Ap Statistics - Exploritory Analysis
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THEME ONE: EXPLDBITDRY ANALYSTS
hical Displays
Gra
lUldSPH,l-ad
• D(>tp!tJr~
•• C'eJ.l[l'
•. R;lr (~lta:f{S
" Chtstt::r:s and
• 1Il's'rClgr::un~
" ()iJtlL\'~hi.
PINS
• Clm1ul~Hjvl7 f:requency
»
G:tps
1\1,or.R:s
•• 1i!L"
• St(wTplqcs
--
...
, .,)lh
here art: :1 va{l('t'!' of w':.q). to Drgmize fUjdc2:rl'Ml~zdata. Much intfwrnatlon can b!,~put 1nto Glh!es hI( these <11'fl.1)'& nfb'tr-t: tigpres renJ tn b,: !'pii:i~iesS)lTJ(.1someI j me'; e\'fn lorbi·dding, Some' j~Jrm {lfgr~l.phi~l display .is.ofren best tor sc£in~ parrcr nx :wd shnpesand hJr presenting an immediate impression oFcve:ryrd1illg abour thc data. Alil(:>ng rhe rncsc common visual repj;~~~1pafi9n~ of da{;~ !'rc. dGtplots, bar ch;ln". hhimg~;'J.J1-H. lJmj srcrnplors. his impOrta'llttCU'emcrf1ber that all gmphjj,'!ll dispLty;;. shouldbe dqrly b.bded, leaving l'ttidc1/J,bt ,>,,that the ~)iC:ttw: wp[es~m,i;-AP St(!r1s(jICS scndi1g guidt's harshly penal};r.,e the-lack ofridJ.::'s :\11<.:1 labels! i
DOTPLOTS DocpMlfi
ual
IZh';H[~
J.tcp-llIrimlurly
'tlsef:~itw:h:h I'eg;~,d ro
i7i1t~g.f);J'1'£lll (or ca:ceg:ory to \vhkh L-:tH,d, tl1d.iyidtf,.'qii4irtt14t1P:e Itari{{bt(.~(, whi~h rake' on
Ilfl.;ri(l6/es, that is, \'<Jft
(jifa/fmtit!{')
1:)dtl11g£,
'1 'his
1:Y
IJI
corrrrasr
n \Ili1t'lflr~1 ~"l.lU;t$,
Suppose:thilt in wGiass of :35 .studems. 1O>choostiba$kiMtiall 'BS H1eir favorff~:wm'l whne 7 pick baseballS pi1;kf00tbaH, 5 -pib-l< termls. 5pt~,~QcCerqi.,~2pl.~ htJckey; Th~~~~i;il
.. ••
..~
l'
.:~
+
.~
;,.
..'" .,
8
~-
,.
••
'!':f.
.,.
Bcseboll
·f
B(}$k~1bo
." 1t
Footbnll
Soccer
Tennis
1
TlHc dDtplotGul also b~ drawn wirh.a \-'erti~1~4X:fs ;md bbflZonr;JJ t'l;:}WS of dots. In bG, in almost all displays, verve-ill and horizontal tan be 5¥iircJu:d depending. upon which picture seems easier to reader simply which b,ect!;:ifirs rhe Pttf;iL
BAR CHARTS f\ common visu..~ displayrocornparc rhe ;,17.£;$of cat~gt:ir,ie.~ OJgroups S~i:,esam be measured as frequepdes or as percents, ''
is rhe
l1(lt
ebart.
in. BSut\lE)yt.ak~f1d~rij:tfllil:e tlrst Wft~kof ~3nuai)l'1~~~. SpoJ6 pflt)p§e:sl!r',;~ye,(j~anleDthe c!rf1tonlmpoofhfh~,ht \1i~1tl:J)90q frhtn~cJia1fjly, ~5% \llrdfll)l:q ~ t9~~n'~i~ with \¥itri~s~9s" 25% \V<;.rit~,'tn-e trfal tQ:c'onilliu~",1moutci.dljr\g Wltr1iF~s:>\mi~ 5%~xp(e~s~q tlOQjJlnrQn. Thes(itja~c,an,,q~'a,s~!a;y~,~ ill ihe IQllo'A~ng,bar than (or,qaig(.sPn)C '
+ 4u1
"
:5,0, f'ei'cer;t
of those
svrveyo.cl
30
20
lO
1 1
~
rContinue with wilness-es
ThECfr.eJatlV€ trequeneies.ol
(jllferentfBsults
n I
I
I
(ofinllue
NQ
b'l1IY~r1h
OpInion
nQ'witrie'S$~s
are 1n€licated by th& tlei,gr;13 or tne bars r€iprB-
}1entif1g!h~~a fa~lbt
HISTOGRAM$' Hi:(1;(}grattJ,J, US:togt':3.:rrls.there<\w.':'ofi:enqltt'stions
on i:l'tieqrr~inggi~'enh!Stt?grat:'))S,
--------------------~======~=O
"
Suppose ttlBH3 flrEF2£lOO iamilles in Cl srnan town and the djsttibutien of childrefl among is 8& touows: 300 r<'IfT,illes areehildless, 4(.\(}have oM thild; 700 have Iwochiidren.300
tharrr
thre~. '100 have Iour, 100 r~-v8 five, and 100 have six. Tl'\eSe'
i(l
,
;
hav-€i !.he .
500
__ .__ 1
•
o SOr'i11iitimes, insteaci more m
l~e
n1€Bflingful
~j
to O$€
populafior1,
'2
JaI.l€iirVSlfh~ VMiC",j axi$'wllh'ftf1fll;len~j!:!s,
!! is more conv~n[S:lit
(fr
roJative treql}em-;tes, thrl! is, 1~{I3n.CWs'dMde{j by tt'ie iotal flumb~f
.
.
400/2DOO.
2
= .£'00 350
nXlf2000-,,;
':'Kj()i;~O(JQ=, 15'C)
~t~r\.9
10or2000
=
100l2tXJO
ss
iuit'o
100/2000
=
1~,
,1
-
..
~r
>
OSD
.050
I
-:'''" ",ן~f-
.050
i
nl
I I
1
, o ,
i.
I
•
I
I
!
I
'
I I
>
I
""
I·
I I
I
i
1
Note Ihalthe sh.ap~ 01 the tirstoqraffi is IM .-~,e·Wnethef the vertical axis Is laoeled wlthfreqpgncies Of with ~e!a:Hv'<3 freQUe1lGles;;Sometimes we show both f requencies an;d relativefrequem:::ies on the<same;:gr"lph.
4
AP Sttttistlcs
Consider' the tofID\'."jrlg~istogfa m dispiey[ng40 large
salari~s pEtid
company,
to the tow1evel
ex€()ulJ\fB.S
ola
··t .
',3,0 ,25
-
,20
I
.:
. ·ReltttiY~ fr~nq
±~05. Whatca:n we,l~.¥r\ f[(J.111 m~s,histagram'{For exafhpl~J nQriEl'.of\heBxooutivi.3s eam..if~ rnqre Hi13n$fjQ.QQ9'dr li?ss thit1:$-<~0,OGO. Twelv8.earr:e4 ti~jw,eBn' $50,{)00 ~mO:$qCl,QOO.,Tw;;lnty~
$SO:o®, N9!e
five P{j!fGel1t ~~'rci'lSdbJ:}lWe€n$40:000 and 1)\ lteina ,(~'?'i.?fl~' f~1JjflgbefiNeen eenain
How tlii,Shi$1ogr
b~f
~ve~~ th$17'{J(Lti!(rrtfli;$$ had '€xactfy two chlldr&n,
Ci,J~idefth.B.f~~~Vlt19 :~t#Q9ramWh€'Ie 1M Iea.f[lfr'Q'rn1.tiI~:~q'gf~mt'
v:eJ;ti~t'~i~h~notbeen,lab~~'t i,flih~~J'l'~f?'
I
! i I
25
26
27
•
&
10 Qete~if'le the actuallreqaencies; Il'.)Wf;lver. we can determine re!:;:ttl'v"e frc;quencfes by nutir::tgth~ traction of the totat ~n?a that 1s overany rnte''l,ai:
AffSVl'er: 11is impossible
ttw:
I
1 ~
-
,
,
-,
.
I
2-1
19
28
V'le can divide the area jnto~:terfequ.~I,p'onion:s. 'and then not~,'that
It qr la% bnh~ ersa Js,
aC'l)\o'e2;:,-2$, .20%jsab~'JI1€t·.~ii-2j .:41)% fs·a..h.ov~27-28. :affe SO%!·js .aPQ:W 213-2~, AltoDugh it is usuaJly riG} posstbie.;{o,diItID.z hlstogtah1SSQ nl~$'lylnl6-ten equaf are,all, thlij prilittipJe 01f€.Jativefr0qU'.encr~s COf'ieSpof)qing 10 T~la{iV,:: aH,:aS still B;P1JfI~. ' ' . .. ;
-,'
-,'
,'-.,
Rd;uive fre,! ucncie -;rrt:iht: 'Us;lill choice when <;:mlJ,;dring disrrihurions ferv':M size pcipuhn inns. '
of dif:
CUMULATIVE .FREQUi;NCY PLOTS SQmerimc$ "-'vC: sum Freqo<m..zieS,and ~hdw the result visl~allyin(l: tJ{.mtila~.iuc{l-(;q[te!lo/ ,/,>ioj {tlho krKfwn as an 0t/ut).
