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~ ~ ~4 DA:~D is* ~ '~ ~ Of ~ ~ ~ Of ~ ~ ~ All .. "~- is intoned to su=~ two related top:~. data =d ~~ Ne~:~er ~s ~ topic mth~:n mathematics, ~ey am both, hower- phenomena ~~: ~ tbe subbed of math~tA ~ ~ ~ ~ ing' =~ and Hi are the Thai finds ~~t ~~ with d~ and chance' resistively. Recent recomme:~s co:~.~mi~ Sc~: cumcula. are una~s in suggest: ng th at =~: ~:~ and: pm babi] :W should occupy ~ mu ~ more pm-minent pl=e than has ~en the ca~ tn the pa~*~4 Howewt ~ =~e of the emph.~:s that t - e reco:~-~s place on ka anal~ ~87 it :5 ~~: t~ ~~ 5~-~ i~ p8~i0~: 85 ~ =~ 0: ~~0i50 Son (or ev= as ~ bag of tnc~. The task of this =~- Is no: to- u~ 8-~n to d~a a~d ~~e in the whoo! cu:~:~_~Nt 370 8~v 81lX=~g attention~'ut to -I this stmnd of I Ideas in ~ ~ that :~CS clear the oVemU themes and strate~es within whIch :~:iv:~:~ topics :fi~ their natum! p:~. A:~.~6 O:~.~ that Is t~,l~c to ln~.~e tea.~g S~ ~~.~: the expenence of teachers and students. Su~o as Or cumculum :~ detached mom that cx~e o~r Sepias hopes ~~t are d:~w Ceil in By ~~Sti05 :O t50 $~5 iS ~Ot UtOpl8~, ~~W Cal presentIv being tested 1s p=~.~y used:! and alds mther ~~n did 6~t O ~~Ct CO~tS =6 S~. ~Q~$S, 1t iS C~V I~ our Am to o~rerIook pmctica:! p - gems and to urge the teaching ~5 .~

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Char ~,~~~4~.SS TO I, CY -of s^~t moor that is A ~ quan:~v or ~. ~ :~s :~t to =~! =~;~n to the d:~ =d Ha ~~e s~ps, ~ well as to -~he adv=~s, in u~ng Ha and champ in ~~e By: of ma In wntlng ~~s e~y -I h~ tned to e~ ~ the p~1 ~~r than the umpian. direction" {~ in tea~ sumacs ::s ~iV due 1:~ -am to rec~on of ~e place thm ~~g ~h data prays in ~~ - Il~ and in : :~.~;.~;~.s at ~s -~.~ngly =~ to te=h momma t~s th~ ~ of di== use, r~er ~ -to0 mp=s s:=ly Us- th~+ i~d tO :~r t~ in. -~.~. smn~= ~ ~ ~ tople News - ~s present Aims economic =d so=~! Aim opin~n pails, me~=T ~mm bow I: ~~es and cIln:~] tn~s =d Ws~ and finance ~. M~v c~s must d~ m~ ~a in morese) on the job. :~ =d ambustn~s use Mop fo~= a~ ~e remIts of a~icultu~ field teals. E~s a~ concemed mth data on prompt pe - ~~:~, qu~, and reli~:~:~* Man:ufa=~g wo ~ sked to ~~ and a~ on ~~ ~r ~~ss mntr~+ The :~=lth sciences ~e w:~h da~ on co~ and c~:~s as -~11 as m~ -Gus mom median! rese=~ Bus:~-=s =ns on date of eve~ new. =~' profits' s~ p~:~, ma - t wsea - ' and moue more. Tte:~e are compelling pumice rea.~.s ~ leam ~:i~ics. As thew eXampleS s~st data are n~ merely =~' but numb w`~h ~ conI=~- The number iO.3 in the ~~= of ~ -I =~s nO inf6~tion, that the bl:~ Aims of ~ bead,.,' is iO.3 wunds e~s us tO ~m=~t on the hea~ ~ze of t~ child. That 15, d=a I: our I- Of t:~r.~.~t ~ -and we =n un~d and interim, h s~~y cams out anthm~1 o~g:~-i=.~. There a~, there~e, stm~ peda~! as wet:} as paretic reasonS to teach Ice in We - ~~* Statistics combines mmpu*~1 ac~ tiv~yf in ~ mean:~ setting with the e:~.~e of ;~gme^~t i::~. ~;hoos:ng methods and I =~:~* Statistics :n the ea~ grades is teach: not Imp ~r its own 53~$ 5~t ~0 it it ~~ 6~ti~6 ~~' t~ 60~ 6~0'p-~tit~-6 ~:~.*~]i~6 ~ 8~7 anthm-~tic and Aching to p=5i6m Ida Teache~ who understand that data are :3~mbe:= in a* context we ah ways provide an ~~e co:~t When posing pr-~ems ~: stude:~:~. Ca:~ati~g the :~an of five *~umbe~*s is an exercise in anth=*=ic~ It statist~~ Ca*~ting the :~n pnce of ~ popu:1~:r mus:c ta' ~ five Ceil 0~10~5 }5 It I;:* ~~-~ combined w:~b ~ look a~t the D *I , the pnces and ~ companson with the price of Other types of

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:~m :~7 :~t ~ Used ~~t ~e practice! and ~~ am -of coke -both data nOt -~.~:.~b- to an =~ emphases on me-- at~- :T~che~ =d developers of Am matena1 mu~ =~se :~ - attain In pmvIdl~ d~ ~~ are Am; to ~~.. {n ~e ~~ =~57 ~3 - = ~t ~0 - ~8 (~ ~ 80~ ~ be used, ~~=tLen~ hardy arm, -such ~a ~0~ : ~:- ~ :~- In the bower ~7 data p~ ~ t56 ~5 t56~ ~ ~~ ~ p~0 ~~ )~ =~ ~~$ =~ 85 g~ ~e cl=s ("Dow Hey chlldren I:~ tn wU: house9~) or ~ as~ =~ 5~t t~ ~07 =~7 0: I 50~0 ~~* The ~~ ~~u Add to provide -~a rather than s~v ** hers ~~Id be -taken Into a=~nt -when planning ln~- 6~ ~ =t j~ ~ 8~ ~~ it* =~ 5~$ tt07 8~0 essence to ~ n~e ~ s~- ~t it :s im:~:~ the the e required to p*~ce Ha not m~ - add the mat~] idea Cat and Ie~.. ln Dant=~.~' *A to D~ ~ ~ On imoO=~t :~S of stOe sch~! ~ ~~s much Amp. m~ 15 ~ :t,{~ ~~;~- ~~:~:~: :~s ~~n t:*,im~g anO c~g atI~:s to pmduce dam m~* ~~) Oral tearoom Em At: ~tics. The disunities ass~ed w:~h -Gil production ~~es ~~ the 5=t :~f =~] w~ ~m ~ As. to c~ *~ *A ~ Cum cul~ maten~ ~ =~:~t p=~6 60~ i~ ~~ 3~6 p=~$ - - =~5 ~t -~ of d~a bY smdent:~. Over t1~07 t=~ =~ -I and share data -ads that prim to their commun:~+ =d school. Com-~:~s aw an :~! means of ~~g and sbanng ~a. ~me phenomena have predicle outcomes. drop ~ co:nm ~ known :~t and the tone ill ta~ to ~~! =n ~ p - ~~ - - m ~~c tner sma~ m~eme~t em>r me 0!~= is ce=~ If we to~ the co:n, on the other han~^ we =~ot predict r it m11 ~~w be~S or talls. min t-~g tS nOt haphaza~. . .~ ~ ~ . ~ ~ ~ . ~ + - ~e outcome -is u:~- Yet Tf we ma~ ~ :~= =:~r -of tosses, the pr-~::~on ot he~s mil ~ Vew close to one~If This lo- re~n,\y ls not just ~ theistical con~t but an ~~d ~= The French natuM~in Bu*~n (~~~) tossed ~ Din 4040 t:*~. Result- 9048 :~- ~ proportion of 204874040 ~ O.~69 of h.~* Around i9-~0 t~e English statisticlan Ka~ PearSo~n herCi=~y t08~6 ~ Dig 247000 ti=-~. ~~* i2~9 5-~*, ~ p=~ of O~3005.

