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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 .~
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
:~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.
.~: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
:~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-~-
~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
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$i it tt~ 5~t W~,(~ I In a.. cia;~m exercise on tiiampling,18 Ids were Milky to recOrd the colo~ of ~ Cage sample of M&M candies and to commas the
~Q:2 ~K~'~ 'To N~Y ~ K ~ 00~6 ~5 ~ ~ ~~ ISIS the -I ~~n of=~ ~ ~ ~ un~- ~ ne pu~e of ~e emus ~s ~ demon~e from ~ = :~ ~~ t~ =~ Mars ~= n~' ~n - Hi um~N restated. yet ~ mme smdew- s~pb bel~ ~~ ~e Co~er Gel ~ co~ bec=~e :t Is on the ~mp=~ ~n ~~ :they~ h~ entered Em ~ut t~ e~Ivenes$ 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 ~ _
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$~C O:t 68~8 I iS tO =~t t~0
{:~4 j~/ A^~= ~ ~~:~Y i~ ~ - ~~s in ~ - ~~ at ~ : r ~e ~~ are =~:~;~e Of some ~;r-~:~ve=~* Tnspe~mn of Ma ~~ -as ~~ ~~. :If the data w~e Dro~d to an~= ~ sn.~-~.~c ~-~.~.~ ts ~ - ~2 In whIch s~ t~ moods ~ co~.~e~s ~~. ~~e tats ~1=d us m : ~e =,3~8 ~6 5~ turf ~~ ~8 ~5 t6~ ~ - ~~ my Tyson=' ~ ~ ~~ ~^ In other c~s we do n~ ~ ~~c ', 1n mind: :~" ~t w ~~ ~e data to ~~ ~~s ~= ~ ~ seek ~ barn . 0~ then~ dew of Sexy -~a a=}ys~s'" an~ of ~ Alp- ~~ un~ ]~ The bests c~ns ~ Ma ays~s ~ new methods ~~r w~ as ~~ and ~~s (= scam- plms 8~6 50~i - t pi=5 i; ~~ p= - i0~t turf Em: ~~= 0~ amples it is ma- to ~ Ma analysis as ~ mile=~-on -of clever mols -and Aim thy I,. g p=~ Ah =3,1\rSCS Of =~,~liOX~ ~~ S~ at ~ daM c~ ~~ 2~ ~~ bY three s ple pnnc1,~. 1- M~ fmm si~ to m~:~- Lam exact ~ s:~e variable to relations - ~~= twO va~S and connections among ma~! Baby 2. When =~ni~ ~~a~- wok hr~ ~r an owr~l pattem and then for mood -~ati=s :~m ~~ pattem. '. M - e :~m -I I to :1 m=~$ ~ 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.
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 $k~:~:~s £~m the existing mathematics cumculum ~ app~:ing them ,~n i*~-~*~rg settings. ~en we h=e -d ~ did Nve mu~ :~Ct It and com mundane our understanding -lo others. Ch:1tLw:n are not natumlly ~~e to Add data a~ =0w ihan lbey a~ ~ ~l0 to :~ad wo~s T~ must be m - t both the ~~ of io:~g at ~a =:d - ~~c ~~s tO be ~~= off The st~ is e~d In t~ w..~d pnnc~+ ~~k ~r p~m, t~n for dI- Thechic ~~es change as we -~-e throu~ the stages ment:~ in the 6:~ pnnc:~" An example ' i ~0 tt~ p ~~s A ~ · . In 1961 Yankee outhelder Ro~r Mans b^e 13~e -I ~~ of 60 bome ~s in ~ single season. H:~e is ~ bac~ back mmpanson of :~y bome Mars hit by Ruth (~n the :~-~) and by M~s du:~g their v== w:th the Ya:~. :~H \~S ~ i. 5:' 54 97-~66 ~ 1 944 ~ ~ v.. :~. 346 :. 368 -I. 39 . .. ... overall shape of Ru:~s di~:~-tion is `~:~v sv~metnc* Thee center is at ~~t 46 home mns, in the sense -God he hit -more than 46