.5
TO
!
I
l-~
20
25
eo
"l
35
40
W!"',at can wa !e<J.fn'from thi$ ~lJmll!ati\teJre.qvenQY plot? FCH examp!.E:;,: going up to tfm graph' ftorri ~ge 5, 'Ire see that. 15 -Or 15";<~of school emoltrnem is belOW age 5, GG1ng oyer to the gri:!~h from .5 on \he vertlcalaxfs-, \Nt'l see th13l 50% of ttre acM()j atTrolimenll$ b~fQWand~ 50Q,~is abr)VG a middle ag~ of 1'f. Going u,;rfrDrTl age';30, W~$:B~ tt!}aJ~95()r $5% of H'!i~entaUmenl is below ').ge 30.snd
thus '5%!sabOVe' age 30. GbJ.ngever trq,rn ,?S ani:! .75 on ihe v(it~ (conli(1l.1t?d)
\ iiGai aXIS, WC:isee that the middle 50%, pl.school fQwer 81'1d n.ndage is at the upper end,
enroilrnent Is betwaan ~ges 13.an{l? a1 Ih",
STEM PLOTS !\hhough ,1 hiiifqgr;:iIll may ~hvwb(HV many scores l-:1il IntlH'ach groupint; ur interlh~' cxacr values of individualscoresarc ofren kx.;t, An alternarivc pkt{wbl {ll5~
V;11,
pL1Y. rCllkd
i1 :'>[cmpiur;,
inform;:nion.
ret;{·'nHhJs.l111,.hvidu::),f
COfls!denhe:sElt{17,1£. tB. 13·;a8~~~, ~1,7Y. es. 5Q,'43, 37, 2.4}of pcircent8g~,oflhf~-. matJe by j,••liohael Jgfdandurjrig'~his t3iY~aTs.\
pOJOi shots f...'tJ
OLglt6T~ad~ value from trJeQrig.ifl.ljjl
pfB.tehoklers fm lC~2D,3D, 4O•.B.lid'50~Lts1ihelSEt
set <1P,sl 1he appropriate 'p1a'cehofdJiL . ." . Il;," rtfsu\l r,5~.stef!V:dC;(,{uJiQ ~aneo a sJ?mabd-ieaf cfiSP!~c.!j cif thes~ data: St'l:!m-s I 2
L~ave$ I
7
7 7
:3
8 3
4,
,3
5
{}
8 4
J
5
7
Note lh!:lt the st~mpfot ;giltes:!hB SrJ3pe (lftbs t~!l;to&!rnfrrarl!:i, llnllke .tJ¥lI'\i:;;t\ilgr~m, rr1l;llcat~s the valuer, at th'e OljEinB! ciata.
by rearntti.glng '[be runubersin each n:)'\l;· in :.:xmding.orde:r; Tki.s' (~rd(i'ed.d.i~ph~J'sho\.vs(l sktoridlt.;;'/~l(}f inf{nma,ti~)tl 'from d1t~ Somcdrm:.\ turther
~~'l:tiUiP":
is-shown
ilrigrnal Mnnplm. . nh- rt''''l:;cd display Df the :data.in Example 1.7 is'~s follq-,\is: 1 2
s 4S-
:}
"7
4
7
B
I
')
'7
3 O
(
7 I
"
g
8
EL
IT
$tJ~POS€ lnedistribuMn
PI
01 25 advertised home prices (in thousands OfdQIW~) in a certaln
community ISgiven by the.set {56, 69. 1135,73.,'83, ;45, HD, 188. 127,77, 1IQ,11·::>,132. 120, 94, 130, 8'~. 65,.99, 154, B6, 12H, 12'2, 10§. 130}, Onf;1;possible sternplot of Th'6S-e d"t(lis
50-74
56
75-99
1;)t)-17A
17 t33 '1;}4 36 -89 Q3 Hl 1? 20 ~ 21 :ia 3{) ~2 .41'5 M 9&,
175-20Q
89
1-00-124 12&-r4~
6<;
73 90 9.4
99
"r,
"!!4.'.
Noi~that the S!o,imSWerB chosen io btdnl!3ivalsofIenQth251'ind ftiatth~ . 1 iir 1G'{lis lell ()~1 . of the l~alies,s'O lhallMy aJik}f'\\iJ:!:J1i!':aIly, ~'
,
CENTERAND
SPREAD
Lnoking ara graph It:cj display, we see
dUi !\-VU
imponnnt
ll~pecf5.
of rh« overall p:n-
rrrn .lft:
L rhe csnrer: which separates che value:" (o,r"fr~;4 under The (LID'" in lilt:' :l hi$t~gnnll) [uughlyi nhalf; ;,tIW 2, the J1Htat:/~ rh.:u is, the scop.e'~}Fdw \ralues.from:smaile.'H.to i<J,tgeSl.
C;l,SC
of
or
Jn tht"h);~l:0gr~,;u1dthe-spread 1~ii-orn 13!Hl It) 'SO%i' in rh~5tefnrh1-1 of Example] ;i{, 'tnt' censer is $11 Q,QOO,:¥ld the-spread .is [rom $')6,0011 ~o$H}9~000.
CLUSTERS AND GAPS OlhcriInpnrtlnt
aspcessof the
0vt,;rilJ RjlITerrt~t6
Ldy :~~,,\~Ihich~h(;\:.;'narurs] s-~hgmups:'lM~)which the-values faU f!{-jf examp]e, [he sabr~£s of reachersin ImaGa,NY, ful! inio du-et (lyer~'lpping clusters, onc for puhlic ::;6h0t}1 teaehers, a :higher DI1~ ((,u IdUIGI C( liege professors, and <111. even higher one fo,r Cornell University prfJft:;i;$'ms), ~fld 2. ,el~ti!; which show IH>ks where Im vsj~esfil1 UtSr example, 1 he Office of dw tJ~'ill5,en,d!i letrers m ~al(kms being l}ut on the honor roll and to those being pm on receiving leuers frqm rheD.k.:m has-a huge middle g:.lp}_
8
AP Sla!/siICs
I 11
~I
i,
iI
11
!
11111
I
!! I j·
I
I
,
II ,
1
I
J
,
..1.'
35 Simplyooi'lnginalth~'Geroef pf llie:rlJsl1ibu.!iem,fS ~itrun~r42~dthesprli;ad'l$ IiornJ i ?2CJearlyriil$$~$ ~oTn~jhinQ'T~~ vatU€Sf*II,~t();t\~(idi~tf~~q~U~luffi 'With~gap~fN~E:;(L
OUTUERS
Extreme v
-
',:'
-'-
-.-
-',
-'
-
:
-
-.;
'-.-'~'-
.-.,'
- .'
'/~,
MODES Somc·d15tribWtlnnsh:axe
·6n~'fJl'rn()re.m
rwo such peaks; the.:dislrlbut'on
CGfisider 1heM~~fam:~f (PhoenTx and H~~~n):
lssaldtobeR/rtim(jikfJot
!he ysar£y a\iBl'~B
.lli.1!f'lf.rdal,
~#i~s~~er 'game
I
5.5
(:1;:..(;r1y
rc.spcctIvdy~
Ilgute$f~r Jetf
Hcir'r1ac~k
I' 9
NotB thar thls,'is' a b:modal drstlibu!iorL The tour.assis! s,vB;ages bet,veen 4 ?nrl4-'5 3clu1lJly correspond to the: tour S8a~O!lS{tom 1994'-95 ~hrough 1997-98, while the fP'Uf be~"(3~i1~ &iid 5:5 Gt)HIi)$pond to rnicH:fla car~rsii~s6trs, ' " , .,
Consiser the. I1js{O£Fw.1 of the ye-arlY,fiI1era.{jB POInt:? per if~meflg\JfBS for KarJ Maloge; {Utah) trom Ht85-8U 1hmugh Hl9t:'-98,' ,.. ,.. ' ,"
NO,~tl~halthis i$;:i unimoj.ald]$tfibuti(.in withiW'q Dl!n~-er5,The 91)tliers.as. migll t?,0 g•.lP,,$ScBd, Qar/esp,hn'd to Matone:'s 1ir?i NO SB:a~o~s es .., pro during which he B;VeG3.gw,ooly 14.9 am:! ~i ,( points per game. Di$r~Qafdin!J these;tw6 D(JlI lers, iNBsay Ihill thB- distnbut1(!Jnisfl!mO~t
sym.me.mc, N()re ~h:;n, \1$ illustrated ;lntWc;il ; . IJ~tl;tltr ins: RKtivl;:' [.6 Inp~f(ir reasons behind t)utlktKand modes.