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.~:3O ^~w :w x~Y e Ash ma~em~man Qua ~~, ~:~6 6v e ~~s ~~g - ~ :~ il' to~d a-. :~O ~~. ~~:~* 5~67 ~, ~ p~-~= 0[ ~ 5067 Pheno~a h~g unwed: :~: ou - ~ ~ ~ ~~r Ace: t0= 0f ~~s ant many re~ ~~ ~~ ~7~ "~= A ~ 8 synods ~ ~ haph~- bm ~ des=~on "of 3 HM of o dI~m 0.m the .~e one ~t :~s =~.~d ~ :Pr~W is ~ bmnch hi.. # ~e ~~ ~ fit - n tn :~ Out of ^~! p~s I=s :~= areas of smence :~n ~ch ~~m beh~r I; Ash ~ ~~cs ~ ~ sch~l a~ ~~n ~ if - V ~~t sol~d In- cou=~. Uncertain 1s of course ~ pewter aspect ~ ~] . ~, ~: :s ~~e o~r ~n ..: that is ha~ ~ ~~e :n cas~ stonings. Den =te :~*, a~ ~f~mili~ ~ m~ stoners' e tithe e~-~e with ~e order a~t of .*: because th~r e~s on e~Iv unTIk~y I~ pn~s. Tame weD~d ~~s of ~e use ~~ p~! =~#z t. ~ e nch haph~'dIv~ s ha ~ show th~ our i: of chance pro~un~s cOntmbi.:~s ~e iaws Of pr~fI*t ~ :~* This ~~t u:~*~andi~ ::s very di~cu:It to =~ct by Anal :~=ion. At=-~s to teach prob~:~v and =~:~$~l tn~0 out ~~e intuitive p~*~:~n are ~ second :~r pitfall in tntro duclng data and chance into schoo! cumcula* ~ ~ Stu~ ~~l to Unde,:~d pm~il.ity and i. be~= of m:~s that am not removed by ~~Y Of fawn I 1Rb.e Indict bCtWCC;n p;~;11~ thtCOor 3~d ~~s died of the wo~d is d~ at Iem ~n :part to ~~ limited conm~ mth Is* We mum ~~refow p=~re the whys ~r tte 51~v 0: ch~ by ~~-~d:~g ex=~= -~:h mn~m behavior eady in the :~tics =:~1~. W~:~v, the stu~ of ~ta provides ~ natural setting ~r such expenence* The :~ of data analysis ow: Army probability awl index= is an impotent pn.~:~e for instmetio:n in unce~ai.~^ Artificial chance devices (=ins, dices sp:~) c~ be used to pro duce data in the cIassroo-m w:~h the i:~:*t of applv:~g data analysis 111s to- d:~ ~~e Body na:~:3~e of thew devices. Un:~aint~Y alto ~~n in d=am sources other than chance devices. Repeated meetly" surements of the ~e quanti*~v (~e by sevem) students~ ~r =ample) iC~ld: V3~g 37~S N31~l, I, 3,99~ if. thC: hOi~t,5, =:~di~;g If-* or in of a* group of people. It is perhaps su~si:~g that

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:~m e ~~.~s ~ carton A. =~! ~~s or ;n ~m on :~- :~als =n be Or by- dance mathem~ that Micros she o~= of chanced ~ices. =~e wlm hare ln ~a ls ~ n=t n~ toward I: t~e conne~n Between S~6cS =6 pro6~li~+ ^M ~ leer see ~e mic ~ rieiitem:te ~oln~z~ion In Swish Resee ~ pro hers ~~s =~. Finally, ~~ scow Ice Us= e Car and flacks of p~babd~W to express the c=~= we =n b^e ~ -=nciu~s Own ~ ~a. ~ mmou~ the uset~ n everyday T~e of~ ~ Pasta o~t ran~ dom~ ~s le6s obvious Can ~e ncce~ of dealing gin data' ~ac- t~ a- ~ teaching ~~ut chase ~ not abbe 3= goal of ins - ~~= abbe prob~ilhy ~ to he~ students un~xand t~ -~e maria mther tMn determm:~c c=~n ex;~3~ins many aspects of the worm ~~ :~= ~ ~~;~ p~r ~ ~ mng I: ~= mace JAYS- of her ~= throws. At the =d of ~ touma~t ~e ~e att~s hVe -A throws am makes =.~ t~* "~..~- ut th:s c=~ explanat~on need not be And; pl~ :~g ~ ~~v of O.? Of m~ each ~~t haS Mimi of about O. id of :~sing t:~e or ~~w of bye ~~* such ~ ~~e can edgy ~ Mmp~ chance v~.~ti=- e unde~s~Ong of prob~litv en^~ us to consider the ~ chance rather than seek ~ spheric cause' Wartimes spurious, ~r - ~~' While the a~nt of ~^ Bilk Ike computing has ho ~ imp t on :~emat~ as ~ whole, it has revolu:~:ized the pmct~e of s=~tics.: ~~n obvious 'amp of ~e r~ Is that mOre =~x ~ ~ vses on Ia~er sets of ~m are now ea~. At the computing In has also bro~t about Ranges :n the nature of Aim p~e. Tn the pa~ natisticlans c;~ed ~~rd but computatio-~V te dious a:~s based on ~ Bibs mathematical mode! in order to d concIusions ~~m adds :: in statistics showed ~ co=~ding emphas:s on :~=ming to ~ om tenths =1~. ~w the parade= statistic an.~s ls ~ dialogue bet:~n m~! and data. ~e dam are allowed to criticize or - ~~n talsiG the once nal moms* Diagnostic methods to ~d this p:~ss aw ~ major 5~::~ of research in sut:~tics. All am compu~tio:~v id ~6 the most sv~v adopted make heavy use of :~:~c Id. :~n add:tio-~-

