The ©©apufaf i@ao of %s are

attached in Tables 1 through 6. Tables of g2. @ avail- * able in saany statistical tessts* and sou© of these tables fcav© been composited in y©f B [ij„

The value of X* c©®puted for a site is ©©spared with

the tabled value for some selected level of significance* usually Probability ^ 0 €>5* and for the proper * 5 degre©s of

freedom'^ If the computed fr©m the samples exceeds the tabled critical value* it may foe concluded that the samples

differ ©or© than sight fc© expected in random sampling.

The results obtained for these 6 sites indioat© that*

©a the whole* the 2 samples from a sit© &r© homogeneous and.

therefore that the sampling tectaaiqu© is a satisfactory one.

2 a Beta lied Results of Analysis

The following valuer ©f X

s as*® obtained in comparing

the 2 samples at the various sites?

Site BA-129

BA-132

HA-120

HA-134

BA-91-T

BA-91 -T

BA-87-31

For sites BA-129* HA-132* HA-12®, and HA-134* X 3 does net exee©d X*Qg therefore the 2 samples, d© not differ excessively

<sn© from the other.

X

s X

s X*

X®

X

s Cfor comparison of

X

s - 3.287

Cf®r comparison of

X

2 -10.685

1 0 2S7 (3 degrees of freedom)

(S o 366 »« 09 95 3 O ;08S to 09 99 1 .442 ft 09 P9 0.096 to 00 90 2 samples 'of 150)

3 d.f.

2 samples of 300)

3 d.f.
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a4
-2 -

i

"1

At HA-91-T* the comparison of 2 samples of ISO yields

a 5C * 0c096,y which is so small ({the probability of a la^r*?

valas© of X s exceeds o99T

=sfSITTt might lead to a suspicion'

.aa s these 2 samples were not random. Sa other words* the e samples

to a agree tooweUp which my be due to unusual luck m defect in^Iesalpling technique. Since there wa*3 an additional sample off 300 available for BA-91-T* the 2 sampleoff 150 were pooled for comparison with it* and the result tar'' %

s ((*3, 287) is not significant and shows good agreement between the 2 samples off 300,,

At HA-87-T* the samples off 150 show larger differences than would be expected,, the large value of i

8 H0»685^ apparently soiaes from the relatively large discrepancy in "other” classification,, In view off the non-komog^neity

these samples were not pooled for comparison with the aam^I

.

off 300 „ Iff the comparison off samples off 300 could be however9 it would undoubtedly indicate lack off homogeft % due to the quarts class,,

For HA—91—T and IgA—87—T* the small classes have been merged into on© "other” classification. This was done not only to achieve uniformity with the other 4 sites* but because

off certain theoretical difficulties in applying the %* test

to classes with very small count® £less than 10 as a minimum'

3o Short-cut Methods of Computing %

a

3a,

The usual definition of originally given by Karl

Pearson [2|, is*

X

s {observed frequency - expected frequent*

e*pec£e3 *requeW~ ' ~ '

3b,

A ©impliflod formula 9 given by R0 A. Fisher [3j t

. for

X^5 in comparing 2 observed frequency distributions Am*
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' ' .."I.; - . £* , . \

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aia&I

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1 *

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•

.

.. Imttml&kes & aa Of ssjiuS$ ' "Z /

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-3-

^•boye the. Claes frequencies are £* and £„ respectively for the

ith class* H aad W

9 sr@ the total founts 2 in ©ash sample* sad

s. is tho number of. classes* The degrees of freedom for

t3 8 » L

3e»

For two observgd distributions with egnal tota l eoenta

(as in f "T^mBi

a

"in'" iFlafjwe ea!^^'is&es“¥o

g

dofo ® s • 1

Sa other word®* CD for ©ash ©lass* take the difference b©~

t*?e@n the 2 counts* square this difference and divide by the

av.s of the .2 ©ou&ts C22> Susa these quantities over all ©lasses.,

This farm of computation is somewhat simpler than the usual

one* and has the advantage that it doe© sot require taking

deviation* from the .average count and therefore eliminates

resulting questions ss to how many decisal places to carry 0 3do

For three observed distributions with equal total counts * there Is^a iTBifaF== sS3rt-©n€T'^

=TrF^'oy' 'eaca cXail* take a I if

3 possible differences among the counts $ take the sum ©£ the

squares of the -3 differences and divide by the sum ©f the 3

counts (2$ Sum over all classes,,

Mary ©0 Ha trel la

Statistical Engineer ing Laboratory

Mational Bureau of Standards

21 January 1955

REFIRMCSS utfU1 :r-i‘xrr- jr/r:^ -.v -

I IS Rational Bureau of Standards Report* MIL SI -2 ”S©ue

Percentage Points of the %

3 ©istribution”* August 195© 0 [21 Pearson* w0a a Crit©rf©sa that a Given System of Deviations from the Probable in the Case ©f a Correlated System

of Variables is Such that . It Can 1# Reasonably Supposed to

Have Arisen from Random Sampling9 '* Philo SIsgo Series V* 5© P IS©©* 157~1?§ 0 t3j R« A 0 Fisher* "On the Interpretation of %

3 from Contingency Tablee sad the Calculation of P'\ ^ ft R,S.S« @5* 1922

„

87=»94 „

-
vl

. ;v;-..:Q:y V.;/; :7i«.S r£i &£& *s

. -

: f

.. i®* *x

:/r £o£?ittftiqaG-.'j> 'i: . £-w»fc fciiiST

' ' * I . $ 4l?fW5 S) iffl «

'. 88 5

’

' k 5

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• i|

:

' slv

. " &£},

.

