AD-769 753

A FAMILY OF DISTRIBUTIONS TO DESCRIBEA MULTIPIOJECTILE ROUND

David Kennedy Hugus

Naval Postgraduate SchoolMonterey, California

September 1973

DISTRIBUTED BY:

National Technical Information ServiceU. S. DEPARTMENT OF COMMERCE5285 Port Royal Road, Springfield Va. 22151

StCsm REPORtCTn 4 P1101AENTA Mi7s POn G _0PJvfl /*1

FMOAT UU21-R GOV- ACCE$E10n !4L' 3. Ps-cila-rst ALG nt

34.~~~ TYME OFAEPORT & PEMO C0 -EAFnmilv of Distribution tttfescril zare. Texs

a tiuti~rojiectile RoE

wA~rd K. R-ucus PC~TATi EY MEN

2.Ppc-iGOItNIZA7TOn %"AE "DMO &ZZES W.UMBSASl_

Naval Po-stgaut Sco..-t

NvlPostg-raduate School [ o11A 17IMonterey California 93940[1.IIBRLASj

It. gCTrM* fl" AHGOFFICE NAME &N AOESUSmtf~Cmrlh OI t2. IL ECURITY CASS N e

Naval Postgraduate SchoolMionterey, California 93940 1e~qlaTI-tNR~GI~ _

W . WSTfIU~lO*C STATEMENT of V-14 SCSIEOJL

Approved for Punlic Rel:ease; Distribution Unliamited.

Mt WMtlrJ~T!OM StAThSEIT fta fte efrfrt Onfa ft Bt W. U d-Wurs'f LOn Ra.Pj&

S& SUPnEUETAVO--F

INFO2RMATION SERVICE

M2 61 .45 caliber- zistold ens ity s ho0t c-hi-squarLe goodness-of-f-itpolar comordanatesgamma31 distrih utio rbivarinate orm± dastri-bUt] _________

T hl i s tPaer is an analci- of th belhavlor -dp t shot-IGQ-6I1 winen fired fron the .45 - aliber itl(u9± 1 . estshot ammiinJion is a lt~';cieroulnd whic', v--nta nm 16 -small p-lsletS. Bo0th the tanard n. fleA barm' eJ

ro rA A 'a-n-flyV of- aast'ribrtf. i-ns selzztea r ac r arzetyretocC e tnB h atte o. fr:--1 essirom theXN3421G n atva~r -.

-ffs~~ -.3

------~m -- I- - -

Ri - - -Z

A Famly- Of Dcstrirr ___ '-orato Describe a utptjctl Ru nd

by

David Kennedy HugusCaptain, Uited States Am

3.5., Pr-,due Uiversity, 1-965

S7ubmittedq in orar nJal t _

reqa zeni for" F Z4i~e ofth

MASTER OF SCIErICE' TM OPERATIONSJ RESEARCU

Naval Postgradnate Schc-'o3Septemt~r 1973

Author _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Approval 'by: t c; P;?Jrnezis Advisor

~'flSecond Reader

7-7

/ Acatiejc Dear.

J-rF-w r __

4~ ~ -~p - - wp-

This paperv is an analvsis of the behavior of density

shot 9iH261 when I fire d from th-e .45 caliber pistci (Ml9l.1

Densitv shot. aiinunznton is a railiproiected roCund which

contains 16 sna-1l pellets. Both th&e standard rfe hrp

F a ~and a smocoth barrel were used to co-.lect data on the behavio~r

of the M1261 round. A tally of disribtins- is selected-

for each barrel type toC- model th*e scatter of pellets fromW

the 3M4261 round at wzrin ranges.

p

Sa

-~ n- -1

QN

TABE OF CONTSU--T

I N ITR D aION ---- -- -

"i. DATA COL*aCTIN .------- ------------- 8

III. SELCTI OF DISTPaBUT PBi O------------------

IV. ANALYSIS OF RIFLED BARREL DATA ---..---.- -- 14

A. INITITAL UT- ............... 1

A. FJBL RES ULTS----------- --------- 23

V. ANALY.1SIS Op SMOOrh BARLDT---- - -- I

Vt. PSC MVn.ME---------- --------- 3 2

APPSIDIX A List of Data .

:.PfI-DfX B Pazameter Ltimat ion by the Method ofM ents---- --. . . .. ..--------------- 43

APPENDIX C Center of zlass as a Function of Range-- -45

APPMD7X D ChAnge ii Distributien Due to Centering--48

APFNDIX S Simulation to Determine the CriticalValues for the Goodness-of-Fit Test- 51

ATPESDIX 7 Variation of Lwdda as a Fun0tionof Range --------........ .... . ---- -5-

LIST OF RFERENCES----------------..57

INITIAL DISIBU"IM ST ..--. --------------------------- 58

S- -

100M DI-1473 ------------------------------------------

3So4

tLIST OF II~~~~~~~~~ -c~i~o 4s jj~~ RL----------- ---- 9a I1. GENWIEJ OF 4ASS-IIX 3~--.~ -- 19

III SZ&LZPERCEINTIL-S FROM 7h SflliON~. .54

-tE

I.Partition. of the Plane f or -the Polar fisitribution-- -2

-=2. Partition of the Plane for the Bivariate 4.

51

ANNI=

- ~ ~ ~ ~ ~ ~ ~ ~ _V - te=- nC--=- , =~---~-~ 7-~=_

investiaatinq three different types of av=-nition for es

with the .45 calIiber pistol (M1A The three tves of1

a~nitIon beinzm consider-ed are te- standardhl W mI -rm ition,

a sgwetedprojiectile which bre-as into sixpee fe

leaving the barrel, and t-%e- aensitv zo X1261 whidch containcs

'16 small pa1liets. USMsSASA is also invuestigatina the ef fects

t-Nat a moth fra bae and dt tairr ued--'h

on each type of amni tin. tASASA is trvy na to deterulmne

if greater effectven-ess car b achieved with the 45S nSlst-

by chanzing the combination ort asm tion and narrel. This

paper will deal only withay theu. density shot a-4-10nt-Ir-. H. P.