~~:~ I: ~~:Y am fe~r half ~e t~. ~~e =e ~ ~~g ~~s - m ~ - ~~! p~^ :~ part~ar, Oh's famom 60~ home ~s )92? do an: mnd ~m the ~~er wtue$' :It :s 3a~'sst e not u=~ in ~e =~ Of h~ =.~. ~ =~' Ma - s reco~ of :~:l ho~ in Age ts an o~ ~~ fans ~~y In: ho oven patted. Cat overall p=~= (~xcl~i~ ~e owner) is again ro~v synod a~ ~~s =~ ~ abom 23-- T~ I 1~s of ~ t~ ~~:~s ~~ Its Anew supenon~ as a homed hdlm ~ ~ ~e ~~ ~~m of the his - ~~n of ~ s - e ~~:~ ale I=m to ~~k for symmetry or skewness. for Awe or m~e beaks ~ ~ ~ L tor We cen~r and ~ Wee ~ ~~ abOut me cemer. dev~a~s ~m ~ ~~r Hem :~lu~ gaps and orders. ~~= that while c~g ~ dismay ~ ~ ope~;n to ~ Ieam~' ~.~ :~o d:~ - ion of real ~a :~s the Ire mane ~m~ of sow mat~. ~~. ~t ~! dI~s am ~~ ~~ as either ~mme~ic ~ Id. T~ much his on candying w~ we wH] frustrate bow :~= and ~~- :~m ~ o6~ ma t0~57 ~~t t~ ~^ ~~¢~: arm ~m ~m - ~ :~ ~: ~= nasally' )~65 t0 8~5 ~ t~Ct W58t ~ 560- 85 W~ ~ ~ that :3Ruth's 60 was not an unu=~ pe~~:~e ~r hums while NIans~-s 6-:! was an out~g a~:t ~r be~d hIS uS~ I - ~~. Inw~g the overall sh~ of ~ dist:~tion ~s an i: pm of Ieami:~:g to look at data. The histogram in n~e ~ did collected ~~a on the Ie:~s of w=~s ~n Pop^r Sc:~m magazine. The :0 20 :m $ ~ ~ A ~ ~ - ~ 0~:~ ~ ~ 3 :e ~ ~ ~ 10: WORD ~~H ~~E 7* stu~d d~ o:n the ')~:~h of ~~s In A ~ mvea) ~ 5~6 ~Y~~#~# S:~£t 8~:~: ~'0~5 3~ =~ 00~#~#~ `~ i0~#~t 0~-
Hi: ~ A: ~ - ~ - - ~ Ad: 420 4~: me: ~e arc: S 3. ~ ~ the mean v - ~ SAT ~m by =~e rewal ~ ~~e '~ ~ ren~s two ~~t tm~g tm~. :~n Amp- arm ~~ ~~ ~~nu And the S¢'c~ ~:n other St~$ O~7 ~ ~ ~~= ~C =~ ~ t~ ~ I,07 i. is - t skewed because - m ~ ma~ Am- to - ~~r warm and fear I~ wo^- (~he am stati~i~ te~:~ e d::~n of ~ s34~s to ~ ~e d~re~ion of the Io:~ mil, n the ~~n in chid mo~ starvations are =~) ~e Am :~n ~u -re :3- sho~ ~ man ~w by' s~te on the verbs pan of the ~~c Apn~e ~ (~. Aid ~~n )s double peach The peak n=r 42-5 wp~s =~$ ~n which most mIl~ bound students take the SAT, ~e 3highe~ed peak repr~ts ~~es :~n which ~-~ students m~ the Amencan ~~ Tested (~) exams tS 80~Yi~g t`O 5333~?rO CO],l,~O ta~ I. Desk Already in ex~$ the Ruth am M~s hom~n dam ~ saw that ~~Mion =n he~ us describe da~. By simple munt:~g (~f more and half 1~) we can give numbe~ that m~e more e:~= the -e in cent~ that ~~ see :n the smmplots~ The natural procession of math~ ematical