SHAPE DisuihUl.1(JnS come in fH1tIJ;;,rHSarcwonhspecial
:U1
tndj1;is~ Y4f1clJ
of $t}~lPl'S.; hQ\vever.certain
common
memiore
l..
A t'ymrncttk dl~tribud(}nhn:nt
in which the rwe h$h;t,'S arc injITorir),~
2~
Ad:lstdburhm
~~cl;.!)rher. For-example,
I~ :flN'{IJ.'!(/ {.fiJjif' iiiht ifirScpf;t":a!nE\)-'LiJi~
t11inl)' ),f>\v::rrdxhe
!'ltht":r>v
5; AdisrributioH
values. For
thehigheI end with kw:
10\'"
vnlues,
4~ A· ~it:1j"'Fjl/.tPtf:i Jisrrihmjl)tJ B;:5ynip1:erric )vlth n center iUl111l1d (lnJ two ~16p:iligt~it,,;,For example, the ~1i:5(rihU(ion 'JflQ scores acsoss thf.'gener-alpopulation isIOUgh.ly symmctri;~\~·ith
is
un{t?n-m if itS hisrogramis
ahOl"liiJutaJ line. For example, many s:pors (p:ip~c) ~pp~ar on rop yields a withl" thruugh 6 flll equally likdy_
t{~$sing>::lf!liJ die.: ~nd noting. nnihmn
distriburion
11Qw
\ 10
AP
sienetice
I~
Symmetric
'/~\ ~
/ 'Skewtod
SkC'wed to the righl
\\ 10 lhlO
left
<; ,/ 0\<. Boll~1hap..'-i!l)iiif?tm
Even~vhenab~gi~ sltilp,eis J10Ied".it is ;n1pbrrill.1[!,l1:kH~;riQt.elfseme.ofthe'd,trn ~r~ from r:his shape.
L--
.
devl-
----.----.-----
Questions
on
>
'.
10PlC
1: Grapnlc,<J. .;,.
"''''')
.~
.
.:
QuestIons on Topic '"\: Graphica\ Displays Mu\t\?\e-ChQ\ceQueSl\OnS, Directions: The qnesrions or llKoX:1plete SUltl.-Wpn,ts rh"'-l f6nQwateea~J61lowed by fiyeSl:!gge~ed:lll.S\f(erS orC:O.mp~e!i(}ns;Ghoo~~' the' responsethar be.~r an vers the question .or completes the"S:tate.mtlH'; 1_ Fnl1vwing "ir~ dl~ SAT math si=Ol:~sfi.)t ~ldl,'\JJ':St4ti8tk~class :of 20 students:
D6.'L ()5H. 6HJ,. (i7L), £i40f 643. 675; 450, 676~'5'70}6601 (?6T~ 510, 667d)?~:~
645, ~,nd6)O, The di1'trib{Jt~ont)f$CfH~S.L'i
(l,l), '67 l. (it;)} (A) symmetric,
(B) <,;kei,v~drorhe
JeJL
(C} skev ed L{) the righr, '-' {D) illld1Jrffi. 0:'1 ',L,
bell-shooed ' ,,' j'W_
L - templors ~H>{:u.scflJJ borh for quantitative and c;llcgoricai dam sets. 1I, SrempfQt$ are equally !J$du.l fors.B1~LUand Y4ry large dat::t sers. lJ L Stem pl(iL~can shOWsY.ll1.meu)',g,.-qps. clusters, ;lnd ourlicrs. (.1\)
I onlv <
(B) 11only (e} lll.only (D) 1 ;md
(£)1
11
8.nd lIT
3: Consider
the Folli)\>".ingpi:cnue; C'.
-falof ~c.le-L
_.
-
--
.~
l 19N
~rh(ch uf the foJJmvingsraremems
1;995.
1S'~6
u'r~.~tiH;r:
J. Total.sales .in. 1995 were two Limes. toralsales ~n 1994. while total sales In 191)6 wercrhre« times the 199,1-r{]lf'.'tL . I]. The choke of l~bdil1g for the ~eniC4J ;'>:i5 results 1n J i''lJislc::3_ding sales
m.
picnlre, A.hl:f(0gr~rn sho\ving rhesJrut:1r1h)fmab\j/1. but [hi:; time with verticalaxis star:ting ,11 $7H ,000. \~'011ldlH: less m.i:Jeading>
.~
10011,
c(mjlI'o);ly (C) III on~y
(D) JI and III CL) None the above gives rhe complete
or
s!?[
of rrue response:'>,
(l
Dit;ofavs ',>
'\"'\
Nt!:nb~r of POf'S"DfU ir-i!ir:1C1{r ~D!
h,hhc-
~tt'rl'pIJ~n (!.;>cc"",; ~c.t..!hf
\'\/lI!\
b of d~\; fD.]Ii",,,,jars, sraicmeors
L Each vc;~r frDm 1?77 to J986 l'
dJ('.
-."
,A
number of indic;tment5 h.l$ ;<;'t:~ldiJv !
11 \);~!htlc the l1ull1b~~rofindluTnems has dc:di,~~~edr;:a.dh Ye'.:Jf,th~;;lllH)Unt of' d~'Cf{';aii~ hA~k"~~Iled.
liJ 'Ih: bbdinf:{
X-
or rhc b6rizDnr-.1J
;Ix,ili
h:r.~l;(:-s).!lt;ed in lErilj~!c.3ding picrme. -./
r nn!y
fA)
n onl:_·
(SI
4) Hi
iiniy
n)) i ;md
IJ
(E.i None
or rhe:J!Hwegiv'
o
the complete set of [me
L Hud1 dorplets ~:!1ds'remplots licrs,
£311
reSpOJ1S(!S.
show f,:yn-imerry, 6'dP-S, dusters;
;!t)d
V""
1L In hisLOgr~lm:-;. nilanve;m:J3 correspond rcrelarivc fre!.}lJt;"Kles, V' / m. inhht6gl'ril1l:';. f)'e']tH.:nci
1 (l rH}'
\,i
CB) I
LI
,lIlt!
lli
J
n» Jl ILl i.n, ;md nr L AJl
Sj'trl mt'I'Jit, h-i:m~gf.;;m13
h:lv,~ single
Psal;$; 'I--
I L All "ymmdl-i( hdl-~hape.dtlJrve~arcnnrrnal, v; IlL All nnrmal G_UlVc::\ Me lYdl~shapn! and sl1TImer~l.c,
(J\1 1 only (8) U
Dill)),
{C) Jll on~y
(D) J
;1J:lJ
1J
'(Et None d' the ilbnvc; gfv.cs the cornph;re:;c'f
Oflfll~
r6ports~s.
our-
•
L. _.. 11'
7. \\/bich f)f the f~)IIo\vlng distributions than skewt;d to the lefe .
robe skewed to the right
ll'1orelikdy
L HtHJRebold 1nC0111(.'5 ~ IL Home prices ~ J It Ag.cJ~of' tcmage drivers
~r\~_jl~'I {Cl I al1d IU {DJ Hand]n (E) 1. H,. ;md
iu
$, WhlCh of dw follo\'
Il,
-,
-..,
.
Dhph\Ylng oudiers is IC5:i prob!ef'ilit:ic \Nhci) iliiu:ghis\'qgrams t.h"...n when
x:-
lJ~;t\g ~:iTt'!)lpl<.!\:5,
lII. Hlsh"Jf/,lHn are more widdy used lhOil1 sh:mplQciqr 40tpk:its because b i~~~ t(tgnilJiS dj~pia y [he v;J1Ut~~ ·of individual.observarions.
MJ l onlv (13) H'{~nly (C) IIIoo1y iD) 1 ;lOt! H (Er 11and lIJ
I
~-
I
I
I i
t
I f
l
I l
i
Il
I
I
l'
ee
j
60
I
•
80~{)
70
~W'bich of the fullov'/in~ srarcmcnts
are
I
i I:
lOO.
trut?
Il,
Tilt: mit1d!:e Imedia») score Vi,',1.:') 15,X . _ Jfthe pa..5-~lng score was 60, moststltdents;'failed.
IlL
More st,udem!;
1,
lQO. ~ {A)
1(01)-'
CB)
U \lnly Ill only
{Cl
@/D ...·.}}i. n~lrHJ n I .M L 11, and m
$('0>.;
I between ~O,~!)~~O
rl,ianbt~\>;.i~·C.J190 and
-"'-~_'Jr~,"
14
AP Starlstics
a hiscograiTL for the agesof aU fam!}y members inhouseholdswhcrr the parenrs have been married less than :; yeil.n; and a fe,,,,.of the famJiies have
.IO. \Xfhi,ob is
srnallcliildren? '
le} ....