White Laboratory, Beel A-ir, ma- land, was coztracted to

provide veloci tv and accura-- data forv the eval-iation of the

six ccmbinationts of' amuni tion and barrel types. Inp Fetr-rvz

1573. the laboratory w-Us a ~LnaJ I reor ictrtaiIdiflc the

data required [ Ief a.

rhe ~r po se of ti s Paper is t^ oc -men- an a t-al--s. of

that sgntof the data nrov d byie . P. Whi e !A-

tha o-_m h K16- m -- = ' .J -= -t as -i=-='

taken at three r-.j's 110 -, ao Ui.mters. -,CNNever. K

t ap-, vot anre Zu ter -- at --t

data did not .eitsel to anallwsis at rt ano - 'rc

T .y~v the analysis- wa_% condncta w ta h r. four o=Ao

6A

______ fi--=LL~-- - -=--- __ --

pea lt -ad-u h o C ---- J dismito--:Uto --o db

ud ffa -- cas siv4 3Qato to rdeem.= the r =

ives s~1 o tE2SAS ASowieaste us; teji aWrca

sh11ee abo t te 40' - Ti thi stxibutrincoii

use di fwr exa I e r== r-iseft usearion toof data

w _i - t Of a COMter Sim =3 M

Canfe ctive t o mcde t:-- t-ca=l-t± tim;=j~ e3- I e4

Ofso-.- ofar Beasn' sh pooeteduseiof thef ditribtions

in ~ther s gu1latian ofAS -i--= tha theotr

the dP--qr-tni darl-u'te to ano aas -O trat

raif ec analic i~tt ied- a-o ati e q ddsts

can n d toe adngle r sttr- ar The

utiliz er -- te sthe :2da t=vt-_ - ,escria d ::; ff the

~e aq~e -5 aszrsu!.eAS

fal ecf =- fw -z t~ .- ne tmat

4IX9 Wve 101ile ex41- 4-cN t30ad1 t- e -- M - -c= t__ I

S tXS' mse r tait- Am th ieoSta1: - a t iets S 1--rC

mzl t -cc -j r- 15 f-- r= -the wwl

czaV off mr1timnzn-1tec le r-- =b WEechpc

jecti- on eac of n tzx - I trt ISG -

a~erci wane rccn~zt in r~Z. -4a = - ~-~

t7=u thre -- r - usd- St=-' -tr mtWta -0=CM

Lav-S fket 5-0 w a elizrs thi stme--a-

co" taIh zx~mfcier-zejr A-amml onflJ

v -1 ramsme

l -ler rm w 4 a smn-

- di At~;tr! -7- kit Z, o= -t 7 A

R~a = -

III. SELECTION OF DISTRIBUTIONS

For the present analysis, the basis for selecting

distributions to model the scatter of pellet strikes from

the XM261 was a letter from the United States Army Material

Systems Analysis Agency (USAMSAA) (Ref. 2]. USAMSAA was

replying to a request from USASASA to analyze data obtained

from firing the XM261 round. The data analyzed by USAMSAA

was prepared by the Frankford Arsenal. As a result of their

analysis, USAMSAA suggested that the distribution of pellet

strikes from the XM261 round when fired from a rifled barrel

could be modeled by using a polar coordinate system with the

angle havir.g a uniform distribution, the radius vector having

a normal distribution, and the angle and radius vector being

independent. (The model based on these assumptions will

hereafter be called the Polar Model.) When the XM26i round

was fired from a smooth bore barrel, USAMSAA suggested that a

bivariate normal distribution with independent X and Y would

serve as an adequate model. This student felt that these two

models could serve as an adequate basis for an initial investi-

gation of the XM261 data provided by H. P. White Laboratory.

The plane had to be partitioned into regions so that a

chi-square goodness-of-fit test could be performed. The

objective for the partition was to allow the( maximum nurabar of

regions consistent with the requirement to have an expected

frequency of observation of pellets of approximately five

[Ref. 3 and 4]. In connection with testing the fit of the

10

-~~~~~ ~~ 'A-- - lW---=-~~

Polar Model, the plane was originally divided into 30 nearly

equally probable regions. Since the angle was asvumed to

have a uniform distribution, the plane was first divided into

=six regions by constructing six rays from the origin at 60

degree intervals. Then four circles were superimposed over

the rays to further divide the plane. Tha radii of the A

circles corresponded to the 20, 40, 60, and 80 percentiles

of the fitted distribution of the radius v*ector. This

resulted in a parti tion of the plane into 30 nearly equally

probable regions (see Figure 1), Since there were 160 data

points, the expected frequency for each of tb: regions was

5.33.

Wher tiesting the fit of the b --riate normal distribution,

the plane was divided into 36 rectangles. The mean and 1

standard deviation of the X variable were estimated usini

the maximum likelihood estimates. Then values for the 16.67,

33.33, 50r 66.67, and 83.33 percentiles of the X variable

were estimated using the estimates for the mean and standard

deviation. A vertical line was constructed on t/he X value

which corresponded to each ot these estimated percentiles.

Then using the same method horizontal lines were constructed

on the same sample percentiie. of the Y variable. The

resulting partition prrvided 36 nearly equally probable

rectangles (see Figure 2). The expected frequency based on

the 160 data points was 4.45. Although this was less than

the desired frequency of five, it was sufficient lRef.3).

1a 1A

I II I - -T* p-A rn w

'15

I 10ra 2

U7

FU I

I1

PA TiTi _- __F T-I PL , ' F-', --r:,,.

T I I t P!iI s i

54- 35 36

!9 20 21 22 23 24

1314 !516 17 18 ;

7 8 9I0 !i 2 -

F I

F- 3

rQW

IV. AN.LYSIS OF DATA FROM THE RIFLED BARREL

I- A. INITIAL " tSU-S

When fitting the bitaria t norm dist±ibution, four

paramzeters, the mean and standard deviation of both the

X and Y variables, had to be estimated. Since the X and Y

variables were assimed to be independent, there was no need

to estimate a correlation coefficient. Because there were

36 regions and four parameters estimated, this chi-square

test had 31 degrees of freedom. This yielded a critical

value of 45.0 for a fite percent level of significance and

a critical value of 41.5 for a ten perc2n Thvel of signifi-

cance. The test statistic had a value of 95.6 for the 10

meter data and a value of 93.8 for the 30 meter data.

ience, the hypothesis t-hat either set of data was distributed

according to a bivarate normal was rejected.

When fitting the Polar Model, only two par.eters, the

mean and standard deviation of the distribution of the

radius vector, were estimated. Because the Polar Model had

30 regions and two parameters were estimated, this chi-square

statistic had 27 degrees of freedom. The critical values for

five L-id ten percent levels of significance are 40.1 and

36.7, respectively. The test statistic for the 10 meter data

was 55.25, that for the 30 meter data was also 55.25. This

led to the rejection of the hypothesis that either set of AV

data was distributed according to the Polar Model.

al14A

Since both dLstributiuns had been reiectad as methods

for adequately describing the data, an additional distr ibu'-n

-war soucht. A histogram of the values for the radius vector

suggestod that the radius vector might be distri!huted -

according to a gam-ma distribution. Therefore, an additional

distribution was introduced. This distribution utilized the

polax coordinate description of the data withthe asmptions

that the. angle was uniformly distribut d between zero and

two pi, the radius vector was distributed according to a _

gamma distribution, and the angle and the radius vector

were independent. This distribution will be referred to as

the Polar Gama distribution. The plane was partitioned for

the chi-square goodness-of-fit test in the same way as for

the Polar Model. The values of the radii of the cireles

corresponded to the appiopriate percentiles of the gamma-

distribution of the radius vector.

If a gamma distribution was to be used to represent the .

dis-tribution of the radius vector-, the parameters would

have to be estim-ated from the data. The maximum liklihood

method of estimating the parameters has some appeal, but

it is cin barsome because it involves the gamma function.