t - ~~s is -A :in the thiM or$~niz~ p~. graphics to numen~ measu~ to math~cal models. {~n the =w of the ~~n of values of ~ single Van~, ~e basic asp=~s to be described numen=~y are the center (~r Iomt:~) andL the spread (or a~e7~Q~) of the ~~utio-~. (The o]~er te~ "~ ten~ 60~7" which is both Ion~ =d Ie~ cIear than "~- or "~tion'- is ra=~y used ~v statisticians and should be ~~-~) There are two- common se~ of descriptive mushes for ovation and spread. the me clan wi~ the qua~i:~s (~r ~~.~-s ~~r percentiles) and the mean with
}~8 : w^~Y the s~ d - ~~. Percenth~ require o~v --I ~ an a: mn~ ~ simileaims ~ li4, ! /2, 3J4 ~r medIan =d quardles). e ~n :s the an~m ~~ So ~e -amp me~' q~, =d sunniest a~ : - ~~s mn be Mar as sty: ~e-~p b~e ~~c s~. ~~e simile :~s ~~ ~ ~M -~- Me cabana:. ~~e web the ~~=i~= be~n ~ Shape of ~s~ and numerical ~~s ~~=s :nu~nber sense. ~~ bmb ~e Assays and the measu~ ~ -elememary~ the amp ~ :~:nau- =1 unkstand~ required to u~ them e~y (= opposed to s~ly calc~at~g ~ mea~) ~~1d :~t ~ undere~-~^ :In -a-= h~ tm of new ~~ng malaria, ~ ~~, ne1~Mr ~~ nor ~ t=~er could - ~~ that ~~ ~se~= -lo -~e n~t =.-d ~ ~ pkicul Lion wnh may t~ serve:. ~ ~e ~~er ~ the -my= ~ ~ ~ 9: ~ ~ · ~ t~ ~ ~ ~ ~ ~ ~ A I'm to e~l'm'ate mea~= bY Io~g at ~ dl~ ~ dlwus~g resul~, ~~s stu~s c~ct ~mr ~ Lest Of sop ~ p~ simple operations ~ munt:~ :~v~ ~ the o~ - fit ~~e medic) a~ averaging ~l the values (~e m=~. Numerical descr:~on of ~ d::~-n by the median, q=~il-=, =d extreme obse - anions I=~ to ~ new Chic ~i~7 ~0 ~~* ^~ exam~ shehow u=~ Fir Chip ~ be~ AS. l:>epartme:~t of A~ulture =~ns gmnp hot do~ ~~to thme ~~+ 600~$ m0~7 &~] ~~- ~O t56~ t-~S Aims t:H t5C ~~f O: ~~$ ~~V COO~9 :~n ~~e ~ three b=~s display the mini of ~~-~ per hot ~g ;;~ .Q Ace: i: 20 ~ arc. ~ SO ~ . . . ~ awl = _. _ ....................... . - 6~:- ~~ p:~$~- i ~~ ~~ 4~ ~~= ~~ ~5 6],$pj];3,~r ~ ,$~.Jl~ 7,60 At' If? ;~6 6~33~5 :~: 0;~$ p3~.6 5~ ~5 ~~$ (asp that ~~ - ~~g to three ~~ tVpes~ -I meM, . one =n cask> see Hi.? p0~$~$ t~ 60~ as ~ ~~p cotta< ~
~:~Y t:~9 ~~g bra.~s of t~e ~~e t~- The ~ e:~s maw the q=~' the i)~6 ~i~.i~ t~ 5= i~ t~6 =~:$ 306 t~6 ~h.~5 ~6 t~ t56 sm~t =d ~t indw:dualse~. We ~ that beef =d ~t hm -dogs am similar ~ ~~t ~~ ~t ~~ as ~ 'up -amp conside - Ad fewer c~.~s =r b~! d~ Tn ~~s bnef d:~io~ ~ ~gI~^e ~~' we have not ~t men timed m~= the s~M -~;~= ~ the~ fin: s~ :-n the pr~e~i-~n m =~1 d1~-tO numenc~ descupti~ to ma~atic~ m~* The s~ In has S~ -dies ~r d~a - -bike. It is -A to c~ wall ~ brim -I is ve~ sensitive to ~ fit ~0 ~~S, =6 is bits tO Victim ants (~0 ~~D or m~.