----/ 11, A
y
/ \",.:,",
.,.. ".
ot
hiw"6gtlln1o'f rheeducarional
levei (in number 'tyears ofiit:hoollng) d-w adult pq,pulidbn BFt:h~lJnit~d3rares';!
If 1, Ske,.,;vnes;)
J
eo rhe
lefr
v
fA} .Ilonlv
(B-J IMd:1!
(()) Hand'VI
r, 1l,and'IU
lE
1
ro
:,~
2. ~l 0 t 35','(1' TYl8\ 9 ;3 ,f}1 1 -2]. 3 3·4 4, :; 6,,{ii6 '6 7 8 4 o 0:o 1 ], i i 1 i:'; ,f'3 .1'4 56
1
6 70
a 9 '9
67 7 S: ~r
i9Ap P:2' f ~;f~;,"'\;~>4 -4:5':1 s e Cl;,~?7 .., 1 7 8 KB"S& en.e l11 ~'2~.3 4,.q ') ',')i567 7~ ~::J~t 12 ')',6
s 79
12
'~'7h~t'I~ th~triiiidle{m~rli~n)i:,fthi~,§ct? (A)
55
(B~ 5.4
(C) (D)
62 6:3;-
{E} ~4
fj 1P)~)'l)'~'9
~?,;i;j g,'9
,
•
PI,
Ouessione on Topic L Graphir:;ai Displays
13, The ['ollDwing hiswgr'\nl ", crn isrs in 1967:
--+
Q...
!
I-
I
!
I
indicates the relative frt{jucn~le-, of ages of U.£. sci-
,
I
.
I
I
I
. --
~
--+--;---
•
\
Approximately age? fA)
®
I
-., -
I
I
I
~
-
,I
I
I
•
40
3D
20
.
-r--
I
.,
.
I 'I .
I
I I
gu
,
60
+ ..
I
65
, ••. -h~.rpe.ce;niage of the scienrisrs are: berwecn 25 am]
4{))w:~re;~,f'
IQ~'b
15lJ~) (C) 2'J{J{)
(DJ 4tWb { F.) .::;.W!ii! Answer Key
.,1
B
A. C
7_ .B
C
).
E
B. A
;~
B
(1.,
C
~.
~,l ,
la. B 1 1. D lL. B
C
13, E
Answers Explained l, {B) The low 5Qtl,K5Qf510. 57)" and 610, farfr-orn the r.e-t'nainin.gscore-f,Whi.i:lr are all bt:nveen63:)lahd
(iSO, fllake rni's.di;s'trihllrltHi
skewed to ;h;;l~ft\
2. (C) St(;mplors are.rior used for can~,-g~}ric~r~dar4 S~iS and are be used. H~Hv~i'y larf:,~
~;'l.la $et~.
.
I()(}
' .:
ull~vieldj: t~·
'.
J. {IW Thc'v'i;'ttjeal ID;:i~.·s[iriiJjg.;u$71,A1(YQ,resulrs lr would he be-tter ro srarrar $0, not$78.0tJO.
in a mlsleild] (jg.!I,ab};p'knv-e~
wh-hdiffcremyeM spans results ina mislead-· ing picture.Tht number of indktnierits per yt;-n i's actually tnn~tng. ..
4.(.C)
Labding--the'horhontal.;Lxi,s
5. (B) In general, hIsmgrams :JCCU:1]
give hrtorma.riQn~tboU(
relative [requencies,
~:t(H
IT-cq1.Hincies.
Cl. (C) Symm~ ..tdc hrstogr.;:lm:; can have "my ntlmtler or p,:aks. All normal C{trv\?s: arc bc1I~1ib1lpettand s)'mmerrk; but nor.all ..symmetric bell-~h~pod curves-are norma].
15
i£
Ap
St-~tjstiC$
7, (B) Incomes and home prices rend t9 have iJ. fev very hjgh scoresthar make the- disrriburions skewed to the righT:.. Teenage driven mostly have RgeJi"rn the lasr I~t;n~geyears. with a sf.4n~:rlftg:of)"ounger dl'ivers, and rhus :n:listrJbudOH
5k.cwcdw the lefl_ 8. (A) Choice nf width and. number ()f classes changes rbe Hpp1·,aJaIl.;::,e of a hismgntm. Displaying ourlicrs j51f)Ore problemaric with his1kJgt(-IITlS. H'i~t9:gtams d", nqt showindividual Db$(~n:,i~rioIls. 9.
(C)
The median score splits d.'1e_~r(;<1in half) ,11;.1id so[h~m-{,'dia,n iSJ10{ 75..,The, ;iie~ berween 5~}and 60 is~.rt:at{:r tha.n the area bcnvecn;90and i no but is ]l:f;S th~J1lh~~1fe;} bdv;reefl
GO ~jd, 1eo.
.
10.
(B) MOH of the fml~'lie.5 proh..tbty have nrx children, white a ),ou:n,g c.hildr~n, bence tht: -smillbumHon tht lower ~ndqf(BJ+
1L
(1)) T'h:ert>\vill be' ch.l~l'eg around 8, U, and 16x(::ars of schQoHng.'mnll.segmenrs .of rhc population \.'1,'111have '{IT!' fe\-v yel'l'S of sch0cilingi a1Jfi so rhe drstrihUtlOIl ,;,;111be skr:v,"cd to the left.
httye \~ery
~w
us
l2. (B) The scores art already ;::n:r~nged fi1ascending ortkt. Cqullting down rorhesixrv-third ~ . ~ .. $C{)fC shows-rhe median to be 54. . .-
'."'
,
-
-.
.13. {EJ ,
'
I -
I, l
! I
! ,1
I
i
I
t
J
lip
or
t £ or! Topic 1:
Ouestions
Free-Response Questions You
Directions:
l'}lUSJ.
sho)"IY' all work and indicate the nt(tth0{is
will ht:gr~ded 011 the )::;iirec-rne',Ss':6fyotlI final answers.
'. -
-.,
-'
}'~lU
'met11~asahd onthea~~~~~Y~N, . ,,',
.. -. "
,>~,
>,
"{-
Two Open" Ended Questions l.
')
..c,
Ao.:wdjng to .rhc /9£)1 IV/iEP Ttitl! S{at-e./issessmei __ lt the <1.n::ragtl mathematics: P'fOhcithcy sto):e" jn Eighth t'T>ld6 f-(}i ~ll ~t~tcs ,'{{:ereas fOUOi;,"S:
COi1~irkr i1i~,f~}J6\'.j1ng ni:I,h--d.er rates (per 1 00 ,0 OD people). 1\J.
A:K'
.f:3,:;j
-
IN lA
'6.2-
NE
2.)
1,6
NV
15.5
l\.Z·
12.9 9)i
NH
.c;,~
9:[
~ S-.,(
5.:;'
i\R
8.9
1 L'J.
l.A
NJ
t5A3
C()
1'3
.1\-rE
~{
;MD
8,2
tiE
,4.2 ;Cl;?
(.:r
-'.'-'
xs
'1-7
lA
NJvt
5A H1.2
NY
Hkl
Ne
1 0,.8
ND
1'/. .•..
OH
2,0
OK
6.9 8.5
RI
se
15
SI}
L9 TN 9.4' TX 14.:r UT VT 'v':A
0:1 :3:,3 ~L8
'«Ih
4.&.
FI.
lLD.
'·MA: :1] Ail Hl6
CiA H1
-l~:4
1VtN
1v15
12.6
OR
5,0
1D
'6j ~A
WV \Xi'J
',tYf(;1
10,4
PA
6.2
\\7)'
Tt
~:t9
~\!lT
4.-8
4;0 J
» B P:;.
c'
:25 7.1
Greotuc» t Disotev»
17
t
£
18
A,P Statistics
Answers Explained 18
235
27
0 1 J 2234
16
u o l' 2 4 Lt ): 5 6 7 7 7
25 24,
{} 9
rs
728
44778
H 7
9
8 99 9
The chU~tare roughly .ymm~:t[k and bell-shaped; rh~y art, (;(n~ef\:d in rhc 26Q~,\\'~-rha spt~ad fmrn 246 eo 2$3, and no OlH!iCfS, clusters, Qr.gHp'. ]., Tht~gidih3; i~~y.·"e11.0 and.18>$e~rm;: reasoH2Mei.aldlOU'gh manyother choici's·)l.l'CP!;')S5;bIE_ ')n the TI4'33,f<jr mWiple, pmtj~ the data inro U .. mmij1~bn P16tJin . TAT' PL01~. pitkirlg the, hI~lQgra:fI1 {rum among the six· type ,chjjc('~; ·dcsigrraL1J1.€\ 'Ll fOf X]JS! 2ticl I fot Freq, lhcn either Hsin.g Zt;>.gl'l'l$U:r ,pi: 111 \Xl1NDOV;'~ ~rting XmiJ1 "'. 0. Xmax « Lx, Xscl "" 3, V min= ()., and Ymax.,,< 1 )H>suin. in [he feilk)\.vil1g GRAPH: I
I
.