Therefore, it was decided to use the method of moments to

estimate the gamma parameters. This decision was made in

spite of the fact that cAe method of moments produced a

biased estimate of the parameters. Appendix B contains the

r~ method of moments estimates and a justification that they

are biased. Even after the para:eters were estimatee some

J.15

SA

- -+

a

difficultr is experie.nced in estt:atinq the 20. 40. 60, Z

and 80 percntiles of a aeneral cam- dia stribution to use a

values for radii of the circles se-arating the different __

regions it the plane. However, if Che shape narameter

alpha is a positive integer, then a convenient relationship

exists between the gara and the Poisson distributions. I

Specifically, if X is distributed gai-na !G1,) and Y is

distributed Poisson { X ) then P(x<-_)=P (Y>X) [Ref. 5].

Therefore, the percentile values for the ganm-a distribution -

with an integer alpha value can be interpolated fran' a tableI

of cumulative values for the Poisson distribution.

One of the considerations for this nroblem was that

the distribution used to describe the data should lend

itself to easy generation by computer si-ulatiun. Hence,

it was decided that after the gaw-na para-reters were estimated iA

using the method of moments, the integers on either side of

the estimated alpha value would be useti inztead of the

estimated alpha value. This approach would permit the use

of the relationship between the gaam and the Poisson f Idistributions, Each data set would then yield tvo chi-snTaare

test statistics for each set of qamma parameters estimated:

One test would use the integer value anmediatelv below the j

estimated alpha value, the second would use the integer A

value immediately above the estimated alnha value. Both 3

tests would use the estimated lambd-a value. Z

Since fitting the polar gamma also requires the estimation

of two parameters and the plane had been partitioned into 30

-a6f

_ z- -- - - -- - -_- - ---_--_ - x w - - . - -

regions, the chi-quare goodhness-of-fit test fo- the Polar

C-amma had 27 degrees of freedom. Consecuently. the critical A

values for the test statistic were the a am as for the Polar

Mcdel. The values for the two test statisrcs generated

using the 10 meter data with kat rifled barrel were 62.0 and

4.l. TIhose for the 30 meter data with the rilEd barrel

were 58.6 and 52.6. Therefore, both of the null hvnotheses

that the two sets oct data were distributed according to the

Polar Gam-ma distribution were rejected. Although the

bivariate normal, Polar, and Polar Gamicwa distributions bad

been rejected as models for te data, the Polar Gamma

distribution had produced the lowest value for the test

statistic for both the 10 and 30 meter data. Tis indicated

that the Polar Gamma distribution might have sone potential i

t , flescribe the data.

Be-ause all the distributions had failed to produce

an accentable description of the data, a check was made

to see if the data contained anv identifiable variation

that had not been accountedfor. The physical character- =

istics of the method of data collection led to the belief

that such a variation might exist. The recoil and rifling 3

of the barrel were suspected of causina he center of mass 2

(sample mean) of a pellet pattern to drift up and to the

left as the range increased. The recoil was suspected of 0

causing the drift up. The mceMent -& the barrel of one

half of one degree is sufficient to cause th center of mass

to move more than 10 inches at 30 meters. Te riflina was

17

___~~~. iT -- ==- =

susnected 0 causing the drift to the left. Since the

left handed rifllna produces a cou'nterclockwise rotation

4Lof the projectilker t4LtJLJC=-i rwJzla -0 -5tto~

left whce. meeting air resistance. if the center of mass did

in fact drift fro. the adm noint and if the drift was ran-dom,

then tis variation in the location of the centers of mass

would introduce an uncontrolled variation ang the rounds.

This would be considered undesirable since the chi-sqruare

goodness-of-fit test assumes that all observations ishots)

come from the same distribution.

In order to determine whether the =ata would tend to

support or refute the hypothesis that i-he centers of mass

did drift as range increased, the center of mass of each

shot was calculated from the data at ranges of 10 and 30

meters. Each center of mass was based -n the 16 pellets

in one XC2ol r.and. An examination of the centers of mass

calculated in this fashion tended to support the hypothesis

that the center of mass drifted up and t'o the left as

range increased, For both the 10 and 30 meter data eight

out of ten I cordinates of the centers of mass were negativeand nine out of ten Y coordinates were positive.Fuand ineoutof en cordiate wee iie Fuarthermore,

a comparison of the centers of mass at 10 and 30 meters

for the sar -ound revealed that eight out of ten centers

of mass had drifted up and- to the left. it was also noted

that the actua! distance that the center of mnass moved

varied siuaificantly a-ng the rours. The calculated Cen-

ters of mass are pro_ided in table 1.

1 iI34

_ - - _ . -ST . . . .____ - - - -W-

TABLE I

Rou. 10 Meters 30 Meters1 (0.25,1.!4) (-0.46,2.25)

2 (-0 .10,Q -1S1 (-1.27,2.80)-t':e

3 (-0.41,2.88) (0.50,8.35)

4(0C,1.3&) (-3.03,2.83)55 (-l. 9 2 ,-1.3l) 1-3.76,-4.30) --

S (-0.43,2;33) (-8.43,3.92)

7 (-0.85,0.64) 1.45,2.15)

8 (-0.53,1L.48) (t9,.l

(i-2fl0,2.27) (-5.37,2.58)

10(a1.29,0.05) (-4.8 20 C33'

OverNall Mean (-0.52,1.161 (-2.82,2.36)

$ I

-- - --

Inaenl orcer te1 o'- ifmir

m the centers of mass, t~h c~ao .a .- s fo eah rl-- of

th A xa-dY coordinates &-1- that shot'2 cen- of s fram

the X and YF co-rd'-tates of each pellet0- inm- thaG-t grouPP. Ti

Ct~i ~ - - -EL ~~ ~ ~.-ae -- v a

center- of: iass at the oriuin.

A.inte-cesting pheomno miv occur wnien each- gcny- of

J4 '.et has a center ofl -mass at te ori-in. TV-e data may

rappear to hav.e an are-a orf relatively low pronhjai.Li;%t-y at the

orig in (i.e., a 'hole' may appear in -het data) . This may' be

Ftrue evenx though the original data may not have exhibiteua

this trait. A possible explanation for the 'no0-le' appearaing

Is t-hat when the centers of mass are not constrained to be

at the %erigin, the -area or ;L io pr-obability %r one sht ay

lie in an area of high probabiliJZty% for a different sh4ot.

The overlap o-f the differentVI patterns fir-r tra different

rokir my Prevent am 'ho1le' from being wbsern4-r. Hmoever,

when the centers org mass are all a'- the origtn,. the areas

of ilow prcbab'Ulty may all coincide to nnodue aoe'in

thedata.