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1 s 8 . sasdf St

(l> I £ S£ ©*©&£

, ,... -

v

;/' '-'v i'3> ? r : ^r:’-p "?r7- . . cV£

,'k :3ilS £14 .

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.

If

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I :$$ l& iviua*i

S . .. it ' 7; 20 .‘ :: 7

* -. rn - I f ..

l . g Id $ mi

, : '. ,' ... /'. >£ cIs^S s?i: *c

• , !•! ^7;\ £ v^::3 i. r.$ts5*.$A .c^SlS

\eaes

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.

'•' I© . <Mf ' 8&1S »

' i i • 1

•

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• Q~f$

ri
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.^sa/ Utah

gsi'iaples H@o SA-129~A

M-12S--0

Class :

Fre~;^

SSsfsple

*“

A.-

snciy

-.srq> ,* s&aple

. B_

• <Ua —t>*

j

'A«?S‘ A*®*.';*'*

•^assVt'alt® 42 Q1W <Sa ft . ‘1

- Oe.SuS

s 11 10 21 s r •; :

- :

€Es©s:$ • 78 82 160 ”4. • 0 1: 4

1 • •

12 10. St2?

f©m . > & w 150 'ci- -^r X ® ;’:•

• Cl 1.257 a Jttejjwt r »
4 1

i
8o 0 Bridges’

Elk Ridge. Area p ptah

Samples H©o B&432-A

HA-132-B

Lithology

Slss©

Fr®<5M©©oy

- Saapl® Saaple

A B A«0© A-B

CA~F5 S

Qssss’tsit® 30 24 S4 6 0*667

Qss&s’ts 13 5 IS 8 3 * 55S

CSsert 104 120 224 -16 1*143

©thOlT 3 1 4 2 1.000

X

s " TOTAL 150 150 30© © 6 o 306

d«f *

<
5L52&SS

ivOvl

.> c^*K£cs»?3^5c=.' 3 m h% ' os

8.' si ...

&*.-f r ^ - 6* O <u r©$|* &»£

-QQ8.Jt & i

' S -

'

«fc 0 o&i

a

*

4*

<

t
'js'AISLS 3

Gre®n Biver B@$©rt Areap mats

Samples HOo HA-12©-A

EA-12©~B

Lithology

Clan©

Freoueaey

Sample §a»pl@

A B A'OB A-B

CA-Bp

•rarQuartsit© 75 so 135 IS 1,88?

Quarts 16 IS SS -8 Q o 2 © 0

?

Gfe©rt 49 S3 112 -14 Io750

©tfe»E» W 8 18 0 o 222

x® - TOTAL IS© IS© BOO © 3, SSS BB8C88 i ~i?i 3BPP»

Samples Eo, HA-134 -A

HA-134 -B

TABLE 4

The Motefe

Elk Risige Aro&j, Htah

Frequency

Lithology

Class

Sample

A

Sample

1 B A4B A«B

CA-8)*„T'

Quarts!te SO 69 129 -9 0,628

Q^&rtss 32 26 58 6 0 o 621

Ck<§rt 46 45 91 1 0*011

Other 12 10 22 2 0„ 182

X® " WS:Ah 160 180 300 0 1 442

do# 0

*•
* d
Buckhorn Draw

Effler y Co *

fits h

TABLE 5©

H&-91-T

g£ I «s0 I*

Fr@gw@n@y . Lithology

Class

BaaplV

Bi

“llarapl®

®2 B

1

4

' B

2

Quartette 03 94 3.87

Quarts IS 20 39

Chert 32 SO 62

Silt 1. se .-. ""1

Limestoss®!

s\ w Clayet©a@|

Saadst@sis|

TOTAL 150 150 300

2

-1 0 o 0C5

O..026

0 o 0Ss

0 0 .OSS

4

TABLE 51?

Eteskhor® ©raw

Ekery COo,p Htsh

EA“9A“T

(Samples ©f 300}

Fg®QM®ncy

Lithology

Class

Sample

A

Sa®pl©

B A=o=B A-B

CA-B} 2

~1&W~

Qiasrtsit© im 18?

\

* 37§ 1 ©.©03

Quarts? 4S 39 87 9 0,931

Ch«rt 48 62 11® -14 1,782

Silo ls,“ 11-? 121

Lim@©tosB©

Clsy@toa©

2

> "Other”

1

> 16 > 12 28 4 0o§?l

Sandgtoia©^ 2. «a

•• X

s - TOTAL 300 30© em 0 3 287

dJ £ * 3
TS

" \.
TABLE 6

Arizona

ha-st-t

of ISOD

Frequency

Sasipl© Sample

B

1

B, B^=s=@g ®

l “®2

«b

1-b2

)

; ’WST

Quarts ite 92 112 204 -20

. . -r.ri- . • :-«=*= 1*961

Quarts U 17 34 0 0

Chert 25 17 42 8 l 0 S24

Cl&y&lom®

*>a 8

«

1

•

4 2

Sandstone >”0th©r'9 g > 16 > 4 20 S2 7,2®0

Sil, 1®., 1 1

Limestone^ 1- cs %

s - TOTAL 150 IS® 300 0 10,685

d,f