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~e~ ~ ~e .. w.*~Y ~ on~ imposed ~< Id ~=ladon has ~ to new meters br-~=e - m even rune smad Ma se~' ~s band ~:~ nature of so is readily repeated ~:~t~:~ ~es, essay c:~1i ~n increased emphasis an Qh~~ m - - d :~:~al Ma ~ # The ~= of common As Ied to sow ~ searching among ma~em~cians' same of w~m -que~lon He nature of a proof ~ on ~ compmer se=~h of possible ca~ mo ~emm ~r Inn ~" At ~ -~e elemental :~1' b~ borers =d ~s asks Seder eariv u~ ~ ~awrs Ail impede uMema-~ Of hem and arithmetic opt M~i~c~' on the mber h=~, have ~co~ cat and co~m ~ a fling ~* =~g sums of shores h=d Ads n~ increase un~and:~g, ~ mereb numbs the hit In ~ ~rmm~s it :s n~ Or ~ mU$~:~ to urge ~e u~ of .~m ~ ~m -ha -inst~.~n - ~t ~a at all I - ~.~. CoH~ m^w of =~ airea~ ~es ~iversaT ~ of ~lW ~ ~ A ~ [~0 00 0~= (~,0~ is 7 ~ coume, ~ cont~m miner than ~ d~n between ~c~ato~ and m-uters as technol-+ con6~es ~s arrant) Hem -~-e ~m basic It ~5i~0 in t~ Milt ~ ~ -i n~:~e ~ preSents ~ ~-~m of-~a = -~e ~ at Chin each of ~ =up of chin ~ke the~ ~ ~ ~ ~ on ~ te~ of men~ ^~y ~s ;= ~ fit wo~ help us predict the Iater te~ score? Qume upon ~ :~e ~ st~nt would be asWd ~ plot the data and then =~-e the ieast ~:~*~s regression line (~e s~6 Me i;n ~e i) t~r mlb the co=~tio*n coefficient ~ ~ ~e 640 Pe;~s ~e plot would be om:itted t-O s~e time. Mo~ ~ms wOwd require at icast ~ ~ ail As ~ this e*~-~cise Mach ~ ba~c lit OnTy ~ ~ wo~M ask much more ~ them. But -it it appa=nt th~ the Ma ~:~e two out:liers, l~:~ed as -cows ~ ~ and 19 :n the plm~ How do these =~s :-~e ~e reunion analogist An -~e software Ante of the kind th~ ~s mdelv Able -en ~} vanities of compilers pr-ovidLes immediate 80070~7-10~ :8D be viably -dimmed if the comp~r has ~hi-= capability. C~e 19^ although far from the re~i-~n it does not hwe a 1~e i: on the p-~on of the :~:ine or the flue of tale conflation r. Caw 18, 0~ t60 0~: 53~7 is tint infl~mi~. ~g this ~t m - ~s the regression line to the Co-ed line in the Fire =~d reduces the m=-~1~n to ~ ~ 335, ~ut half its original ~1~" Thus the evidence th~ ~ at Fist word predicts 1~r Cilia scores is much weaker if caw IS -is

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A:: is: ~ 1:~0 : : ~w fog: ~ - . ~ ~ Hi: : ~- - :: ~ ~ =~s - - ~ C~e - - ~ > :0 10 20 30 40 so 60 = to 03= 0~ ~t ~ ~ ~t =~5 0t 2t ~ t~ ~ \~ 50~) ~ i) A~ tAve smm the r~k of ~ =~ tm ~= ~ ~ t~r 3~# ~ 8, Is p=}0~dy m0=~! In the =~= ~m deletIng As ~lot =~IV =0~= ~e ~ ~e v~= 0t ~#~ =:~= 5~*~ 35 ~ =~* ., , _. . . _ _. . _ .. .. ~ ~^ (~= ~ 876 ~] it 60~l l~ 6~5 3# l: 8~6 3~ ~ 4 off hIoore,:3 most of ~e figures ~:n th:s essay am I the -I ~:~s our energy ~r ~ d~n of the da~" {t is -~ for the d~-~n to take the ~ of ~p pr~- iem mwmg: "Is anything -~:~:~9 OutI>i~ ~+ THAW impo~am are Chid W's t~+ ~g ~e analogs a~n without ~em.= We are :~n encouraged to seek actions infer ~t the comext of the dam-to am' ~r - - ample': :f the child of -a t S is so slow to-n talking as to be out of place in ~ ~ ~ no~! child I. t 68~0 ~0 {~5 ~5 t~ 8~ ~.t ~5 8~ ^~ it question that Ieads to new and impotent s~ct matter in stati~- ^~-~d calculation ~s ~s to concentm!e on other =~s O pTO6~= Sobering rig 8O ~C =~' Int0~CtiH~ the results :n their cont=~' and asking new mathematics questions s gested W~ an exe~. But it is elm tme that automated cal=~on can hide the natu~ of the work that is camed out and impede ~t ~t - ~r the wok ~s appropnate to this cite probing. Too often, students believe that computers Amp into us about the Amp 8$i it tt~ 5~t W~,(~ I In a.. cia;~m exercise on tiiampling,18 Ids were Milky to recOrd the colo~ of ~ Cage sample of M&M candies and to commas the

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~Q:2 ~K~'~ 'To N~Y ~ K ~ 00~6 ~5 ~ ~ ~~ ISIS the -I ~~n of=~ ~ ~ ~ un~- ~ ne pu~e of ~e emus ~s ~ demon~e from ~ = :~ ~~ t~ =~ Mars ~= n~' ~n - Hi um~N restated. yet ~ mme smdew- s~pb bel~ ~~ ~e Co~er Gel ~ co~ bec=~e :t Is on the ~mp=~ ~n ~~ :they~ h~ entered Em ~ut t~ e~Ivenes$ Hers is ~ m~or Item him p-itf~t 1n t~g logistics' ~ using pianmng to Aces and com - ~= m.~o ~e cu~ Aces Me Of i;~om arm I ~s Amid :~[ nu~ =e to ~m ttelr Ovaries I; =~= to bel~ ~ ~ "magic b~- B=ic ~~w ~~:~s a~ n=~d ~~r ~:~! anionic and esteem t~' wash are Ike :in cheerio automated c~^ Wu IOnctmn =~ p=~ c~l - ~r the O~ ot ~~:~' brim mu~ be requested ~e ~ one, while automate o~y the a:. ,~ ch~d mast At. ~r ex.~, the ~~. between divisor and ~-~ ~n order to use ~ ca1~* ~r Tong divide* ~ Maid must in- begs tO =~ ~ with ~ ~ ~ 1 c~ there~e - .n to u~ ~.~ In ~~= saw of Ma as soon as the orations ~ u:~. ~^ ~ -I that mI! mm~e the .. ~ ~ . . sampte me =c stanuar~ ~~on diedm Beam Ma be used to bypass mut~ al~=hms already :~. At ~ more advanced I - ~~: some :his~gra3~s sh~d ~ maw by h=d w fuming ~ att:~lw software that chooses ~~ps and ~~ws hip di.~V - m ~e ~ ~~. Pe~s most imp eX- =~ce mth Hi ciLan-ce devices and phylum simul^~ns such as braving colored b~s few ~ ~x show p=~ com - ~r s~-. "~wodds~ .~ have nG mutation. ~~h :~, ~ ~~- tend to bel~ th~ the =~er pr~ts reality. t~.~nion Dom physic tO ~ is vm~ I- A:~V ~~ The practice of =~ed use his easiest when calculator am mmputers are pan of the no~! Cl~-m I..*: tO be u.~. as :~, not :~ ~r ~ Cecil ~ r~ ects or u~r ~~s ~ From ~~a ~ Inference There are w~] ding principles th~ help us see :he my i=! studY of data and chance as ~ cobe:~t - ~~- ~~e su~ principle is t-hepm~ress'~ fib* data analy;lS ~ cat-a pro~on tob- ~ t~ i~f l66 ~i50~5i00 id alit 058~ is 0~i:~6 :~ t6~0 same st~. ~ A ~ _