\ t
I
I I I
\
I
I I !,
i
.;
I I
:6
. \
i
, ;
I
i
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15·
Th~<<.U~tJ::~btlt~nn. h, T19f ~YiI1m~m0hut rather Ius 2; slight skew w' 'tht Iighr, IndiC1lt1r~gthat :there are'~ ftW'kt:lt# '\vlrh much higher tl)'LlrJcrt~t€~th:ttl fCiunditl tlje ~rrtttthi!,!g bUlk drtl~SXfHt"5,
hical Displays
Gra
lUldSPH,l-ad
• D(>tp!tJr~
•• C'eJ.l[l'
•. R;lr (~lta:f{S
" Chtstt::r:s and
• 1Il's'rClgr::un~
" ()iJtlL\'~hi.
PINS
• Clm1ul~Hjvl7 f:requency
»
G:tps
1\1,or.R:s
•• 1i!L"
• St(wTplqcs
--
...
, .,)lh
here art: :1 va{l('t'!' of w':.q). to Drgmize fUjdc2:rl'Ml~zdata. Much intfwrnatlon can b!,~put 1nto Glh!es hI( these <11'fl.1)'& nfb'tr-t: tigpres renJ tn b,: !'pii:i~iesS)lTJ(.1someI j me'; e\'fn lorbi·dding, Some' j~Jrm {lfgr~l.phi~l display .is.ofren best tor sc£in~ parrcr nx :wd shnpesand hJr presenting an immediate impression oFcve:ryrd1illg abour thc data. Alil(:>ng rhe rncsc common visual repj;~~~1pafi9n~ of da{;~ !'rc. dGtplots, bar ch;ln". hhimg~;'J.J1-H. lJmj srcrnplors. his impOrta'llttCU'emcrf1ber that all gmphjj,'!ll dispLty;;. shouldbe dqrly b.bded, leaving l'ttidc1/J,bt ,>,,that the ~)iC:ttw: wp[es~m,i;-AP St(!r1s(jICS scndi1g guidt's harshly penal};r.,e the-lack ofridJ.::'s :\11<.:1 labels! i
DOTPLOTS DocpMlfi
ual
IZh';H[~
J.tcp-llIrimlurly
'tlsef:~itw:h:h I'eg;~,d ro
i7i1t~g.f);J'1'£lll (or ca:ceg:ory to \vhkh L-:tH,d, tl1d.iyidtf,.'qii4irtt14t1P:e Itari{{bt(.~(, whi~h rake' on
Ilfl.;ri(l6/es, that is, \'<Jft
(jifa/fmtit!{')
1:)dtl11g£,
'1 'his
1:Y
IJI
corrrrasr
n \Ili1t'lflr~1 ~"l.lU;t$,
Suppose:thilt in wGiass of :35 .studems. 1O>choostiba$kiMtiall 'BS H1eir favorff~:wm'l whne 7 pick baseballS pi1;kf00tbaH, 5 -pib-l< termls. 5pt~,~QcCerqi.,~2pl.~ htJckey; Th~~~~i;il
.. ••
..~
l'
.:~
+
.~
;,.
..'" .,
8
~-
,.
••
'!':f.
.,.
Bcseboll
·f
B(}$k~1bo
." 1t
Footbnll
Soccer
Tennis
1
TlHc dDtplotGul also b~ drawn wirh.a \-'erti~1~4X:fs ;md bbflZonr;JJ t'l;:}WS of dots. In bG, in almost all displays, verve-ill and horizontal tan be 5¥iircJu:d depending. upon which picture seems easier to reader simply which b,ect!;:ifirs rhe Pttf;iL
BAR CHARTS f\ common visu..~ displayrocornparc rhe ;,17.£;$of cat~gt:ir,ie.~ OJgroups S~i:,esam be measured as frequepdes or as percents, ''
is rhe
l1(lt
ebart.
in. BSut\lE)yt.ak~f1d~rij:tfllil:e tlrst Wft~kof ~3nuai)l'1~~~. SpoJ6 pflt)p§e:sl!r',;~ye,(j~anleDthe c!rf1tonlmpoofhfh~,ht \1i~1tl:J)90q frhtn~cJia1fjly, ~5% \llrdfll)l:q ~ t9~~n'~i~ with \¥itri~s~9s" 25% \V<;.rit~,'tn-e trfal tQ:c'onilliu~",1moutci.dljr\g Wltr1iF~s:>\mi~ 5%~xp(e~s~q tlOQjJlnrQn. Thes(itja~c,an,,q~'a,s~!a;y~,~ ill ihe IQllo'A~ng,bar than (or,qaig(.sPn)C '
+ 4u1
"
:5,0, f'ei'cer;t
of those
svrveyo.cl
30
20
lO
1 1
~
rContinue with wilness-es
ThECfr.eJatlV€ trequeneies.ol
(jllferentfBsults
n I
I
I
(ofinllue
NQ
b'l1IY~r1h
OpInion
nQ'witrie'S$~s
are 1n€licated by th& tlei,gr;13 or tne bars r€iprB-
}1entif1g!h~~a fa~lbt
HISTOGRAM$' Hi:(1;(}grattJ,J, US:
on i:l'tieqrr~inggi~'enh!Stt?grat:'))S,
--------------------~======~=O
"
Suppose ttlBH3 flrEF2£lOO iamilles in Cl srnan town and the djsttibutien of childrefl among is 8& touows: 300 r<'IfT,illes areehildless, 4(.\(}have oM thild; 700 have Iwochiidren.300
tharrr
thre~. '100 have Iour, 100 r~-v8 five, and 100 have six. Tl'\eSe'
i(l
,
;
hav-€i !.he .
500
__ .__ 1
•
o SOr'i11iitimes, insteaci more m
l~e
n1€Bflingful
~j
to O$€
populafior1,
'2
JaI.l€iirVSlfh~ VMiC",j axi$'wllh'ftf1fll;len~j!:!s,
!! is more conv~n[S:lit
(fr
roJative treql}em-;tes, thrl! is, 1~{I3n.CWs'dMde{j by tt'ie iotal flumb~f
.
.
400/2DOO.
2
= .£'00 350
nXlf2000-,,;
':'Kj()i;~O(JQ=, 15'C)
~t~r\.9
10or2000
=
100l2tXJO
ss
iuit'o
100/2000
=
1~,
,1
-
..
~r
>
OSD
.050
I
-:'''" ",ן~f-
.050
i
nl
I I
1
, o ,
i.
I
•
I
I
!
I
'
I I
>
I
""
I·
I I
I
i
1
Note Ihalthe sh.ap~ 01 the tirstoqraffi is IM .-~,e·Wnethef the vertical axis Is laoeled wlthfreqpgncies Of with ~e!a:Hv'<3 freQUe1lGles;;Sometimes we show both f requencies an;d relativefrequem:::ies on the<same;:gr"lph.
4
AP Sttttistlcs
Consider' the tofID\'."jrlg~istogfa m dispiey[ng40 large
salari~s pEtid
company,
to the tow1evel
ex€()ulJ\fB.S
ola
··t .
',3,0 ,25
-
,20
I
.:
. ·ReltttiY~ fr~nq
±~05. Whatca:n we,l~.¥r\ f[(J.111 m~s,histagram'{For exafhpl~J nQriEl'.of\heBxooutivi.3s eam..if~ rnqre Hi13n$fjQ.QQ9'dr li?ss thit1:$-<~0,OGO. Twelv8.earr:e4 ti~jw,eBn' $50,{)00 ~mO:$qCl,QOO.,Tw;;lnty~
$SO:o®, N9!e
five P{j!fGel1t ~~'rci'lSdbJ:}lWe€n$40:000 and 1)\ lteina ,(~'?'i.?fl~' f~1JjflgbefiNeen eenain
How tlii,Shi$1ogr
b~f
~ve~~ th$17'{J(Lti!(rrtfli;$$ had '€xactfy two chlldr&n,
Ci,J~idefth.B.f~~~Vlt19 :~t#Q9ramWh€'Ie 1M Iea.f[lfr'Q'rn1.tiI~:~q'gf~mt'
v:eJ;ti~t'~i~h~notbeen,lab~~'t i,flih~~J'l'~f?'
I
! i I
25
26
27
•
&
10 Qete~if'le the actuallreqaencies; Il'.)Wf;lver. we can determine re!:;:ttl'v"e frc;quencfes by nutir::tgth~ traction of the totat ~n?a that 1s overany rnte''l,ai:
AffSVl'er: 11is impossible
ttw:
I
1 ~
-
,
,
-,
.