Another phencaenon may oroduc a 'ho]0' in the data even

when th-e dist-ribution of the data does nolt ehibit an area

at low prc-babilitv. For- examrpe, fr one =tew paikrs of

numbers ft= a urd-form d-Listrilion an n-1o+tsA the i-33--tn'e

of each point fro the center of ma s of its par of po-i nts,

then he would obsFerve an ara of low pret- ailiy at the

-- - -

- - - -, ~ = - ~ - __ _ ___ __7

-OR Cal, a

r-e art F-aue a~ll m- ~ te '- --i iimnts ea- sin-m

st salv for iltsinnc isthics Prr-c-m- t -ntr O fleS adt

flt-IL IS occurringd tht zzp to- atth

'Ut-rke a - diefre orn tz cente ora f =- tiemsno

Alh ro- nc -ey odificatio of tne dat subsd convenent

toabif. .sana~sswvs4 e.lettsnec4 thi ro elre antouce thee

cener f m-, b- wold-o-es h -F cth -e~e

of mass kfcam &'-e aem. ponrsr Mit caiC r--j the cet __o as

Jv'-4 utlin towb cretr odti ed to mare th efmaiect vss ohe

aciofthes roun. rr-' *b- rbzi dosr1 iht P-=--,C

arrnlysis estiate o~flec the sffctte of thes abounth

of mas fljro the accnt Uon. h drift of teCenter of ss

worl navuoe to be rentoue an aCs.altooh

actins ofl thi-s tfLsh'nd Fave t doe so migtit. Tproduce

ero EcnS irtes of-c the Ce -~e ofms- a:--ta h -

Pro cdre to, a- otfrthe driftD--- on!cha-f thes '-ene of msast

arlxteo-noet in Aredj- PC - eaidlfenth

whenomm al is of shots abova. ther sa-m orin measurng

may t 4oft-gtre di allina cohaactertc of f mae scate

ij~21

-t f~i rc ntt zr za 1::ie Value Zra

L; sa ste --n - a setea

a nellet wi-ll strike vditin the sam distance of -t'he -e~nter

of mas- See axni =l M a rate ics o ftis

taIC t Since messur ing amte -S5e jaS does n~f

the distribution of t1 data~e a bivari ae norma± -_

disri uti n4s used, one- w-%- a susnset tr=- a e n a

S___ ri-=4, .. LOLOS .1.c-L

phenonono micht occu-.r wttn 0hrr "ons t-i t' tcp

en t e rin g th t e Pellets at th1-er -oespecti.ve etrof mass

may appear to increa~a Ue urobabiit-wr oEf sitrikin 0 Cse

to the origapn. mI

The~n fia prbempoduceci nv modifin the data '--nerfls

the degree to chich the data conf orms to the asl szti ens

recuirec for the -hsur oocs-ness-of-f it tl;est aferi

has been odlified. Even if the data b-ad been ix--enendent -

'A

iits orliinal form, it is inoortant to note- that after the

viata has been nmdified, the coordi-na tesoth es i-- One

shot are no lan1cer :ndevendenut. This is so because al I '!7

coor'dinates of the pellet Iocationc denapend on the coorda-tC5

of the Cen-r of mass. Since thue cez-ter Of _mss is a

function of althe data poilts- in- oe c-o'n al-I of the

pellet -crLd.ates in a singde sihot a-re statist canv

2~.

e v. -- ft

t_ te .ia 12e= crU t4th sa13as 7bleassa- - as the best c to

7. aa s c- -el -AU - r = Z' - _-~ a-

Set O l-d- 1y- -l t byac-eFac st o dta -a sujetm tot=e- ssOf a:- -

p__- o e th - -4. 1 -1-mitt tierattis-i m Sbr - h

or-rat test s-ta are odred fr= os-nzIlzc toIgLOJ.

tesoivesa ale was scetlete as the test: rtanistcero

-rnlt I Ae alue of Vsne 900 =~n 95 -tst stmsc wer

Sach se ofe data.e w y he cte t ci-i-cdbu-ines v-of fIev

pxa-Bcae offv r-=z- tn tetsttsicwste. Te

'dsvera woamyne nea~iles ofbm s statistic war

prted h e vrorates ofnte 9of mass nd 950- a- sietzss of th

used s+_rb t ioe r oa vales tolli te-- th e ctests-imte Us -

da.t e 'LOd and ~-0eter -vt tetdsriuinwth2lo

-- gmate d~-'iaere used e ve-n at-wsto warna ostaat f-or

Z3

- g

IM met t-k .-L-s- -M-4 C-i Dcx t e pi:

E, - 4C-

sirtv-- the~ v -l les - h

ati= _ wra-- 3 a=Q44- f-c- I&A= .O i d ~th - -ai andth

A smia etr emr wc- n. t- 31 - e aa a

~~zua1t aa equal- Si5c andtiS -~

the~~~ Pola r3=:= iitn e1 R+=-

tenua to 0.-7 rened a:t- sto-nr-i of -3

troa £o -1.r 7 G M est conEsti o' 2be 12ted -tas-5

AuMctv scflrnatepr trg- whereun ZOM- 3CRe dt

distribution wa!s;ch has4 ?hzt Ror ethe Po-Iar the wasi 3P.

the:n- Polar_ Gcedsriuteion wih-~h- 10b eMi 15oth

PCI-p e e to M

I

distribution. For the 30 meter data the Polar Gamma distri-

bution with alpha parameter equal to 16 would be preferred

above all other distributions. However, when choosing

between the t-wo Polar Gamma distribution for the 10 meter

data, an additional consideration should be included. It

seemed reasonable that the distribution selected to model

the scatter of pellets at 10 mtzters should resemble the 30

meter distribution as closely as possible4 The addition of

the consideration favors the selection of the Polar Gamma

distribution with alpha equal to 16, even though the test

statistic for the other Polar Gamma distribution is sliqhtly

smaller. Therefore, this student would select the Polar

Gamma distribution with alpha equal to 16 as a model for the

scatter of pellets fired from a rifled barrel. Additional

justification for preferring the Polar Gamma distribution

over the Polar distribution is found in the characteristics

of the different distribution. Since the Gamma distribution

is defined only for positive values, using a Gamma distri- J-. 41bution to mode] the distribution of values of the radius

vrector is appealing. The normal distribution, on the other

hand, -is urbounded in both the positive and negative direc-

tions. This always allows the theoretical possibility of

having a negative value for the radius vector. if tle mean U

of the distribution of the radius vectcr is not large with

rcspect tc the standard deviation, using a normal di--:ribu-

tion could produce a significant nuiber of negative values.

25

-~ WE-

Moreover, the histogram of data values for the radius

vector appeared to.resemble a gam.. diaribution more than

a normal distribution. Finally, the gamma distribution is

easily modified to allow for variation in ranges. Only

the scale parameter lambda need be modified to reflect a

A change in range. Possible procedures to acount for the

change in lambda as the result of a change in range are

discussed in appendix F.

NI0

NI

-NN I

inn

26

'-_ s'o = .t

__--_g - ==

l£

V. ANALYSIS OF SMOOTH,1 BAR-EL DATA

The analysis of the smooth barrel data was sim.ilar to

thr analysis of the rifled barrel data. ThMe paramfters for

t'N. Garaa distribution were estimated using the wethcd of

moments. The three distributions, bivariate normal, Polar,

and clar Gamma, were then used to ieteraine if they couldaequately explain the data. All three failed to provide an

acceptably siaall value tor the test statistic. For example,

the 10 meter data- produced test statistics of 145.5, 291-0,

359.2, and 24S.5 for the bivariate norma±, polar, and the

-two versions of the Polar Ga-.maa distributions, respectively.