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103 ~~ =~' - :~h I: o~ des~bi~ :~M sum~ w d~, us~v to =~r scow que~s ~~t mme ~ in ~~ =~ 6~n ~ mn~mne~+ ~ Tn~, th~ ~ of :~s - m ~~* Th1s pm~Sion Of topics I- ~~ ~e T~ ~~: of the :fi - am ~e :~ of I- of ~e m=~-. It ~~= gives . A cal =~^ 5~6 3~: in ~ cum=~:~. ~ cO~' the i=er three headings mi: a;~: inch ~ .: the =~t ~ ,~t,3 = - i ~ ~ ~ d=~ Din particular expenen~ m:h chance o=~escan ~~ gin in ~ -~13~ =~s. S:~:~:~, informal concIus:~s band on Ha I: ~ encou~ ~~m ~ =~Y stew The Baby Back ~ ~:s outline is ~at it ~= n~ commas:= ~at pmb~-~:~ ~s i; in its ~ right' n~ memos ~ ~ pad of Akin tles~ :~th to =~t Of pro:~il:~ty and b~c m~ematl~ ~~s ~=t prob~-~li:~y =n be i: in elementmt - ~~ as soOn as ~~s aw unhasty. There is, howe~ ~ natum! place Or pr~-~ty in the p-~io~n of Ii ideas. S~M des~s ~r prOductng ~~ are d ~, the Illiberal u~ of c~= ~n rehem ~~g Al rarer =~tive expenm~- H=e is an Oppo~i~v tO prO vIde mme eX~e w:~h : and to. adv~ ce to: ~ ~~y of random Elation in ~:~al summaries (~ch as the mean- of s obSe - ~tio:~- Bulb ~~sice:l random =~tion and sim.~.~icn =n O~ the other ha~d, Army statistics ::~ce requires wme under~ standing of pi There~fbm it makes =~= that the ~~n on p-~:lity be be~n ~~se ~ pm~ng ~ta and ink Because Ot t~ =~t CO~t I t6~t $~S C~t in p{~iitY =d in pr~il:ty based i:~:renceA fo:~1 mathematical treatment of the~ s^~s should probably be an elective mther th~ ~ core cou=e ~ secondary whOo-l, DATA ANTI - ~S Data analysis is descriptive statistics :~m, with n~ methods, r emphasis on ~,~;hics' and ~ cons:~:t philosophy due to John It t<~8 ~ 3~6 ~ 0: 7~5 ~~ ~4 Kiwi his wnt~ ings in this area.8 ~ reviewer ~~commend$ paper l: in Volume ~ as ~ good starting I:. .. .. {~C CS$~C O:t 68~8 I iS tO =~t t~0

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{:~4 j~/ A^~= ~ ~~:~Y i~ ~ - ~~s in ~ - ~~ at ~ : r ~e ~~ are =~:~;~e Of some ~;r-~:~ve=~* Tnspe~mn of Ma ~~ -as ~~ ~~. :If the data w~e Dro~d to an~= ~ sn.~-~.~c ~-~.~.~ ts ~ - ~2 In whIch s~ t~ moods ~ co~.~e~s ~~. ~~e tats ~1=d us m : ~e =,3~8 ~6 5~ turf ~~ ~8 ~5 t6~ ~ - ~~ my Tyson=' ~ ~ ~~ ~^ In other c~s we do n~ ~ ~~c ', 1n mind: :~" ~t w ~~ ~e data to ~~ ~~s ~= ~ ~ seek ~ barn . 0~ then~ dew of Sexy -~a a=}ys~s'" an~ of ~ Alp- ~~ un~ ]~ The bests c~ns ~ Ma ays~s ~ new methods ~~r w~ as ~~ and ~~s (= scam- plms 8~6 50~i - t pi=5 i; ~~ p= - i0~t turf Em: ~~= 0~ amples it is ma- to ~ Ma analysis as ~ mile=~-on -of clever mols -and Aim thy I,. g p=~ Ah =3,1\rSCS Of =~,~liOX~ ~~ S~ at ~ daM c~ ~~ 2~ ~~ bY three s ple pnnc1,~. 1- M~ fmm si~ to m~:~- Lam exact ~ s:~e variable to relations - ~~= twO va~S and connections among ma~! Baby 2. When =~ni~ ~~a~- wok hr~ ~r an owr~l pattem and then for mood -~ati=s :~m ~~ pattem. '. M - e :~m -I I to :1 m=~$ ~ speckle a~ ~~s of the ~u to =m,~= mathem~i~ models Or the Feral:] fi nd ~:~:~ pn:~-~es sub th~ I=~ng abo ~ ~ mth display1~g the ~~tion of ~ slope van~. My such ~~a are -either =~= :s how qu~ve van~es such as =~or become numenc~r measur-~s mlb units. Smci6c methods for data dies play -God advance in parallel mth the development of =~v qua3~titat~e con cepts~ "~0w many of-e;~ch color :n ~ bag of M&~:7- can be deter mi:~d'b~ counting and dip Blah smelts of colored bloc~ ~~r ~ stempl.~t ~ Wo~di.~t =~s can reinforce the dist1~-~n between the ~ O's and the ~ ,~ p:~ce in whole num~. ~ ~em~:lot of two~ d:~:t data lists each Is died as ~ "~- and ~~s the obsewat:~s by placing their i,5 digits as "~- on the appmpnate stem. He:~' : t~07 iS ~ ~~Ol 0; tt~0 nu - er 0{ home :~s Babe Ruth hit each -of his ~~= w:~h the W~kees.

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Ace: : 25 : 45 6:: :D ~ ~ 66679 449 -~:1 i~r we =:~e to hi;: lo con~= amp -of Ma mth :~= -~an ~ ~w ~~= =~s an. -~:~ng of "Wmee~= =d the ~:~>Y to =~p nu:mbe~ as we~ ~ sin in making =d m~ Is :~n ~~. ~Ch ~ amo~ ~ av~le vanadons on ~~ and Is =~ :~e ~me:= as the -numbed ~ng up the ~~a b=~e :~s Up :~ ~ :~= mth -saw ^ts oh~ ramp A in or t=~, be* examine ~~ =~= m~ se~'e~ d=~] places ~~o -cased ~r ~ Am requires ~ ~ unde~g of order ~r ~~ ;~umbe:~- Cam~! planning ~s ~~nt -to Aid gins adYe~y -sting students mth Was t~t go be.~d the:r Or sk:l]~* Bm :t :s so-so -God th~ ~:~ ;~s :n ~e ele~ grades =~ r-~:~-= IBM: con~s and $k~:~:~s ~m the existing mathematics cumculum ~ app~:ing them ,~n i*~-~*~rg settings. ~en we h=e -d ~ did Nve mu~ :~Ct It and com mundane our understanding -lo others. Ch:1tLw:n are not natumlly ~~e to Add data a~ =0w ihan lbey a~ ~ ~l0 to :~ad wo~s T~ must be m - t both the ~~ of io:~g at ~a =:d - ~~c ~~s tO be ~~= off The st~ is e~d In t~ w..~d pnnc~+ ~~k ~r p~m, t~n for dI- Thechic ~~es change as we -~-e throu~ the stages ment:~ in the 6:~ pnnc:~" An example ' i ~0 tt~ p ~~s A ~ . In 1961 Yankee outhelder Ro~r Mans b^e 13~e -I ~~ of 60 bome ~s in ~ single season. H:~e is ~ bac~ back mmpanson of :~y bome Mars hit by Ruth (~n the :~-~) and by M~s du:~g their v== w:th the Ya:~. :~H \~S ~ i. 5:' 54 97-~66 ~ 1 944 ~ ~ v.. :~. 346 :. 368 -I. 39 . .. ... overall shape of Ru:~s di~:~-tion is `~:~v sv~metnc* Thee center is at ~~t 46 home mns, in the sense -God he hit -more than 46