I
2-1
19
28
V'le can divide the area jnto~:terfequ.~I,p'onion:s. 'and then not~,'that
It qr la% bnh~ ersa Js,
aC'l)\o'e2;:,-2$, .20%jsab~'JI1€t·.~ii-2j .:41)% fs·a..h.ov~27-28. :affe SO%!·js .aPQ:W 213-2~, AltoDugh it is usuaJly riG} posstbie.;{o,diItID.z hlstogtah1SSQ nl~$'lylnl6-ten equaf are,all, thlij prilittipJe 01f€.Jativefr0qU'.encr~s COf'ieSpof)qing 10 T~la{iV,:: aH,:aS still B;P1JfI~. ' ' . .. ;
-,'
-,'
,'-.,
Rd;uive fre,! ucncie -;rrt:iht: 'Us;lill choice when <;:mlJ,;dring disrrihurions ferv':M size pcipuhn inns. '
of dif:
CUMULATIVE .FREQUi;NCY PLOTS SQmerimc$ "-'vC: sum Freqo<m..zieS,and ~hdw the result visl~allyin(l: tJ{.mtila~.iuc{l-(;q[te!lo/ ,/,>ioj {tlho krKfwn as an 0t/ut).
.5
TO
!
I
l-~
20
25
eo
"l
35
40
W!"',at can wa !e<J.fn'from thi$ ~lJmll!ati\teJre.qvenQY plot? FCH examp!.E:;,: going up to tfm graph' ftorri ~ge 5, 'Ire see that. 15 -Or 15";<~of school emoltrnem is belOW age 5, GG1ng oyer to the gri:!~h from .5 on \he vertlcalaxfs-, \Nt'l see th13l 50% of ttre acM()j atTrolimenll$ b~fQWand~ 50Q,~is abr)VG a middle ag~ of 1'f. Going u,;rfrDrTl age';30, W~$:B~ tt!}aJ~95()r $5% of H'!i~entaUmenl is below ').ge 30.snd
thus '5%!sabOVe' age 30. GbJ.ngever trq,rn ,?S ani:! .75 on ihe v(it~ (conli(1l.1t?d)
\ iiGai aXIS, WC:isee that the middle 50%, pl.school fQwer 81'1d n.ndage is at the upper end,
enroilrnent Is betwaan ~ges 13.an{l? a1 Ih",
STEM PLOTS !\hhough ,1 hiiifqgr;:iIll may ~hvwb(HV many scores l-:1il IntlH'ach groupint; ur interlh~' cxacr values of individualscoresarc ofren kx.;t, An alternarivc pkt{wbl {ll5~
V;11,
pL1Y. rCllkd
i1 :'>[cmpiur;,
inform;:nion.
ret;{·'nHhJs.l111,.hvidu::),f
COfls!denhe:sElt{17,1£. tB. 13·;a8~~~, ~1,7Y. es. 5Q,'43, 37, 2.4}of pcircent8g~,oflhf~-. matJe by j,••liohael Jgfdandurjrig'~his t3iY~aTs.\
pOJOi shots f...'tJ
OLglt6T~ad~ value from trJeQrig.ifl.ljjl
pfB.tehoklers fm lC~2D,3D, 4O•.B.lid'50~Lts1ihelSEt
set <1P,sl 1he appropriate 'p1a'cehofdJiL . ." . Il;," rtfsu\l r,5~.stef!V:dC;(,{uJiQ ~aneo a sJ?mabd-ieaf cfiSP!~c.!j cif thes~ data: St'l:!m-s I 2
L~ave$ I
7
7 7
:3
8 3
4,
,3
5
{}
8 4
J
5
7
Note lh!:lt the st~mpfot ;giltes:!hB SrJ3pe (lftbs t~!l;to&!rnfrrarl!:i, llnllke .tJ¥lI'\i:;;t\ilgr~m, rr1l;llcat~s the valuer, at th'e OljEinB! ciata.
by rearntti.glng '[be runubersin each n:)'\l;· in :.:xmding.orde:r; Tki.s' (~rd(i'ed.d.i~ph~J'sho\.vs(l sktoridlt.;;'/~l(}f inf{nma,ti~)tl 'from d1t~ Somcdrm:.\ turther
~~'l:tiUiP":
is-shown
ilrigrnal Mnnplm. . nh- rt''''l:;cd display Df the :data.in Example 1.7 is'~s follq-,\is: 1 2
s 4S-
:}
"7
4
7
B
I
')
'7
3 O
(
7 I
"
g
8
EL
IT
$tJ~POS€ lnedistribuMn
PI
01 25 advertised home prices (in thousands OfdQIW~) in a certaln
community ISgiven by the.set {56, 69. 1135,73.,'83, ;45, HD, 188. 127,77, 1IQ,11·::>,132. 120, 94, 130, 8'~. 65,.99, 154, B6, 12H, 12'2, 10§. 130}, Onf;1;possible sternplot of Th'6S-e d"t(lis
50-74
56
75-99
1;)t)-17A
17 t33 '1;}4 36 -89 Q3 Hl 1? 20 ~ 21 :ia 3{) ~2 .41'5 M 9&,
175-20Q
89
1-00-124 12&-r4~
6<;
73 90 9.4
99
"r,
"!!4.'.
Noi~that the S!o,imSWerB chosen io btdnl!3ivalsofIenQth251'ind ftiatth~ . 1 iir 1G'{lis lell ()~1 . of the l~alies,s'O lhallMy aJik}f'\\iJ:!:J1i!':aIly, ~'
,
CENTERAND
SPREAD
Lnoking ara graph It:cj display, we see
dUi !\-VU
imponnnt
ll~pecf5.
of rh« overall p:n-
rrrn .lft:
L rhe csnrer: which separates che value:" (o,r"fr~;4 under The (LID'" in lilt:' :l hi$t~gnnll) [uughlyi nhalf; ;,tIW 2, the J1Htat:/~ rh.:u is, the scop.e'~}Fdw \ralues.from:smaile.'H.to i<J,tgeSl.
C;l,SC
of
or
Jn tht"h);~l:0gr
CLUSTERS AND GAPS OlhcriInpnrtlnt
aspcessof the
0vt,;rilJ RjlITerrt~t6
Ldy :~~,,\~Ihich~h(;\:.;'narurs] s-~hgmups:'lM~)which the-values faU f!{-jf examp]e, [he sabr~£s of reachersin ImaGa,NY, ful! inio du-et (lyer~'lpping clusters, onc for puhlic ::;6h0t}1 teaehers, a :higher DI1~ ((,u IdUIGI C( liege professors, and <111. even higher one fo,r Cornell University prfJft:;i;$'ms), ~fld 2. ,el~ti!; which show IH>ks where Im vsj~esfil1 UtSr example, 1 he Office of dw tJ~'ill5,en,d!i letrers m ~al(kms being l}ut on the honor roll and to those being pm on
8
AP Sla!/siICs
I 11
~I
i,
iI
11
!
11111
I
!! I j·
I
I
,
II ,
1
I
J
,
..1.'
35 Simplyooi'lnginalth~'Geroef pf llie:rlJsl1ibu.!iem,fS ~itrun~r42~dthesprli;ad'l$ IiornJ i ?2CJearlyriil$$~$ ~oTn~jhinQ'T~~ vatU€Sf*II,~t();t\~(idi~tf~~q~U~luffi 'With~gap~fN~E:;(L
OUTUERS
Extreme v
-
',:'
-'-
-.-
-',
-'
-
:
-
-.;
'-.-'~'-
.-.,'
- .'
'/~,
MODES Somc·d15tribWtlnnsh:axe
·6n~'fJl'rn()re.m
rwo such peaks; the.:dislrlbut'on
CGfisider 1heM~~fam:~f (PhoenTx and H~~~n):
lssaldtobeR/rtim(jikfJot
!he ysar£y a\iBl'~B
.lli.1!f'lf.rdal,
~#i~s~~er 'game
I
5.5
(:1;:..(;r1y
rc.spcctIvdy~
Ilgute$f~r Jetf
Hcir'r1ac~k
I' 9
NotB thar thls,'is' a b:modal drstlibu!iorL The tour.assis! s,vB;ages bet,veen 4 ?nrl4-'5 3clu1lJly correspond to the: tour S8a~O!lS{tom 1994'-95 ~hrough 1997-98, while the fP'Uf be~"(3~i1~ &iid 5:5 Gt)HIi)$pond to rnicH:fla car~rsii~s6trs, ' " , .,
Consiser the. I1js{O£Fw.1 of the ye-arlY,fiI1era.{jB POInt:? per if~meflg\JfBS for KarJ Maloge; {Utah) trom Ht85-8U 1hmugh Hl9t:'-98,' ,.. ,.. ' ,"
NO,~tl~halthis i$;:i unimoj.ald]$tfibuti(.in withiW'q Dl!n~-er5,The 91)tliers.as. migll t?,0 g•.lP,,$ScBd, Qar/esp,hn'd to Matone:'s 1ir?i NO SB:a~o~s es .., pro during which he B;VeG3.gw,ooly 14.9 am:! ~i ,( points per game. Di$r~Qafdin!J these;tw6 D(JlI lers, iNBsay Ihill thB- distnbut1(!Jnisfl!mO~t
sym.me.mc, N()re ~h:;n, \1$ illustrated ;lntWc;il ; . IJ~tl;tltr ins: RKtivl;:' [.6 Inp~f(ir reasons behind t)utlktKand modes.