Following the technique of the previous section, the

centers of mass- w*ere cal.ulated usinc the data po-ints in

each round. These calculated centers of mass were then

examined. As with the rifled barrel, the smooth bore data

centers exhibited a random drift. However, the characteris-

tics of the drift v4re slighLzy different. Since the barrel

was no longer rifled and since the rifling was suspected of

producing the drift toward the left, on- would expect the

smooth barrel to fail to consistently produce a movement to S

the left. ..e smooth barrel data was, in fact, free frcm.q

this tendency. Essentially half cf the centers of mass

drifted to the left and half to the right. Th'e.--re ccntiznued

to be a tendency for the centers of mass Co rise as V,-&

range increased. As a matter of fact, all ten centers of 4

-Mass rose as the range increased. In order to remove this

_ 27

V= random rift, the data was again modified by subtracting the

coordinates of tte center of mass from the coordinates of

each impact poin: in that round. The centers of mass for

the smooth barrel data are provided in Table 2.

However, evm wieh the data -.o'Jifi ed in this way, the

values of the tast statistic were unacceptati-y !arge. Onlythe Polar Gaz distrbutior produced a test statistic less

than 100. WAile this stuAent was preparing a histogram of

vaues for thc:dius "vector, he noticei that rounds 6, 7, Iand B proance! -an inrax-dinate nrnter of large values fer the

radiua vector. This caused him to suspect hat these ror.As

might be classified as outliers. ::onsequfi(ntly, he calcula-

ted the sam,:le variance for the tree rounds in questxor and

the sam.le variance for the remainings seven rounds. Whe2.

the ratio of these sample variances was calculated, an F

statistic was produced. The F stetistic for tue 10 meter

data was 21.9, for the 30 meter data 3A0.3. The critical

value fcr a]ee ofsis .fiaen.oe of five percent and

degrees of freedomh 46,112 is approximately 1.55. Since the

test stttistics were large with respect to the critical

value, this student classified rounds 6, 7, and P ogtliers

and remirved them frcr. tle data set. This variation among

the rotunds may have been produced by variation in the nanu-

facture or performance of the ammunition. In any e%.ent,

there pat-IS to be sufficient reason to desire a repeat of

this exp':riment to see if this variation is c'bserved again. 4

-4 V-

TABLE Ii

CENTERS OF MhSS-SIM Or , BAR&REL

Round 10 Meters 30 Meters "

1 (-0.16,2.2-1 (1.11,3.85)

2 (0.303,3.431 01,.5

3 (-0.98,2.40) {-2.65,38-66). I_ < °]

(-1.73,2.581 (-349,8.35)5(-0.30,2.53) .&-1.23,6.21I) ]

9 (0 .11i, 356) (i. 4-0, 0.73)-1-

I. 2

B9 61II

IiNIIN

r29ma-_---- - - -- -f.-~

If 30 regions had been used for the goodness-of-fit test

after the data se had been reduced by the rcluion of the

three ou-ling rounds, this would have resulted in an expected

- frequency of only 3.73, much less than the desired f"ve.

Consequently, Cm nwnner of regions used when ttng the fi t

of the Polar ad- Polar Gaj-a distributions was reauged to 24

biy superimosing only t-ree cxrcles over rihe = r5.--j frmf the

origin. The radii corresponded to the estimated 25, 50, and

75 perce-Mtiles of the distribution of the radiuc vector. The

use nif 24 regions raised the exoected frequency to 4.67 A

which is acceptable [Ref. 31 .

After e11--ina the atypical rounds, reducina the number

of regions in the partition of the plane, and reestimating

the ga parameers based on the re--ining seen rounds,

the three 'is -ibutions -were tested for -fit. Te bivariate

no_-al aLd Foiar distributions resulted in relatively large

test statistic all were over 30. The Polar Gamna nroduced

a test statistL PF 23. for parx eter values of aipha itA

equal to 4 and Lamxa equal to '4, using the 10 ter data.

r The :LOWestE.-t statistic for the 30 me ata was 2.

; obtaird t he Polar Gamma distribution with parameter

estimates of alpha eraa to 4 and il ,t ezua i to 0.5-352-

Since u. ooth barrel data had been modified r. the same

manner as the rifled barrel data, the same problems were enconz-

Stered. Since th e P ar Ga: distributon with an aipha value

of 4 produced thr 1-est value of tn - test statitict for

both sets of data, it was recomended to model the behavior orf

tc smooth barrel dat-a.30

V aam

=.- wa Che caser~ = wih h ife. aa h ai

distibutionwas ---eer-ed Orerthe ~olr dzst-lbuton l-e1aws

was th e a e wt h ilddaa i od

bii

Of ass Or e-nc ru*.5Tw eNDam-t S 7U~ -m.eaaSold- est iJaaIsdf = h nat SU

i lTe drift of the center- mass as afucinoraw

h e c -- ntzo~duceo in-o simulationls ofi t -- Se ro=ns. hc

en-nd n the Ont - - a- ____l

fmcn -nro te d-e

aJA4-omldata- shoul bn taken- to- allow a recX.rae

investiga-tion of t-he Ai str-ibution c-f the center mass Orf

a single rouand'a a functaon of -rnge. Aditional data

a should &l c be taken to aliro a car zete investicat-ion

cf the variation of the scale Parameter iat *da a function

of range.

ACT

2S

APPENDIX A

A-CCURACY DATA

S hBoot Bare rre.1

Shot iact at 10 Yues 'rct at 30 mers

a -0.1 +0.4 +32-2be -C--4 -0.5 - '1 -c l... 4-0. 6 4i -1.7 .9 -. 41.7

e -0.8 1.- +5.8 +2.2_ f -0.4 i3+292.

g 4~ 1~ +1.4+-0.3 -2.3 +0.4 +3.7-1.0 42.1 -2. +4.7

j-0. As+2.8 - +-4,8k+2.4 A- .74

2.0 +- -2.± '7.5-+1.7*~ 1. +7.7n-3.9 +3. • 94.9

OF 04 .9 -1-2 -0.5c -. 1.. . . -0.

2a.-1 --- 3,-. +.7 --

d -0.6 +1 0. +J0.e -0.9 +2.6 +2-3 +2.9

C' A -4 5,

--U . 9 2.9 +0.: +5.4A- 5 - A + -. 4I

A1 ~~--..4. 74

-5 7 - -+1

Se2unce cf sub-oectiles list- at- 0 m-ees is m t

n-e cePs sariy i e nticzalI w i th t e A s -me n Ce -Iis - - d a t I u .- ters.

- -A-A

*-.L + .i -- a£f

Sno -Imact at, 10- leesia t at et ersXlin2 Yn XWnjYin) I

3-3-

-1.5 +1.4-3.6

5 -7*7

h -0Z-2.:-.

:2-3.02 9

4a 2

32_ o -3.

:.242 (D* ±:-2

esaa ze - ii. ea a3 ----Ial -e -1.3 s =c d t-0m"e9pa sa Z r %

imac a* gee- -npaCl at 3m C M. e t e -r s-

-Z

c ~~ -!.L.