~nof the ^~ d~iatlons of the ob~ti=s fmm thel: mean is Able on ~! them mums-) ti )5 ~ ~ ~ =~nt :~ ~ ~ in) ~ i e ~t is the natural m=~e of spread for normal distnbut:~* Nor mal -I p~e an =~e ~ ~ compact math:~ Aims of the ove~] ~em of ~ d~:~;n of data. The are m~hem~i~ IBM th~ ~ not catch the dreg of mal data or Sciaticas (t ~ 0~0i0= ~~} =~05 3~' ~: ~7 p0~- ~~. ~~ =~m matenal~s i- ~r ~.~.a'] studentS ~~D ~~n of presenting none did his ls tme' ~r exam~Ne of th Quantitative Literal' sexless :~- ~ ~ ~ :5 ~~ ~ointI~ b~ the ~~rican Styli-- Association and the National Counc:! of ~~= -of Math C=~. O~C t=50~ ~~ bC the trad t1~ price O]f O~31 co-ed O1;~0 6151~li-~S 85 P706~: ~liO~^ lO 6C 60~I~eJ fit ~~t = s:~le ~~dy of probability. But it is :~ot necessa~ to -he ~~ m~ prob~ili~* to su - ~t that the heights of ~ Ia~ - ~p of pe^e of sim:~r ~ and and are mughIv Anal or the the stopping Bins of spinner is mu~' Aim: owt ~ circle* ~*0 ~ 5~5 a Atom of ~e {sty 7-~t ~ I : 947 Ii: 5~:~5 in bird bind with the no~ curb that =~-~mately desc~es the distribution of scones. It shows quite cIeady how ~ norms:! cu - e provides an idealized mat:~-~:~tic~ m-~l :f6r ce~n diStnbutio;~s --I Moving - m particular ~-~tio;~s to an :~ized descnpti~ of 44;at}l obw - -A is ~ mbstant:~! ~~tion. ~~ ~~e of ~ :~the~ matical mode) such as ~ normal or --am did to formulate this Aide is ~ ='bstant.ial step towa~ understandl~g the power of math-emetics" Computer simulation is qu~e belp~! at this point.
: : \: o:~:E ~~ ~~Y s-~:~s ~~ ~~ ~ ~~ ~~ ~~ ~~ ~~ ~~: s~ ~~ ~~ c~= : to ~e : Of the bel~d no~ Cu - ~^ St~s :~n fO~e ~ "~n mo~' on the basis of ~~r eX periled ~6 Aid. 0~: t~t ~~} into ~0 00~77 8~6 5~¢ mode} provides ~~w ex'nence mth Bin and ra - -A ~e bas:c I- of nomad =~' ~ idea ~ standardizing ~sewa Hi.. to t<~30 sc~e of ~~.;:C} dears ation u:30:~ a~:~:LL~t ~~e m=n ~ ~ u~ of the sm~ no~! t~e tO ~~e missive - Din ~ be dcI~d in the setting of models for ror pattems in data. ~~ di~:bmi-~-s in ~e m~:~! =~= =~e We p gm$sion of descn.~.~ m~s ~r s:~ewada~e dat,3..~ t.~.~y mmt ,8W ~ar :~:~er ~~e even when it is undem~d that distributions can appear be~m ~ ~~: introduction to pr~v M~r,~e :~C With seve-~e dam w-~Id flaw ~en ~lvancing as ~~ts delop the ne~ mathematics! co:~s =d skids. The begonia study of Able dam comes tater than examination of ~ sledge van~-~' in ac*~e with our hm principle, but usable mathematical ~~s are more accessible in the Ace c=~. The bas:c `~ph ~r two van~le ~ta is the 50~17 ~5 p:= Vides ~ Setting ~r fling coordinates ln the p;~.~" C:~s (~e and male StudentS9) and outlic~ in ~ watte~ot proVoke din cu$sion~ The ~1~ ove:~l pauem is ~ linear God The ant: malice mode! that ~~s ~ ~~e description of ~ I:~r pa:~m :s ~ SttOig~t iiOC bitt itS C4~tiCO~ ~~i =~S it =~$~S