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128 ~ A~:~:~ ~ ~~Y ~ the Bali an ~ :~ th ~ =n =t of prob~ilitv ~ ~ cxp=~ to such ~~ or =~e pmbab~= ........ What is new be= 1S not the ~~emadc~ wh;ch r~:ns the ~~' but the :~mrp=~:~on o~f Area b~V as mp~ ~a su~w a~t -of un=~y ~~r ~~n ~ Ion~:n :~at:w ^=D.~- ~e ~~:~;~ ~~= of the s=:st:tc :x ts now -~.~d to ~e con~t p - ~~s of ~ - ~ ~ ~ . , . . ~ 4% ~n ~ ~ue Ior A. ~ ~~bti~ ~en combln~ me pnOr tn~on ~~ t~6 ~6 ~~ Ah:: ~0 00~ Jo "6 ~ ~ the ~a (~e d:~;e ~ of ~:s ~~ation uses ~ s::;~]e it -I Ii pm~s ~~ as ~a~7 t5~em' ~ - ~~t Bs:= school. ~es :ts Came ~ ~e con~ns of incnce an =pr~d :n w~s ~~ - abbey statements ~~t the u~ p= :~f t~ pability :s 950~ -Gil ~~e ~~e ~~ ~~ t~ ~:~ =d 64~7 ~:~ches The Brim =~n :s ce=:~y easier to =~ th~ the ~~i~ s=~:~- M~ p=~r 1.~-~ is ion :in m.~ pry atist~ns ~~Iv amen that Basest m-~ Ad be -need vn e pnor p-r~ili~ d:i~n of th~e pammeter Is ~-~*~. ~~t :is dimmed is whether u01e pnor d-~:~ns are atways ~e, as ~~sia:~s comend. ~~:~n ~~;~:s do not think that m-v A- J~\e a~=t is ~~vs use~: info~ion and 50 are no! wIlll~g to A. ~~ u~ of su~e prier -I The ~~y clear =~:n of a Brim =.~Ysis =n ~d stro~v On assum:;~ns any the pnor direction that cann~ 'I Ch - ~d ~ the CAMP ~r in: inaction. ~~t i..-: Ba:~;n methods hwe s~l di~:~s - . . ~ # # Th~ :equl~ ~ Ems ~sp of con~dit10~-~ prow ability Indeed, ~~s mud underhand 7;~e d:: between the condit1-~! diminution of the sm:stic gl~ the pamm~r and the con~ Aims distntutio:n of the pamm~r ~~n the a=~:y obse~d ~~e 0t t50 5~ti5~0P 'I is ~~> 5~. The su~-~w id of pnullity is quite natu~, but it dlve~s attentionm randomness and chance as obw:~d phenomena in the ~~d whose pattems can be d-~ed mathemat:-~- A:n un~ of the behavior of ra dom phenomena Is an important goal cf teaching ~~t data and cha:~, pmbabilltv unde~od as persona:! aswsSment of u-~-~:~ain:ty is at ~st :~t to achieving th:s go~. c Kline ~~m d~a ana:~s through randomized designs for ~~a production an'd pmb~il.i.~) tO 1~e ls cIeare~ren Pi inference iS the ~^ Two t>~s of :~:~, con:fi~-~-e id and signifiable te~' figs ure in intr~otnst:~ion in ciasSIca! ~81 in~-~- The rea~ coning behind both tVpes of i.- can be i.- In~v :~ ~i50~.~8 40~t 68~84 ~~: t=~: 806 specific me~s sbould

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If:: be w~d ~r u~e seCO~ Go: tO p - ~~ =d ~ - tl~, a~ no : should ~ m~e to ~~m mom ~~n ~ ~ ~~c As. Andy :n ~ ~e of s~= ~;em ~ ~~ ~ loach o~s tote ~~= ~ suth an e~ the it m~ ~ better to ail ~~s ~ te~ ~~.~s all~. ~e m~ beh:confident ~~ts -is Eve ~~ Cat Wh~ :-s more' news :~s :of ~~:~n po~ ~ Weir ~~s of ~r preside a Cay sew of ~~es ~r -I :Yow ~s :t t ~ ~~e ~ o~y ~ )~0 ~~e can =~y mp~t ~ Opinion ~ i85 In.. I? Ran~m sampling pmV:~= ~ pa~of We ~~, =~g Ire pmv:~e the rest =d co~e :;~s ~~n - M ~e A... of c~r m-~- eX~e mth sim:~n Of ~~ dl~. ~ I between population and ~~' the imp of random ~pl:` and the =~n -of ~ -Amy distrib~:~ am ~~ to in:fewnce. aims -A ~~s ~e 06~! i: -of con5~= intervals dunning t~ emanation of A: and sampling din~~. The ideas of ~~ i ~ ~~ ~ b~ ~,~,=t aria apt . d~~ Y Of simulated =:~0 ~ mme ~~ appmach ^~s f~ili~-v -em no~ _. ~,jf;~ jar Same that :n ~ cage county 300~ of him school ~~;~s ~w -em to: school. As~ ~ ~~e ~~m =~w of 250 ~~s - :~r theV throve to ~CO] w~y puces 230 Ian observations' ea~ m:th probability ~ ~ -of being ves~ ~e p~ni~ ~ of ~s =~= in the =~le vanes~m -do sump:- S:~=e -id' 1000 ~m:~es ~ ~ he Gil; ng ~ ~ ~~:: on of ~ ~ T~t is app w~h mean O~ 3 and st= ~~d den i ~~ ~n ~ K 0~ 9 ~ 9K Hi it! ~ I ~ K at , if Repeated simulations at sampl~ o: ~~s s:~s tram :~s po~n demonstmte that ~e center ~ the w=~:~g -I r-~ai:~s at O-3 a~ that the spread ls contr~m bY tte size of the =~. Tn ia~ ~~CS (~:t lOOO or so) the W1~s of the ~:~1e ~:~tic ~ are teddy =~:~d around the populat:~n -I ~ ~ O.3. ~ in ok as i*: . ~d: K ~~ts can See empincally that =~s of ~~s ~~e allow good ~~s about the entire populations But~ how g~d am ~~s based on ~ ~~? ~ =n I the ans~r b~ describing how the Arid ~ vanes :n repeat-cd sampling. It ~s ~ basic fact of normal Is th~ ~~t 955S of all -~tions lie within two st:~d dev181:ions -oh I:: side Of the mean. so in :~6 58~li~g7 93~ 0; 3~ samples of 2~O ~~s ~Ye ~ sample