+0 al.. %F.. -4

C 20 +-8-A

4, 2 .s --- 7a

ol r. -

F0 t. #±

+42 +12 __

l7nU - j -

I~ *2

e 410.4

+j 4 j~

44 - S

ase- Z -

JC

4.4

Sar I v mt

- A---

- -- gL

7 - -. C

is a -77.

~t

- =4

-- -- - ~ ~ =

_ - -. -- - -~

S

-r.~ . a .. ?J rt tt~Z. ~ at -

- - - -

I. a---- ~ ~. - - I

-; rr- ~ ~ ~

-- C -

-~ ~ t = --

. t -~

-- 3I z-i;? - - -~

I~37 3=

-~ - -~

----

- - - - ---- -I.4. ~. - -~ -

.r .4. - -. -

- .4- -

-s -1~ e -r - -- -~

-4. -S 4

w - -

a I;I- - - -- - a

ft -~ ~,

- -~

a .j~

- 4..

- - = - - 4.-- -

I i:t' I[ _

IrI I

- r~~-- -

Shot Impact at 10 Meters Impact at 30 MetersN umb ey X (in. ) Y ( in. ) X (in. ) Y (in.)

3~a +3,4 -7.4 +35. 3 +6.3b -0.9 -8.6 +33.4 + ,3.5

c -1.8 -8.1 +33.0 +33.4d -3.7 -S.5 +18.3 +36.5

e -7.4 +1.8 48.1- +38.4f -6.5 +3.2 +10.3 +4.8g -11. 3 +1.8 +2.7 +34.3h -12.6 +3.7 -1.0 +40.0i -5,2 +5.4 -21.4 +5.j -3.6 +13.1 -23.1 +2.5k +019 +13.8 -37.2 +1.31 -;4.-8 +15.7 -40.2 +7.2m +2.2 +15.7 -2.7 -28.5.A +4.0 +1.9 -4.9 -27.0o +12.3 +3.0 -11.6 -31-6p +14.6 +4.3 +9.1 -23.1

4a 412.4 -7.G +29.5 -t32.4b +12.2 -6.6 +24.0 +35.7c %'7.0 -10.0 +21.5 +29.5d1 +7.5 -4.6 +16.2 +15.9e -4.0 -9.4 +38.4 -19.0 if -6.4 -5.9 +35.1 -16.0g -8.0 -6.6 +21.1 -12.3h -10.9 -7.5 +6.1 -10.1 Ii -90 +7.4 -7.5 -32.3

j -919 +9.8 -25.2 -- 23.8k -4.7 -12.2 -38.8 -26.51 -13.3 +14.4 -30.3 -39.7m +8.4 +11.i -31.2 +21.6n +8.0' +9.5 -33.5 +28.6o +10.3 +10.3 -33.5 +26.2p +7.1 +4,7 -38.4 +35.1 Z

Sequence of sub-projectiles 1isteP at E t r w rs is not

necessarily identical with the seouence %Ae at 10 meters.

39

Shot Impact- at 10 Meters Infact at 30 Meters :Nun~ber V~in ) (in.) X (in.) -In.

-5a +12 8 -1 +12.0 417.2

"b 7.-4.1 +2.5 +24.0c+1.4 -2.5 +18. 6 +0.1°

d +7, . +0. 2 +14.0 +21.5e -5. -1!1.5 +2.5 -6.7

-i" -15.6 +22.5 -9.5

g-4.8 -8.3 +402 -39.5h1 -25.3 -5.9 +13.8 -35.5i -14.8 2.0 -4.4 -33.5

Sj-16.7 -22 -14.;2 -37.0

k -23+1.5 -14.1 -294 .01 -0.1 +13.0 -35.8 -47.5 :m +4.2 +15.3 -42.0 +7,7

S+3.6 +7.3 -30.5 +32.5U +6a0 +5.4 -26.7 +30 2

Nu+6e0 +20 -23.6 +31

b +9.5 -0. 2.4 +37.2 SC +7.9 -0 3 +23.8 +2.9

d +7.9 -0.8 +14.0 +21.5e - 2.9 -1.2 +24.7 -6.0

f -3.9 -. 6 +28.5 -5.6q -4.8 -6.5 +4.2 -39

h -I.3 -96 -. 5 -21.2i -10 . +4.4-30.0

j -. 7 2+4.2 -142 -3.79.

k -12.4 +4.9 -31.6 -2.7--1 +4.0 -35.8 +9.2

1m +3.65 +_ .3 -30.5 12.5 '

n +6.0 +13.0 -44.6 +12.2a +13,2 +14.8 -44.3 +42.6

p +5.5 +15.3 -351 +320

Sequence of su-.-projeiies listed at 130 1ters is -ot.

nlecessarily identical with tte sequ~ence listed at 10 meters.

. 4G

-C 7

Shot Irapact at 10 Meters Impact- atC 30 MetCersNumber X(in.) Y in.) XMin.) Y (in.)'

fa +9.5 -30+116 +32.6=b +6.0 -3.0 +6.1; +38.8

c +4.2 -3.8 +11.0 +41 5d +0.9 -10.3 +.29.5 1±t-9e -3.1 -94.2 +28.1 -8.9

t-4.0 -9.3 m1. -.g -4.5 -0.4 +411.0 -9.4h .7 +. +24.6 -34.7

i -1.0 +?.8 +2.6-3.+e 31. +.," -15,8 -i4.0

kc -7.7 +3.5 -12.0 -25.91 -1.3 +9.8 -29.3 43.1at +1.4 -rlO. 1 - 33.8 +8.4

+.-+12.3 -36.)1 +10.4o +3 3.4 -23.8!9p ±03 +0.3 -4.0 +30.2

Ba +7.0 -4.7 +23.3 +20.3b +6.2 -7.3 +14.6 +210.0o+3.'7 _-7.0 +16.0 6.

d +1.0 -7.4 +i5.6 +13.0e -6.2 -5.1 +21. 0 -14.8

f-7.8 - 3.2 +19 .9 -- 23.6

I -7.9cc a-t+3.

Sequence -2f su-rjcieslse +t34 -25r s 5ot

k1 -A* -2 .-

Shot Impact at 10 Meters Impact at 30 1ieters

Nuer X(in.) Vin.) X (in.) Y (in.)