SHAPE DisuihUl.1(JnS come in fH1tIJ;;,rHSarcwonhspecial
:U1
tndj1;is~ Y4f1clJ
of $t}~lPl'S.; hQ\vever.certain
common
memiore
l..
A t'ymrncttk dl~tribud(}nhn:nt
in which the rwe h$h;t,'S arc injITorir),~
2~
Ad:lstdburhm
~~cl;.!)rher. For-example,
I~ :flN'{IJ.'!(/ {.fiJjif' iiiht ifirScpf;t":a!nE\)-'LiJi~
t11inl)' ),f>\v::rrdxhe
!'ltht":r>v
5; AdisrributioH
values. For
thehigheI end with kw:
10\'"
vnlues,
4~ A· ~it:1j"'Fjl/.tPtf:i Jisrrihmjl)tJ B;:5ynip1:erric )vlth n center iUl111l1d (lnJ two ~16p:iligt~it,,;,For example, the ~1i:5(rihU(ion 'JflQ scores acsoss thf.'gener-alpopulation isIOUgh.ly symmctri;~\~·ith
is
un{t?n-m if itS hisrogramis
ahOl"liiJutaJ line. For example, many s:pors (p:ip~c) ~pp~ar on rop yields a withl" thruugh 6 flll equally likdy_
t{~$sing>::lf!liJ die.: ~nd noting. nnihmn
distriburion
11Qw
\ 10
AP
sienetice
I~
Symmetric
'/~\ ~
/ 'Skewtod
SkC'wed to the righl
\\ 10 lhlO
left
<; ,/ 0\<. Boll~1hap..'-i!l)iiif?tm
Even~vhenab~gi~ sltilp,eis J10Ied".it is ;n1pbrrill.1[!,l1:kH~;riQt.elfseme.ofthe'd,trn ~r~ from r:his shape.
L--
.
devl-
----.----.-----
Questions
on
>
'.
10PlC
1: Grapnlc,<J. .;,.
"''''')
.~
.
.:
QuestIons on Topic '"\: Graphica\ Displays Mu\t\?\e-ChQ\ceQueSl\OnS, Directions: The qnesrions or llKoX:1plete SUltl.-Wpn,ts rh"'-l f6nQwateea~J61lowed by fiyeSl:!gge~ed:lll.S\f(erS orC:O.mp~e!i(}ns;Ghoo~~' the' responsethar be.~r an vers the question .or completes the"S:tate.mtlH'; 1_ Fnl1vwing "ir~ dl~ SAT math si=Ol:~sfi.)t ~ldl,'\JJ':St4ti8tk~class :of 20 students:
D6.'L ()5H. 6HJ,. (i7L), £i40f 643. 675; 450, 676~'5'70}6601 (?6T~ 510, 667d)?~:~
645, ~,nd6)O, The di1'trib{Jt~ont)f$CfH~S.L'i
(l,l), '67 l. (it;)} (A) symmetric,
(B) <,;kei,v~drorhe
JeJL
(C} skev ed L{) the righr, '-' {D) illld1Jrffi. 0:'1 ',L,
bell-shooed ' ,,' j'W_
L - templors ~H>{:u.scflJJ borh for quantitative and c;llcgoricai dam sets. 1I, SrempfQt$ are equally !J$du.l fors.B1~LUand Y4ry large dat::t sers. lJ L Stem pl(iL~can shOWsY.ll1.meu)',g,.-qps. clusters, ;lnd ourlicrs. (.1\)
I onlv <
(B) 11only (e} lll.only (D) 1 ;md
(£)1
11
8.nd lIT
3: Consider
the Folli)\>".ingpi:cnue; C'.
-falof ~c.le-L
_.
-
--
.~
l 19N
~rh(ch uf the foJJmvingsraremems
1;995.
1S'~6
u'r~.~tiH;r:
J. Total.sales .in. 1995 were two Limes. toralsales ~n 1994. while total sales In 191)6 wercrhre« times the 199,1-r{]lf'.'tL . I]. The choke of l~bdil1g for the ~eniC4J ;'>:i5 results 1n J i''lJislc::3_ding sales
m.
picnlre, A.hl:f(0gr~rn sho\ving rhesJrut:1r1h)fmab\j/1. but [hi:; time with verticalaxis star:ting ,11 $7H ,000. \~'011ldlH: less m.i:Jeading>
.~
10011,
c(mjlI'o);ly (C) III on~y
(D) JI and III CL) None the above gives rhe complete
or
s!?[
of rrue response:'>,
(l
Dit;ofavs ',>
'\"'\
Nt!:nb~r of POf'S"DfU ir-i!ir:1C1{r ~D!
h,hhc-
~tt'rl'pIJ~n (!.;>cc"",; ~c.t..!hf
\'\/lI!\
b of d~\; fD.]Ii",,,,jars, sraicmeors
L Each vc;~r frDm 1?77 to J986 l'
dJ('.
-."
,A
number of indic;tment5 h.l$ ;<;'t:~ldiJv !
11 \);~!htlc the l1ull1b~~rofindluTnems has dc:di,~~~edr;:a.dh Ye'.:Jf,th~;;lllH)Unt of' d~'Cf{';aii~ hA~k"~~Iled.
liJ 'Ih: bbdinf:{
X-
or rhc b6rizDnr-.1J
;Ix,ili
h:r.~l;(:-s).!lt;ed in lErilj~!c.3ding picrme. -./
r nn!y
fA)
n onl:_·
(SI
4) Hi
iiniy
n)) i ;md
IJ
(E.i None
or rhe:J!Hwegiv'
o
the complete set of [me
L Hud1 dorplets ~:!1ds'remplots licrs,
£311
reSpOJ1S(!S.
show f,:yn-imerry, 6'dP-S, dusters;
;!t)d
V""
1L In hisLOgr~lm:-;. nilanve;m:J3 correspond rcrelarivc fre!.}lJt;"Kles, V' / m. inhht6gl'ril1l:';. f)'e']tH.:nci
1 (l rH}'
\,i
CB) I
LI
,lIlt!
lli
J
n» Jl ILl i.n, ;md nr L AJl
Sj'trl mt'I'Jit, h-i:m~gf.;;m13
h:lv,~ single
Psal;$; 'I--
I L All "ymmdl-i( hdl-~hape.dtlJrve~arcnnrrnal, v; IlL All nnrmal G_UlVc::\ Me lYdl~shapn! and sl1TImer~l.c,
(J\1 1 only (8) U
Dill)),
{C) Jll on~y
(D) J
;1J:lJ
1J
'(Et None d' the ilbnvc; gfv.cs the cornph;re:;c'f
Oflfll~
r6ports~s.
our-
•
L. _.. 11'
7. \\/bich f)f the f~)IIo\vlng distributions than skewt;d to the lefe .
robe skewed to the right
ll'1orelikdy
L HtHJRebold 1nC0111(.'5 ~ IL Home prices ~ J It Ag.cJ~of' tcmage drivers
~r\~_jl~'I {Cl I al1d IU {DJ Hand]n (E) 1. H,. ;md
iu
$, WhlCh of dw follo\'
Il,
-,
-..,
.
Dhph\Ylng oudiers is IC5:i prob!ef'ilit:ic \Nhci) iliiu:ghis\'qgrams t.h"...n when
x:-
lJ~;t\g ~:iTt'!)lpl<.!\:5,
lII. Hlsh"Jf/,lHn are more widdy used lhOil1 sh:mplQciqr 40tpk:its because b i~~~ t(tgnilJiS dj~pia y [he v;J1Ut~~ ·of individual.observarions.
MJ l onlv (13) H'{~nly (C) IIIoo1y iD) 1 ;lOt! H (Er 11and lIJ
I
~-
I
I
I i
t
I f
l
I l
i
Il
I
I
l'
ee
j
60
I
•
80~{)
70
~W'bich of the fullov'/in~ srarcmcnts
are
I
i I:
lOO.
trut?
Il,
Tilt: mit1d!:e Imedia») score Vi,',1.:') 15,X . _ Jfthe pa..5-~lng score was 60, moststltdents;'failed.