. of the :: and ~~d Of each wn~l ~ lone ~ ~ des~ptlon of {~= I: =d pe~s the (~= =t ~ ~ ~~ ~ ~e sol of I:~ar ass~atmn. The I. =~ciem., I - - ~ . . ls tied ~ tm dither muffin ~~s and me - ~ - ~~e a~:' ~ ~, :~ d~ until ~ q=~ advanced stage Of st~+ Me ~~wn = lichen in- - ~~v real ~ ieast sale- Essay t~ :~$' =~ ~~s ~e thresh of a: =~e ~~d of Cat - line =~ciaton. :~t as the standard ~~ ~~ld be del~ed unn! ~~ Ins grace ~ ~ mnt=~~ c~n =d le~ Auto Dressy need :~t m~e 5 of ~~cs ~ us ~ =~- M~u,~ ~ d~ =~s - ~e u~ ln it - - -m V ~-~ - sch~ By- - ~ Am yews of *I sc~! of the Fiery e~ to de~~elop quant:~ skill.- and mas<: A* In ship ~~ s~t I~.~.~s can bc fig. ~ ~e ~ by s:~e me th at are cOmputanonallv easIer ~ an 1~t squ== =d ~~= r~t to e:~e I, The Quant:~ Overly mama cIear explan~i.~. of such m~ ~r uSe in the m:~e m~* ~~ ~~-~ ~: ~~ ~~ Ma - ~~ p~ Or (~;*- don and lea~ ~~es m=~$~:~. Thew income the I ~n eXpl=~ and ~~e van~, the ~~n of ~~n to. =~ ~~n, and the e~s of un~*~*~wd "~ vanables~ on an subset asm..~. These ~~s a~ subtIe but nOt mmput~ional' thev aw be~ grasped by guided expene:~ce with a~ I-* ~~t =~ ~a' ~~ ing ~ ^~y of display. =d Ale-. meth~, and th - - me cIoselv rela~ to an un-~.~d,.~.~ of the k:.~.d.s of ;.- If by the I: and SO=~ sciences. I~n teaching ~a < -in ~ gene~ whoo! All t~= .: be chow~n :~t ~r their ::mpo~= :n the ~~e of Aims ~ Or their lm~edi~e =~*ce ~ ~~- thelr I: in ~;~ng:~ng I unde:~, and their I to d ing reasoning ~ut u:*~ai~ data. StM:stIcs Is I in -are ~ rim=~:~e :~t than calculus in most ~~*~:io-~d that Lime po~ce As: be =~ed in ~ su*~:~al e~-cti~ cou=e :~:n the in: per ~~W Vea~ ~~t includes mo~ ad^~d ~m ana.~is as \~:~.! as data I- probab:~lit~, and :*:. N ~ . ~x a. P-P - as, - ~ ~ ~ as. i- ~ ad- . J^\ ~ ^ ~ , , :* , ~ ~ A , . ~ Cood data are as much ~ p.~ of ::~t huma~ e~ as are If: d:~c p:~= and hv5~d com~ Chew are s - ~~) :*: w:~v producing data 1S an impo~. pa~ of tcach.~g ~~! data and chances
:~2 ~~ TO N~y Den Brie Is moM I: ~ Out On ~~ m~ - - we mt~at~y I ~r Bigamy sumac- bo~ ~~ed stratums ~ t~k ~r and explan=~= ~r ~~.~ ~~. ~~: ~~s fbrpm~:ng data to speck que~s am ~e mnce~al bake t~g ~3 8~5 t~ ~~t p=~ ~#. ~ no I'm ~r ~e :~e my.. am:. m m~ tm~th I ~d=ce 1s 060n =~ed ~= # ce ,!~t - ~s m~ ~ qu - ~n =d ends ~ =~ based on -I th~ ~~ oume - S h~ ~~ D~ use :in the -~ of Is m~m sever s~. M~ of 1t -is ~~ Ace. nu~= S~y pmvI~ ~ ~e - ~~r -= the teM. WI~ mnce~-~d e~ to Amp. ~ ~ - ~s ~m ~~ dents expended or :~;~s ~ m ~~d In~, pmv:~ ~ =n odor ~a ~ biting ~r me~n 80 :~ 85 ~~] it: ~: ~~.~ .