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=~t art: ~ N~y prop: ~ m:~n 85~ O-*~6 0f ~ =0 pi 0~3 - 0 ~ - ~ ~~. ~e simulate=: Sh~ ~ ~s is m. ~ pw~ th~ ~ ~~e Of:_5~0 Indents in an= ~= c~ 6~s :~5 ~o drove to s~1. WE ~~s th~ :~e ~ ~~n ~ of all ~~$ ~n ~~s c=~y - he -no ~s~! ~ c1~= m~p ~ tO~O ~~ O-~. If (= ~s t=~) the vary ~s ~~ut the smog as :n the co~y we dim ~ i:~5 '-by= ~6 0~ ~~ = 95~0 0t we are By% ~t ~M the u~ wMadon proposal= ~ I~ in e :~! O.42 ~ O-~. M~ generaDy~ ~ Swerve ~ ~ O+~ :s a ~~% ln~] ~r the -ale ~ of ~ =~ ~~ :~ ~ heart . As- Spewed ~~= of =e 230 :=e ~, som6 of ~e In v~s ~ ~ O~ ~6 -am the tree pm~dion Of p' wh~ others do nm~ :~t ln 1;.~e ]~.g =~ 95~: offal! ~~. pm~e ~ 1,~.-~, ~~.~g ~e tme p* ~8t lS, ~ probability that ~e m~om i: 9 ~ O-06 contains is- Odd* AS 1S ~~b ~e ~e in crania ln~, ads- p~y =~= to the pe - ~~e of the mealy. :~n an in~it~V i~ =~r Of wpeat~ oomph. The ~ ~~n of the =~t ~~ belon~ t:o ~ stu~ of ~:~ pllng and ~~:~n and ls es=~v an em~l ~~:~:n of ~ ~ , ~ ~g t=~r30~t}~ S Ot S8~= thm the size of Char p~tion. ma- ~s that eme~ - m S~h I~ de~:~s ~ much more tm~t than the ~~! dre~ we ~w there in the s=~nd sta~ of the a=:~t ~e seco~d st~e helon~ tn- more at study of in:** ~ _ ~ ~ _ are ~ ~r <~ ~~'4;;7 ~ ~ ~ ~ ~- ~~~~~~ _ we ~e qu.~it~ive concIuslOn ~.~t mo~ =~le res~s lie cIose tO the tmth 18 made quant:~=ive bY ailing an :~wal =d ~ te~ of =nfi~^ The natum of this concIu~n =d i~ li.~tions both need emp~i:~. ~t are the group:, Of Our =~= State~9 There ~ ontY ~ , - ~ ~ two pO-~31 )~- I . Me interim O-~2 ~ c, 06 contains the t=e ~~ulati.~-n pi it* 2. Our simple random sample ~ one of the :f6w mmp:~s :~* - ~~b ~ is not ~~::n O-~6 points o ~e tme p. ~niV 5~D O all Samples ~,:~r~ 5~6 ~ ~~:~C =~. WE can~t know whet:~: our sample is One of the 930~ ~r whiC~.h the ! catches ~ or one of the unIuc~ 5~. The statement that we are 95~- confdent that the unkno~ ~ Its i:n O.42 ~5 O-06 is sho:~d bor "We ~t ~~e numbers b~ ~ method that gives co=~t results 95% of the t:~' As ~r the limitations on this :~oni:~g, remember that the ma~n of e~r a.. ~ =~e l:~:~T In~:~s onIV mndom ~:~ling en*~:~.

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:~31 ,,- - ~ \ SMIPUNG o:~U ~ 10~ O:F ~ \ - ~ :c It ~~ - - : ~- - ~ - ~ - - ~ - - ~ - - ~ ~ - . ~eK :. A nGURE ~ The : of ~ mn660~= im~ 1:n : - m ~e =~e 'pulatIcn. ~e Oo~l cu - e lo t =~ng dl~-n of ~e =~^ prim ~ cenwr~ at ~ pop~n :~= p~ ~e do~ aw ~e ~~= of ~ from 25 =~:~` w~ fin :~ - ad :~ ~` ~~ ~ ~ s^* I~ the Ion i; mn 95% of th=e ln:~s wIll cOntmn ~ In pm~e t~e am other sources ~ e~r that =e not a~d ~~. ~ ~ olis are my? (~6 ~ t0~.~= using equipment ~at dies resident:~ I Imbue~ randOm. Te~e m - eyS ~~t hou~s m~t phones. :~ wh: ste~ ohen find that ~ m=y ~ 70% of the =~ns who answer the phone a~ ~~. Men will be untLe=~:~d ~n the sample un- l=s steps =e t~n to =~t males. ~~e ~~S of ~! Licit Il Irk mme bias into op;~= polls and other ~~e Su~* 51~fi~e :~ ~e p-~e of ~ confidence interns is to est:im~e ~ popula:tio:n mmeter and to accompany the estimate with an ind:~n of the un 00~)t 600 t~ (~0 Dim; :~ tt~ j3~- Ii tests 60 ~~t

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I3:2 ~~ ~~ ~~ ~~: ~ u~ p~r but - ~y as* asw~t of - ~~er ~ ~ or Florence ls patent ~ ~e pop~. ~e mere ~~= ~~t :~h an a~t ~ nee~' ~m not ~ ~~ out-* comes -I ~ rem I c=~' ~~ ~~s n=~] mph~ tIcat10~* ~~cs ~ sclence felt ~~so ta~ ~ the am ts Who 5~ pt6~6 ~~ 6~6 ~ ~ - ~t ~ tt~ 60~ - 4~7 b~r ~~' ts ~~ as C~i~C~ ~e role of ch=~e va~n * ~ ~ not '*I ~" I As Is ~ ~~ of Is the qu*=~-n Is ~e Ob~ ~6 ~ 1~ ~~ =~ ~~ ~ =~ - ~ ~~= ~7 Hew ~s the reasoning of s=~cance te~s -~ IBM ~n the wn1~ -of ~ :; =;~3Le. . amp: Me Wet:~m era:, Con~s :~d ~~t ~~g men should ~ ~~n at ~~ ~r w-~ice :~n ~ a~- The 5~t ~ah Iottew was held in 1970. Firm ~-~ ~= ~ ln ran-* dom o~r and men wew -~ :n the c - -~* -in - ~~h their A- ~ ~ ~ ~ ~ is. .- ~ . . .. .. Atwt me A*, n~s omn:~ tI=s -~ ~at m~ bow Iate ln the year were mom ~ ~ ~ :~o get low~ ~ numWrs ~ so to ~ tnduCt~ D313 anal sAs (~*e 10) ~ s~ ant ~~n ~n bl~h ~~-e a draft :number. ~ sadistic that m-~*es ~e st*~h of the as m=~*ti~ b~-~n -my O:~r (! to 366) and bl~ ~~e ~~ to 36-6 beginning mth I=~ ~ ~ ~s the =~n =~:~" ]:n t7 ~ ~ ~226 for the 1-~7O Iotte~. Is chic ~d ~~= that ~? ~ IBM test ~~s Me 158~e by asking ~ ~~I'h.~, I- Age- SUppOSe ~: the =~ Of s`~mlent that the I~t0~= t=~>r :~ dom' what 1s the p - ~~li.~y that ~ =~m I=~! would p - ~~e an ~ at :~t as far fmm ~ as the Reseed ~ ~ ~Q-~9 /~- e I -bililv that ~ mndom I- mI! pmdu~ce an ~ chic ~rm ~ is :~ss Ohm (3-Q0~L. ~~. Si0= ~ r' 35 If* Hem ~ as that observed in I970 ~6 8~ ~~t 00~[ in ~ =~= i0~7 ~ t3~0 5~8 - ~0 that the -3 lotted was not ha;. ~~m tO did the scane~-~t of dra~ numbers ~si~-~d to each both date b~ the INTO drain Iotte~* I:t is diffi-~t to see a~ systematic 8530Ci3ti0~ or bind d~e and I-~Y number :n the Id;-. CI - ~~r ~~s can emp.~e the as-~ciati.~, as in the h~. Bm -Iliad calculation is needed to :~-~= whether the obse - ~d aSso-~.~- tlon is Ia=r than Ale reasonably be attnbut-ed to c:~:~ce a:-. In ~ random assignment of OCR for page 95
~ 2 art?? ~ = I: .. C. al!' ~ 4~ :2: ~ ~ ~ =4P 41i ?.li hi: ~ #k .4' 46 e'. .~- S. $~,.4, ~ ,?~4' -~. he ~ .. 4~ ~ ~ "a' 4~ ~1:~0 200: 30:0 4~ ~ ? p7~ 0~3 ~: .: ~7 07 ? ~3 ~=RS 10 Dam-m ~e I97O dm~ ~ew (~ ~ ~t ~6 00~7 ~.~ t)~t 6?~?~5 ~t ~t 0~6 0; t5 )~: =~ i~v ~ 5~ low - 0 ~?~ )~-?~^ =~6 =0 ~ ~ - ~ =~0 ~> mat ~g tt-0 =~ ~t ~ ~ ?~t =0A~^ e pI0t of mo^IV m~ns conn=ed b~ amp..? 5?~.~5 ~ 6~7 t50 t~' ~6 ?3 =-~-~ tt~A'?~, ,5 ~. 0~ t~l 0~6 IRKS p~5 t~ 5~5 ~ a t~ ~0 I've l ~ ~ 70 ~ ~ 22(i ~?~g t58t =~n 50= i~Ct in ~t 708 t0~:~?~6 t~ ~t i~ 4~ ~?~ - ?0m~ 50~ ~:~6 6~t 6~t i: ~ 0~6 ~?5 t6?~t ~0 i0~?~ ~5 ~t :~ ~fi-~: all the 00~ati0n in ~ :~= i0~-li &~ n~: 50 0~Q 0. p~?5 that ~ ~ Kid ~ ~ ~ 6: i~ I t50 =~ 0: =~5 tt~ 00~] pi=~5i5~) ~ 0 0~0 t~ 053~ Ill OK To reSolve his uncertainty we comp~ th0 obse-ed ~26 to ~ mf~ erence do: the ~m:~li:ng di~tion 0~?~ in ~ t=17 random imte~ ~ We find that ~ tmIv random Iotte~ w~d ~m~t never p~ ce an ~ as ~r-m zero as the ~ observed in I97~D. The iv ~:~.i0~ t0~5 ~5 - ~t ~ 50~0 0'0~. ~t ~ a ~6 is ~ i - 0 0~.~, ~ 50~g 0~t in: ~ t8~ i0~. emit 00~i~?5 ~8 t5~ t5e ~ 970 ]~?--wa8 6185~+ J~i00 ~i$~6 tt8t ttC C3p s ma ~ ning the biro dates had ~e:3: hIled ~ month a: ~ lime and not adequately mt~d. ~r dates remained nea: the top and tended