9a +0.9 +1.7 +3.7 +4.INb -1A-9 +i.3 --7.1i +5. !Sc -0. 8 +2.8 +8.4 +7.9

d -4.6 +2.8 -2.8 +8 .6e -1.4 +2.9 -13.8 0e.er +2.2 +3.0 +7.9 8,7g +0.4 +3.1 +.2 r9.0h +2.3 +3.2 +7.2 +9.5i +2.0 +3.3 -.3.7i +2.1 +3.4 +4.9 10.3k +1.2 +3.5 +3.2 - 1.61 -!.3 +4.2 +. 2m -0.2 +4.3 -4.5 ±12.5n -2.! +5.2 -4.3 +15.7o +0.8 +5.7 +. +i.7p 4-2.3 +6.1 +8 A6ir4

10a -0.5 -1.] +1.5 -9-4-04 +0.3 +2.8 -3.7c 0+I1 9 +249 -l,23 27 -1.3 +3.7

.0+.8 3 4.4f 0.8 +2.9 -37 +

- =. +3-0 +6.9 +4.zh -2,1 +3.1 -3.4 -4.9

- . A-4 .5 +5 ,

~~~ 4±. 46j-I.S +3.@ .t-

-i L!. '7 -4.2 ~*. - .k -A-8 +2.7 +-

n-3.5 +56 -4.7 14. 2- -2 .2+ 5 . 8 -3 .7- + 4 3

S- 4+ 5 1- . 9 1 6 . 3 !

Sequence of sub-projecta:.s i at 30 meters is srot

necessarilt, identical with the sc ence listed at '0 meters.

- 4

II

F4

?AA Mi. EzTIMATION,, BYl~ YZM-hOD- OF M-i4EflWS

Xi~a xX: for s-ach i I,.., x~ii

The expected value of a oau-ta. Aist-ibated randot, variable

is a/A, the varianc.c at/-.2 To use the methof. of mn-'ents

set the sa:tple :Eiear. eqxaUj to the theoretical mean -and -the

Sample variance equal t, the t-hcor.et-ica± variance. Thu- as

-I a / end S A.

Slubstituting xfor it- equivalen-t exnrecSSion in the reltnU shp btven s~andthedistritution nararmetev yi'eld:Beve s IIi P

Substituting 23 value fcor 2aLda in the exnr-esSion foDr z

X 0

Thlerefore, the -jetloc or noens ro-2uces &h..flThin

esttzatces -for the parame-e~rg

Since &aa.,.& both denen on x and 2s*, -e are petvi

cxrrela ted.

The bias b (9-i of an estimate iLs defined a lb (8)

E (T) where T is the stati~stic used t%-o esti-ate -A, the

true parameter value. Approximate th3,e bias by using a firs t

order Taylor- expansion:

43

= X~ X-2:x (s-04)

2 6

C; 0

2 'ad

22

A ~ Zcv (six)

ED)SI <Q c (,ic since aa ar~e positivek-V

-= correated.

~~44

APPENDIX C

CP =MR OF MASS AS A RXNCIEWCN 0F RANGE

The information iout t.e center -f mass of any om round

is limited. r location of The center of mass is knVwn at

th-ree ranges: 0, fl, and - meters. At zero range the cewnter

of mass is at the origin since the pellets would not have had

an opportunity to dispere. -he location at 10 and 30 meters

can be calculated fr the data. Wth., this limited informa-

tIon it is difficalt to acc-urately derict the ction of the

center of mass. Additional experimentation IZo rovLe more

data on the motion of the -enter o r Mass would allow a more

thorouh examation of thle mtion of the center - -ass as a

fucaen of range. A desirable situation would ba to have

sufficient data to allow an analyst to determine a distribu-

tion of t-he center ,f mass. in t.'-e absence of furtler

info0mation, the data at hand sh-Uld he used. Therefore, two

possible anor-oachos to model the motion of the center of ra-s

are presented here. Users of the results of this stu---

should deteIrine what procedure best satisfies their view of

the drift of the center of mass.

One meathod to mdel the drift of the center of mass is

for each one of the coordinates to fit the lowest degree

polynomial that appears reasonable. An exam ination of the Z

data w uld sewa to eliminate a ccnstant or a linear relation-

ship between either coordinate of the center of mass and

range. Since a parabola contains three parameters and coordinate '

_ 45

NA

$

values are available at three ranges for each shot fired,

it is possible to fit a parabola through the coordL.nate

values for each coordinate of each center or mass Usin the

parabolas determined for the two coordinates of a center of

mass, the position of the center of mass ca- be determined

for any range. However, the procedure does not provide ary

insight into the physical pilhenoenon procucing the ntion.

It does provide a method of m ideling the motion of the

center of mass as a function of range which will reroduce

the experimental data.

A model of the drift of the c enter of ass can -h based

a the physics of motion if one can accept three assemptions

the forces acti.g on the projectile (the reco-i an airresIstance) act for only a short period of time, once the

covering on lhe pellets is aerodynamically stripped away

air resistance is negligible, and gravity does not signi-

ficantlv affect the pellets their maxim= effective

range. Under these assum-ptions, .e forces causLnq the centtur

of mass to drift can be visualized as acting instantaneously

as the projectile leaves the barrel. After leaving the

barrel, the projectile would travel in a straight line. One

can then use the two data points at 10 and 30 meters to

estimate the parameter of a straight line tkrough the origin

- usiz.: the least si-ua.-.es method. The line rast go through

the orjgin because at zero range the center of mass is at

the origin.

46

UK. CR T ---

T-hi m--odel has thie asxeal o-f being b-ase-d on rohvsica

principLleso motion,.Uteei has the Aisadvalnte

-not rCDr-dC2cfl the- C actu-al- e3-merimn tal resultsecas it

-is unlikely that a le squares fitted liewil1o2 roz

either of tlhe data points.

Once thne usr has se lec ted a m-tbo-d of moe'g th I e

motion of t-he cen-ter of n=-ass as a functi-;on or- ranoes, he can-

use th-at model to cnrtarutinfor each- of the extanr-

mental rounds- When a rom4n is efireciaL in a simlulaton of

one -of these functions hasad _an Den-..otz1A

randw-.lV Selected to%. XrvdG u nsUno the center of

mas at* the range desire-d fr t'. sirnulation r =L

the drift ofa the cente-*r- of nass can b-ein'cue to the

Ss:*'fat1 On.

__ __ 471

it'.LIL 1&±Z coLiz,1-p-- -t fs t rie ri-qn' daab

rtlnr' frcxm "e ~JLates -. r e-k --- #_ e nt_ rr=t

Wc--fLa have- Cx the 2stribr-on of pe ee stnxkes.

Atbnucn g the bosa--mri4ate no ea-. L ~~ir~v aescribe

the~_J~ tabt± it was afcinl I- cw- tt ' -

~~~~~~~~ nn__ c-rW-e :ix ue ~cetei na Gflra the na5,_ - thr e --- % -.

ttae cr-ana r-oielzto~s-=--~x use e--..

no~ua ~'n eCar. us fy hiTheni~ 00 nnLIL cr- -is a±=.

distrbton.