$ ~~ ~ -A =~ u=~! mth olk chUd~o have ~e wider knc~ =d e ence to Bream the c=~;t of the ~~- Inte~ :~-~-~n ~~t s =~ ~ ~~e them=~s can. ~ put ~~ ~e cast.' =d he dime and e~n ~~d ~ ~ well u~. ~\,emm=t ~~ on< sea; t0~5 0: ~87 ~t 0~' 0~ ~~ p~$~8 in p~ 1:~n, house= ~:~' and I.. that aw I. ~ ,~ .~.~d ~~ ~ ~~' is m~ in th~e cTa~ =~d ls w! a~ p~ to stu~ts in ~~e cia~ withoW ml~-~he question of whether mnciu~s ~~t some ]~r -~,10n are ~~ oass Ha provide ~ natum] Brig ~ tea~:i~ dau analveis~ which has ~ sim.~r w~n on ~e ~~e -of tts =~-~. Simple questions are ~ - ~~* ":Ho~ ma~ children 1~w :n w~r bouw9- I: much money do wu h~e :n your pOcket9- The :~st que~on pmduces ~le number data' the s~d - ~~ - finals. JPl=ning ~e pro~ct:~n of data i- th;~g ahead to the 8$~8 ~~t ~~.~} t~ caned ~~ ~ reminder as :~nt to pm~:~s a~ed with mfWa~ as to teaches e~ attem~ to whether thelr students stould ~~e =~s or decl'm Is =:n also prods= cia~ dam~ w:th ~ tape mea=~#; find the shoulder width an-d amaspa3~ Of an the I ~, then make ~ scatte~-~t an] street Rho :-~ionsh~p reveai~. ^~ 8~ ~ ~.~ 50~6 0[ ~~# E:~penmentat:o:n is active data -~tion* (3~, ~~:~-~r qu-~g or -messunng~ seeks to co11~t dada without cha¢~g the p=~e or th:~s obse~- In an nm-ent wee actually ~~ly mme ~~:~:us in order to o~ - e the -Gus sponse. ~e Ii- between exp:~-^ and response va:na~les an essential any: of cau~ e:~;~3~sis cieare~ in the setting of an -I The -~:~;:me:~s most Similar in basic sc~, unI~e
Ad. ~~y ~3 the qu0~.~: or m~s th~ ~~ Cla~ ~' :~O ~~e mn~ ciusiOns ~ applV to: ~ wodd ~ ~. when ~~s heat ~ :~d volume of ~r =d watch ~ balloon 0~$ t50\ ~ 8~6 to unde~6 t 3~ ~e b~r of the -~e ~~n -~m al~ ~ ~t of ~ on :-n ~~. This r-~t ~ ~~ ~ ~ 0~ t00 ~^ hidingm mass data to Idly -angry Sa'70$ has ~e =M ~ pant ~ t=~n -~m -~a abed th:s one cia~ to ~m ~~ =~t ~ i=~r :po~at:~. How ~ ~~e ts a, tOplC W~W ~~$ Wlt6 I ~ ~ t53~ =~ generating m ~r analys:~- SO also has mu~ -lo ~ abut how to erimem** arch the a~ce ls not r~ ~ mmt eXpenment~s in .e s=~= The - ~ ~ ~~S and ~ expen~s ~ ~ I.* twic in the ~mM:c stu~ ~ dam pmdu~on. But =~r top~c =~s 6m, ~~ logon 8~6 :n ~~=penanced 85kiOg que$~=s and measure to; pmduce ~~s data bmb amid the I: -of :~^ ~~ " me=~re ~ ch~:e means to r~nt -it by ~ ~~. This c notion ~-~-~ mtm~s an I-- Thinking ~~t -my w:~:t i=~ at Acre to ~ mature =~ of w~ smog nu - ~~s are Chum and others ~ in -or n-~- Aims~ iS ~ palm (~e or I w=~, to measum ~ pm:~r charm tensti~ Bmn mth mn~le physics ch=~stics. ~~h is -~we 8~ t~ ~ =~: ~~ 60 it" ~~= is 5~77 560~0 ~¢ ~~ ~0 hi- - ice that 7~0 can p~ be~ the :~ sh~es possible ln t~ dimensions as ~ -aft ~ mIer besl~ a~ Ien~. we man co.