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:~o ~ ~ ~~r (~ ~~ mOre de=~l ~~t the in: ~h , lnd~ :~ s - ~~ Amp: of ~e :~: Questions h~ "~Is ~:~s ~ la~ Cam ~ "Is ~:s ~ ~~ :~ t. =~e up onen In anteing ~. It is qulm ~~ lo Eye an 3~r key =~ ~e if o=~me ~ ~ =~n ton, ~ we if =~w ~e b~h ~~t o~f ~ child ~ t~ di$- ~n of bide we~u of Al ch~. Sm~ ~~d =~ai~ be `. ~ ~ ~ ... . .... ..~ .. . ~ . ~ ~ w. ~ cucQu~c To Temper: ~ role oT =~e Van=~n and to ~~s "s:~- m603~= Islam ~ -~= lnd)~ Ou^~e to a. sui - ~e fee dI:S~: {f probity =*d c~er $~IMion are b ~ 6,~r0~6 t~ I, 0~^ 50 by; )~ t56 ]~= 0{ pDim am ~~g ~s~s B~m ~~ ~5 05 - ~~;~ 7, areas in ~ wh~::l c~:icul~- ~m =e :~ ~~s ~r ~~. TO m~s of scarf ~~ ' =~g ~ te~ =~, ~ co.~g Am. t~d ~~*~s y c~ ~e ream~*~:~g of s~i6~- te~. ~ :~x~:~ : seli :s some - ~t deft =d ~~! ~ s~. .. ~ ~ ~ . ~ . . ~ ~ E~e examples of :~e u~ o: s~= te~s ~ m~ remo~d tram e~' experience ~= opi~ni.o ~ polls and Si mIlar - ~*~S of co n6~n Ce ~~e nts. .~ fling of dab and chance' =d the ~~nt of q=~:w =~:ni:~g ill: fib 1s berm* se~ m! con~i~g ~e study of stat:i~ 05 in ttO 506~5 `15 ~#7 ~~:~g ~~uti0~S, 00~00 inte - ~~' and ~ mnt1~uing emph~i.s on u~g these tOols in =~g about uncertain data-. Statistic =d pmbabllilV aw she sciences that deal mt:h unce~aintv~ m~ variation :n natural a~d m=~e ~~= o~f ~ - kind. As such theV are :~= than simply ~ part of ~e :~.~ticS cumculu.~ they At wed in that setting. Pm~ili~ is ~ :~ld Ethic* math emetics. Statistics' like physic or =~*~' is an ~~t d~-i~ :~::~:~:~e that makes hawk and =~tial use of mathematics. Stat~= has mme cIa:m to be:ng ~ ~~! me:~od of ::~^ ~ gener~ way of thinning that is mOre i:~.t tha~ an-~ of the s~ him fac~ or technigues that mom up- the d:~lin*~. {f the pu~= of education is to d - ~~p bmad :~:~! s;kills~ ice meets an es=n tial ~~e ln teach i ~ and I~) ng. Ed ucat lo n. sb.~u ~ ~ 1.~u ~ ~ ~~:~ts to lima-~ and hill methods, to the pol:~} and social analv~is of :~:n soc:~' to the probing of natu~ b~ expe:~menta1 s=~ and to the power of abst:~ctio-n and deduction :n mathematics. Reason:~g Prom ur:~=in cmp:~ data :~s ~ aim: p~} and pe~ astve i:::*:. meth~+