th2- bivrate rcz=i - idectndent* Variales. 4 A Without

for ro%-th var 4 ab1 es was asslzed t bezc=adteviae

{ -- *its a .I'a~ of : s ze La - .a -

'litrbui 'nwit

ASA

At

2-C --- v Ia -

A -rI +V

diacona]_ and- t

ahe~~LV.4' ptJ6 e)n for, lt V rit'xha eC

CLe r ~--Xz C- .i e r. --_ _AassA

VXY VA X

1 ~ (.,zI=CcaI7ZNEach± t r-. te g -asteult

r Lre"-e assc theraanrc - of 4p t snriad cp ncr ; Henceztant2

2-

t~~~~~~~ t.. S2=tt j --

mv ' Arc - :r~ewr v 70,

!.I. 7 - i Nd C -0 0.rCIM teS S

at=± --d LAA Is oi t-r o--nD

n -

wc *-_ rzn ohl' a vc-real ptrrtrt

it-: foorw al .o a'meoe ' _ _ - the abov ro_

lo a X=- %rw -Q U

f reae all K>-. thcte t-a--iv.- ththat'n

wit-h' ------ amw 4 - -t heo -I A -- %r

hl -r Catrt of Z__

shou -af - C a D Q -- -;7

A~4,

Wrcai o- samew disz-3 atn-ze 7-c tz fran I M' e f arg -

A4=;taha *'-a gr Pz -,s- - --

th dta i rdued e n tala inc et-&a-:.X-. -.. .-- dat~'are U or-

T.Aent gir-v -Sa rc-1nt D-er-sac a ' ete s

e~~- x; ~ x~ PC'c t~ e dv iuet r rcen s or mn s~t~ s and -Jrasws e

assises h o- f tne th-tsls nav *-sAe madit

to have tei centers of ms cicie a Thl~may be

preOtc Q

n -ne

r- rc...A

-r~ ~~qs- - -- - ,

~ M-~-~m

The data ct.3J by RIZN-ill. Vr_'SZICt

nTh exAct dtenlt-ated be ....ne a... ::ae U-Sice

ral data, the cxeviat .on was net;- S~-n- * icet

~ZiZ n~i istitit had pnroc- ce

tZ ir 41 1tion was bae on a P ola_

rS-- t_ L-

Were proaducea. A aa. r .. A&rf nZZWS

the relaaonswt-s: X=z- CO- -- and - ~

data. n'-* the" ac p-- 7- -

thr u eic ts paibS of2 am2 Zda- z

L-: ah ar I .*. a

*the radii Of the ocice- were then rea !rzm a re-ctor.Th

valuesselecte, were base-d o the val-' of o tne aJ 'a aaee

produces- hrO t-.nin h a7- alu~ at the d-ecimal 7.V -m

PO,

-=4-. -7;l

- -. --- ~ 4--- -4 ~ 4-- S

am-.- em th a=Ead______u-_= =

- -a t7~ o~- bee -z 1. z t C* Vt

ZlzF-- te m~- t - t - -

the t r b-;n ew .-or -h ur-~e- 1= 0= as

Se i b =- - y-._ =_-n~ snaC- ;CTlmC

_ -ate SCA____

tim ___s i-_ -Z=u

- - ore

nse the~=

3--s- 4--~ r- -e - - 4

on-e ~ ~ -ean q -onM - -t -- M~t

ec~ns-o -o e ~ s~aiU 7__ a- Ih~ a 0ecXno.t.

wa s s t Calec

- - = cF

AZter 1000 runs through the program , the stored test

statistics were sorted from smallest +-c, largest. The 5, 10,

25, 50, 75, 90, and 95 sample percentiles were printed. The[

90 and 95 sample percertiles were used to represent the

critical values for this goodness-of-fit test. The sample

percentiles are given in Table 3 as a function of the alpha

value.

rI53i

TABLE IV

SA-.MPLE PERCENTILES FROM GOODNESS-OF-FIT SIMULttTION

SAMPLE PERCENTILE

ALPHA 1 10 25 50 ?5 an 95

3 21.8 24.5 33.1 84.5 157.2 135.3 199.24 26,0 31.2 42.2 65.3 105.1 135.5 162.85 20.0 22.2 28.6 36.8 45.8 56.0 65.36 20.0 23.0 28. C 36.1 46.2 57.5 65.77 21.8 24.5 30.1 41.0 54.8 60,.0 75.18 n-.5 24.5 30.1 39.5 55.6 69.5 76.29 i6 22.2 27.8 34.2 43.2 54.5 63.5

10 19.6 21.5 26.3 32.0 38.7 45.8 50.711 19.6 21.5 26.0 31.5 36.8 42.8 46.637- 19.2 21.8 25.6 31.2 37.2 42.5 46.613 18.5 21.1 25.2 30.8 36.5 42.5 47.014 19.2 21.5 26.0 30.5 36.1 41.7 45.1

15 19.2 21.1 24.8 30.1 38.7 41.3 46.21 1- 18.5 20.7 24.8 29.J 35.3 41.3 44.7 "I17 18.5 20.7 24.5 30.1 35-7 41.7 45.119 177 20.0 21-5 30.1 35.3 41.0 44.7

19 19.2 21.5 25.2 30.5 36,1 42.5 45.820 18.8 20.7 25.6 30.1 36.5 41.1 45.1

54

APPENDIX F

VARIATIO2 0-7 LAMFDA AS A FUNCTION OF RANGE

Since data for only two ranges was analyzed, only two

values of lambda are available at different ranges. An

additional value for lambda can be deduced by observing that

the variance of a gamma random variable is ct/@ and the

variance of the pellet strikes at zero range is essentially

zero. Since a is a constant in the models of interest, A

must increase without bound as the range decreases to zero.

This is the only way to produce a zero variance at zero

range within Lhis family of distributions.

Since lambda is urbounded, it is more convenient to

determine the reciprocal of lambda as a function of range

and recaire that function to go through the origin. This

sit3ation bears a strong resemnlence to the prcblem of

deter--inn the drift of the center of rass as a function

of range. Is in that case, it would have been desirable to

have additional information about the way lambda varied as

range increased. If data had been taken every five or ten

meters, a more detailed analysis of the var'ation of lambda

could have been accczplished.

A parabola can be used to fit the three values of the

reciprocal of ±ambda. This method has the advantage or

exactly duplicating the experinental results at the known

ranges. R'mu-ever, it offers no insight into the reason why

lambda should be expected to vary tn this hay.

55

An alternative methed can be nased on the dimensionality

of the reciprocal of Lambda and the moments of the cyvnma

distribution. Since lambda varies with the ran-e and the

reciprocal of lambda times a constant is the expected value

of the range, then the reciprocal of lambda must have dirn-

sions of length. Since the dimensions of the reciprocal of

lambda and range are the same, this suggests that a linear

relationship between the reciprocal of lambda and range might,

be appropriate. Therefore, it may be reasonable to use the

least squares techunique to estimate the slope of a line

through the origin and use that line to determine the recipro-

cal of lambda, and hence lambda, as a function of the range.

I56

LIST OF REFEPENCES

1. . P. Uxit-.- Laboratory Reporth a ri c-

Tests of .45 Caiber Ball, Segmented-and Shot Cartrides,p.,2 February 1971',

2. U. S. Army material Systems Analysis Agency Unclassified iletter to U. S. Army all Al s Systems Agency, Subject:comparative Effectiveness of the Cal .45 _14261 Shot -Rouand when Fired fr a: a Rifled Barrel Versus a Su-othBarrel, 6 September 1972.

3. Hoel, P. G., Port, S. C., and Stone, C. J., Introductionto Statistical Theo y, p. 95, Houghton Xifflin, 1971.

4. Meyer, P. L., Introduito, Probaility and StatisticaI J5k icapns 2a -ed., p. 333 :dison-Wesley, 1970.

5. Hoel, P. G., Port, S. C., and Stone, C. J., Introductionto Probabiily The__ry, p. 130, Houghton Mif fl- -n, -1971.

57