IlL
More st,udem!;
1,
lQO. ~ {A)
1(01)-'
CB)
U \lnly Ill only
{Cl
@/D ...·.}}i. n~lrHJ n I .M L 11, and m
$('0>.;
I between ~O,~!)~~O
rl,ianbt~\>;.i~·C.J190 and
-"'-~_'Jr~,"
14
AP Starlstics
a hiscograiTL for the agesof aU fam!}y members inhouseholdswhcrr the parenrs have been married less than :; yeil.n; and a fe,,,,.of the famJiies have
.IO. \Xfhi,ob is
srnallcliildren? '
le} ....
----/ 11, A
y
/ \",.:,",
.,.. ".
ot
hiw"6gtlln1o'f rheeducarional
levei (in number 'tyears ofiit:hoollng) d-w adult pq,pulidbn BFt:h~lJnit~d3rares';!
If 1, Ske,.,;vnes;)
J
eo rhe
lefr
v
fA} .Ilonlv
(B-J IMd:1!
(()) Hand'VI
r, 1l,and'IU
lE
1
ro
:,~
2. ~l 0 t 35','(1' TYl8\ 9 ;3 ,f}1 1 -2]. 3 3·4 4, :; 6,,{ii6 '6 7 8 4 o 0:o 1 ], i i 1 i:'; ,f'3 .1'4 56
1
6 70
a 9 '9
67 7 S: ~r
i9Ap P:2' f ~;f~;,"'\;~>4 -4:5':1 s e Cl;,~?7 .., 1 7 8 KB"S& en.e l11 ~'2~.3 4,.q ') ',')i567 7~ ~::J~t 12 ')',6
s 79
12
'~'7h~t'I~ th~triiiidle{m~rli~n)i:,fthi~,§ct? (A)
55
(B~ 5.4
(C) (D)
62 6:3;-
{E} ~4
fj 1P)~)'l)'~'9
~?,;i;j g,'9
,
•
PI,
Ouessione on Topic L Graphir:;ai Displays
13, The ['ollDwing hiswgr'\nl ", crn isrs in 1967:
--+
Q...
!
I-
I
!
I
indicates the relative frt{jucn~le-, of ages of U.£. sci-
,
I
.
I
I
I
. --
~
--+--;---
•
\
Approximately age? fA)
®
I
-., -
I
I
I
~
-
,I
I
I
•
40
3D
20
.
-r--
I
.,
.
I 'I .
I
I I
gu
,
60
+ ..
I
65
, ••. -h~.rpe.ce;niage of the scienrisrs are: berwecn 25 am]
4{))w:~re;~,f'
IQ~'b
15lJ~) (C) 2'J{J{)
(DJ 4tWb { F.) .::;.W!ii! Answer Key
.,1
B
A. C
7_ .B
C
).
E
B. A
;~
B
(1.,
C
~.
~,l ,
la. B 1 1. D lL. B
C
13, E
Answers Explained l, {B) The low 5Qtl,K5Qf510. 57)" and 610, farfr-orn the r.e-t'nainin.gscore-f,Whi.i:lr are all bt:nveen63:)lahd
(iSO, fllake rni's.di;s'trihllrltHi
skewed to ;h;;l~ft\
2. (C) St(;mplors are.rior used for can~,-g~}ric~r~dar4 S~iS and are be used. H~Hv~i'y larf:,~
~;'l.la $et~.
.
I()(}
' .:
ull~vieldj: t~·
'.
J. {IW Thc'v'i;'ttjeal ID;:i~.·s[iriiJjg.;u$71,A1(YQ,resulrs lr would he be-tter ro srarrar $0, not$78.0tJO.
in a mlsleild] (jg.!I,ab};p'knv-e~
wh-hdiffcremyeM spans results ina mislead-· ing picture.Tht number of indktnierits per yt;-n i's actually tnn~tng. ..
4.(.C)
Labding--the'horhontal.;Lxi,s
5. (B) In general, hIsmgrams :JCCU:1]
give hrtorma.riQn~tboU(
relative [requencies,
~:t(H
IT-cq1.Hincies.
Cl. (C) Symm~ ..tdc hrstogr.;:lm:; can have "my ntlmtler or p,:aks. All normal C{trv\?s: arc bc1I~1ib1lpettand s)'mmerrk; but nor.all ..symmetric bell-~h~pod curves-are norma].
15
i£
Ap
St-~tjstiC$
7, (B) Incomes and home prices rend t9 have iJ. fev very hjgh scoresthar make the- disrriburions skewed to the righT:.. Teenage driven mostly have RgeJi"rn the lasr I~t;n~geyears. with a sf.4n~:rlftg:of)"ounger dl'ivers, and rhus :n:listrJbudOH
5k.cwcdw the lefl_ 8. (A) Choice nf width and. number ()f classes changes rbe Hpp1·,aJaIl.;::,e of a hismgntm. Displaying ourlicrs j51f)Ore problemaric with his1kJgt(-IITlS. H'i~t9:gtams d", nqt showindividual Db$(~n:,i~rioIls. 9.
(C)
The median score splits d.'1e_~r(;<1in half) ,11;.1id so[h~m-{,'dia,n iSJ10{ 75..,The, ;iie~ berween 5~}and 60 is~.rt:at{:r tha.n the area bcnvecn;90and i no but is ]l:f;S th~J1lh~~1fe;} bdv;reefl
GO ~jd, 1eo.
.
10.
(B) MOH of the fml~'lie.5 proh..tbty have nrx children, white a ),ou:n,g c.hildr~n, bence tht: -smillbumHon tht lower ~ndqf(BJ+
1L
(1)) T'h:ert>\vill be' ch.l~l'eg around 8, U, and 16x(::ars of schQoHng.'mnll.segmenrs .of rhc population \.'1,'111have '{IT!' fe\-v yel'l'S of sch0cilingi a1Jfi so rhe drstrihUtlOIl ,;,;111be skr:v,"cd to the left.
httye \~ery
~w
us
l2. (B) The scores art already ;::n:r~nged fi1ascending ortkt. Cqullting down rorhesixrv-third ~ . ~ .. $C{)fC shows-rhe median to be 54. . .-
'."'
,
-
-.
.13. {EJ ,
'
I -
I, l
! I
! ,1
I
i
I
t
J
lip
or
t £ or! Topic 1:
Ouestions
Free-Response Questions You
Directions:
l'}lUSJ.
sho)"IY' all work and indicate the nt(tth0{is
will ht:gr~ded 011 the )::;iirec-rne',Ss':6fyotlI final answers.
'. -
-.,
-'
}'~lU
'met11~asahd onthea~~~~~Y~N, . ,,',
.. -. "
,>~,
>,
"{-
Two Open" Ended Questions l.
')
..c,
Ao.:wdjng to .rhc /9£)1 IV/iEP Ttitl! S{at-e./issessmei __ lt the <1.n::ragtl mathematics: P'fOhcithcy sto):e" jn Eighth t'T>ld6 f-(}i ~ll ~t~tcs ,'{{:ereas fOUOi;,"S:
COi1~irkr i1i~,f~}J6\'.j1ng ni:I,h--d.er rates (per 1 00 ,0 OD people). 1\J.
A:K'
.f:3,:;j
-
IN lA
'6.2-
NE
2.)
1,6
NV
15.5
l\.Z·
12.9 9)i
NH
.c;,~
9:[
~ S-.,(
5.:;'
i\R
8.9
1 L'J.
l.A
NJ
t5A3
C()
1'3
.1\-rE
~{
;MD
8,2
tiE
,4.2 ;Cl;?
(.:r
-'.'-'
xs
'1-7
lA
NJvt
5A H1.2
NY
Hkl
Ne
1 0,.8
ND
1'/. .•..
OH
2,0
OK
6.9 8.5
RI
se
15
SI}
L9 TN 9.4' TX 14.:r UT VT 'v':A
0:1 :3:,3 ~L8
'«Ih
4.&.
FI.
lLD.
'·MA: :1] Ail Hl6
CiA H1
-l~:4
1VtN
1v15
12.6
OR
5,0
1D
'6j ~A
WV \Xi'J
',tYf(;1
10,4
PA
6.2
\\7)'
Tt
~:t9
~\!lT
4.-8
4;0 J
» B P:;.
c'
:25 7.1
Greotuc» t Disotev»
17
t
£
18
A,P Statistics
Answers Explained 18
235
27
0 1 J 2234
16
u o l' 2 4 Lt ): 5 6 7 7 7
25 24,
{} 9
rs
728
44778
H 7
9
8 99 9
The chU~tare roughly .ymm~:t[k and bell-shaped; rh~y art, (;(n~ef\:d in rhc 26Q~,\\'~-rha spt~ad fmrn 246 eo 2$3, and no OlH!iCfS, clusters, Qr.gHp'. ]., Tht~g
I
.
\ t
I
I I I
\
I
I I !,
i
.;
I I
:6
. \
i
, ;
I
i
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15·
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