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:135 ~:~s :~s ~t to =v the ~~d :~on in I n~:i~al :~- ~,,5 ~r t~r ~ s~e ~~d be prom1~t ln ~ Sch~ ~~. c~ ~~:~g, bm~y underwood shad ~ pm of the ~~ =uipm.~t of ~~ educated '~^ can summan::e the ~~e eleme~ of-. ~~ ~ Mows~ A VIA An ~e'- r=~d " the ~e mbiMdual =e Va~* The dommn Of ~ strict: Bins ~ nature and ln b'~= aF tal~ 1s qu~te In:* 2- ~~e n=d ~r Am: ~~: p:~^ Stad~= ~ Bedfast empt Wick r=~r ~= same - e. Looking at the data teas host pnon~. 3. The USA of ~m =~n with vanatmn in m~. ^ of mu=~ of un=-~:~ Saabs we wo:d sel~ ~~ ~:~-~s and i: on co~:~n in expenmen~} ~~- ~~M we lntmduce pl=~ wnat:~n tn~ ~~ pro~:~n bY u~ of r=: ~ ~ . ~ OomIZ=~:~. < . ~ ~ . ~ . INK A: 4- ~ q=;~n of van~. ~~m variation ~ de$cri~d m~:~matic~v by probab-~y ~0 ~~ 05 \~ti0~* I 3~\t~$ 5.~5 ~6 ~.~_ ~16 6~3 6~ ~C =~= V8~7 0 Hi am meas:~- . A Magi cal thinking ls not rec~ dne or re~ f rom - ~~V expe ne~* But it mil :~ot be deVel~d inIdren ::f lt is n~ Meant in the cu~.~- Students who min. thelr ~~n mlb wiling and ~ t~ t~ Ii t50\t ]~= quic~ tO 0~ 0~0 808~: tO tO ~=t 3~6 01~= ~~g7 8t i085t W~ t56 answers ~e numen~ 6~. Darien ~s unexp~ and un=:~ . Lis~ to A~r N;~6 :~g the expenen~ of his ma - t research 6~ mth wphistIcawd marketing ma~no ~ ~ ~ 700 must 5~5 =~0 ~~ 04~) \~ t~ ~t ~~ p=~6 0~ mono Th~+ =.~t numbers as =~s T=~h aria 6~d 1t dI~t to ~~k tang tt~ ~pt 05~>. 750~ 60 ~0t ~ ~ I 85 ~ ti~6 0t 5~6 ' ~ =~g ~ ~ b~ ~ d 0$~S O~ ~ 301~! keg ~ ~ it heir s~f~7s~ 5 ~ manufacturers 6~0 thmu~ r~t stoms~ AX . ! once decl~d thal we would d~ all cha~s to- shOw p:A06~6 m~e around the number : ~r example`, =~s are e:~: up pC=~t Of ~~ ~ =~t O hi; I~ CrW60~. t,S tO=66 CUt tO ~t 0~0 0 =~ ~~{ MA 0~( ~~$ ,)~; 00~l,6~,~t ~~ ~~.~ ~.~5 tryst. ;~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~A The ab-il~ty to deal in~:ligentIv with v~on and uncena~ is the } of ~inst~tion ~~t data and chance~ Chew is mme C1r~;C h i ~~ act;~>r :~mp=~r68' this ~i:~yA N:s6-~t 0t 3~- ~0

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TO =~- ~ lying EMS salmon , ~ ~ ^~ ~ ~:~y ~ . Ol _~1~. Hey nose 1bat ~ ~p~b~~ Ed Ins Be Alder lo c_ S1_ ~^ I!. an Men beg i~ion~ of ~ Anon kind -1 Ages nodded 10 ~~ Philips ~~ ~ untrod ~1~. ages ~~ a Up: [So #= ~ ] apt ~ a ~ lip Imp ~ -~ dig ~i~1~0~ ~1 wits ~~ a ~_~1 Em ~ egged Am our sanding ~~ on be ~1 YiSil. In -0 Abe no b~=d~ iffy glitch amok aft ~ Is paled #~ rely pon~Us$i~, ~1 Her sob as am he car c ~ ~1~- or <~r fins ~= so bail ~ul~^'1 Loire up ~ ~~- Surges to bag In one Sl~l~iSli~ gum me arson age :~u~e0~lsll~ Lions =~ ~ = ~s~ am- oaf e~l=1 imps, ~ b~ Ace _ jug 1~^ 1be 6= (me,- ^u1 20 In ~ Ace Ni^1 Ed his Allege find it ^^ 1b~ ins1~ion of a quite so fiend dog bag an egg on Bin gout ~~~ oc~n~. Tbe of is s-~r into Ding 0~ 1be Ilk of Lisle ideas in eggs line as ~1 HI goad Bank go. figs is agent ~~ we an in ~~ Baling ~~ ~ ~~/ ly Angle intelle~u~ said. Bigly ~~ ~~~ Herb Going 1b~ lining in dee~inisl~ic Splints earn ~ Be Huge 1~> dog not mid Ida ~~# S~1iSli~ Paining. His is evidence 15~ we ~ denim ~~ ~ ~ If inflect mead. gay Cab 1 ~u and cb~n=? Satiric and p~iIi~ty a~ ups ~1 in patio. Data analysis in Paul blips ~o legging of basic m ^ emalics. Bu1> mod impo~1,ilis because Pistil 1bi - fig is an independent and fun eggs inte~1k~u~ method lbetit dk@erves allenlionin 1be olcu~iculum. 1 - -- --- ~ . Bobs. ~~ ~~ ~~ ~ ~~> Ones at, 1 9S9, p. 30. 2 Beat, Vic. ^~/e ~~' ~ ~ ~ Beg By, MY: Joan ~~ ~ ~` 1982. 3. E^n, Barley -ampules am 1~ thaw of stations: Id 1ho untbin~bl~.- ~ ~ 21 { 1979\ 419~37 4 Hon~S SPEW E. dominion and ~ia1 amid: ~c 1970 dam lollc 171 (1971). 255~261. Gamma, ken and Plan. Andy ~Di~cul~i~ in lca~i~ basic ~ncepls in ~ ability end slati~io: lmplicalions~r Me ~~F~ ~#~6 ~~ 19 {1988), ~3. 6. Oas~i~b, Joseph. awe ~~1islical precision of ~mcdica1 mania p~codu~=s: AD Elision ~ Yaps acid AIDS a~nli~i~ last day.- Ifs) ~^ 2 ( 1987~. 213-27~ at.

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:< -aid 137 . In, Ma; ~~> ~~ Ha, Jag. ^r / ~~. Pro a, a: he ~~, Pubes ~1986. S. an, LV all. ~~ ~~ ~ ~ ~ ~ I ~~ ~~ /~ 7~7- ~ ~ ^~ ~ ~ ~~= ~ ~ 7~ anal, a: gab ~ /, 1986 9. ma, Ha. gad =~d Aria he ~ anion of in~.- ~7 ~~ art afar I, 83 ( 1 98S), 92~9-~0. 10. ~~ go, Jan ~ as Ann ~ ^~ go ~~> C^: he War ~ 19~. 11. Aft, Jug; ~~ fin; -^ Jag ~ ~ ~~ #~- ~ ~~ ~lo, ~ ~~ Seymour Page 19~87~. 12. H^~:i=1 ~~ aeon a. ~ ~ ~~ ~~: ~ ^ ~ ~~ ^~r I Riot ~ Munch. _~n, a: Agony ^ Pa, 1990. 13~ , D^id ad age, G. ~~ ~10, ad: he ~~r aria. 19$6. 16. Dillon, Bug C., Jr. lit in meting.- in ~on, O.; Rag, V.; ii=' ~ C. {Ed$.~: ~~^ ^~ ~ ~~ i~ #~ I. ~ . ~ ~ . as, . ~ i: Mae ~ Cab ~ e Abe ~ Bang, 1986. 17 fit, Rib E.; and Oe T.; firm=, Can OR.; ^~ Pay W. _~- 23$ (19$7), 62-S1. ~18 Rain, ~d=; a, Balm; Rota Ann; Dupe*, ilium. gelling ~ ugly Ha Gail ~ tag $=i~i~ =~$ in big ~b-.- afar CZAR Act. Amps alibis Otis, 1# 61i 19 Rubin, ate and Rota, at. tic` misunde~di~ in S=i$1i~ ~ Bait Bidet ham ~ ~1~1 of innov~i~ modes.- In Skins, man (Ed.~: ^~ ^~ ~ ~ ^~ fling of an l~iona1 ~liSli~ IN ~~- 1~, 20. ~~, ~= and ~=m=, Spiel. Calf in 1be ~ of ~11 =~.- ~ 76(1971\ 1~11~ 21 ha, Amy ad Enemas, final. ~E~$io~ ==us insipid I: Tbc ~Junclion flag in p~bili~ j at." ~ ~ ~^ 90 ( 1 USA. 293 315 22 ~lone, Rip ad ago, egos 4~c ~1 bad In be: ~ Abe mica ~ ~- ~~ 17 (19B3), 296314.

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