~m ou:~S with undeml=ding the -~k=~stic to be mea=:~d, w1~th dI; ~ satin I i~$ 8~0 ~~t ~0 0~- ~~t ~~ ~ tidbit :~5 t~ -abler untts. Even ~ physical measure-~:ts the stud~ 0~f ~~se ques tion:s ex~s thro~:t the school ~= bmb in mathem=~-~s and :~n sci.~" But the Widow of physics m-: :s simple =~ed Erich the meaSur-~:t pr~Ie::m.s o;~-~e s.~.~: and I sciences. What is ~ g006 W87 tO =~6 60~# =~6 ~ {~> is 0t tteendtin~ss 0{ ~ fellow sm-~-~9 ~~t do the Iowa ~~s o~r the A~ and S#AT college entrance cxaminat:~s really measured ~ detailed examination of such I ~6 i6~ t00 ~t 850~. 8~t 5~5 860~;~6 6,0 0#~6 always to ask w:~r data are :n ~~t Did ~r the proposed I ~~= m~er 63 ~~rs of ~e are I'd in more ~~! accidents shim dnve a~d ~ ~ and :~. So teens a=~t so n - + aher all? Nothere are many more d~ o~r 65. Thee rate rather than the count of acc:~s is the
: Il:4 ~~.~:~ TO N~Y Brats me~:~t =d the ~ acm th~: :~: ~ ~ the ~e semnd :~or aspect Of ~ qu~t of me~ a~ ~ il~77 t~ ~~ ^~ =-~0g p=~ =~ ~ ~ by as when ~ s=le ahoy m~s 3 ~~s I~" B~ ail: ~ so ~ 1.~a On~ - ~n t~ "~e ~~- ~~ m~: ~~d M~ 05 cIe~y u~. ~~e ~~s In S~ s=~: 1s ~ m~.: ot intense ~e, ~~ce no "-~- v~= is a.vali~ie ~r =~+ ^S um~' DISK crease 1s :~ mom s~ t6= bavm~ or m~! m-~+ dA un~ pro~s ^o is )~om th~ is, ~~ed meas ments of ~ ~ q~:i~ ~ not Bare ~~ remits. Me vary n mmmon myomas say as b~m ~~= ~ ye ~~s are . . s~] wI~ to the ~d,: .~' ~ we ~ ~~d to~ l~:~ng ~ . . ~~= th~ de~:~e ~~t vanatmn =e n=~. Wpum~ ~~s ~ ~~ be~n s~e Fig when m~ Ah or weight, or to est=~e ~ Ie=h or count by 6~7 ~~= ~5 ~ 80t of At m=~ ~~e ~~ - titan can ~ ~:~6 am 6~0 ~ ~ ts descn~ bY the =~r of ~e ~~.~n of m=~ a~ \~n ation by t~ spread*` M=~nt actw1~es Lowed b~ d~io-n Of the ~m the He increase spend' $: ~ t66 ilk Ot t66 ~~: O: : su-~. Here ~s an example for ~ =~= con* The ~~*~or a~ed each ~~nt to measum =d r=~ h:s or ber pulse :~e (heartbeat per m: on ~ }awe of paper. A -I of Me co~d data showed ~ =~r that ~~ ~~:Aped from ~ gmss <>~r, thou= ~o one Bard ad*- m:~t having r~.~d ~ seated ~:~.~e m~ of ISO. ~e stemplm a~o so ~ suspi=~s Con.~tion of pulse rates en~g in O.* ~ ~-~ re~¢n t~3t s~=l ~~ts na<:t teamed in aerobics cIas~ to count beats i* ~ seco~s and m.~V by :~. This Ied to ~ ~~ssion of the measurement meth~s used. Mo~ ~~:~:~s h~ COunted ~~s ~r 60 seconds. The class de c~d that th:s :s :~m accurate than the aemb:~s 0~3455 ~~, but :t Offs ftom papa;! beats at the beginning and end off the 6OwS=~.d pC=~* ~~C SO~d timing exa=~:v 50 beats w~h ~ stopwatch am calculating beats :~r m:;~e Am this time. ¢~s waS ac~d as ~ more accurate p~:l =~ men: method.
115 ~~ of S~e w~s and e~s )s ~ com talc ln =tiStM and ~ malor * ln =~- oata analv~s ~~= A; strong t~ spe=~c ~m at ~~- ~w the Ha ~ ~~d as repre~ sender ~ ~~r ~~. It ls ~e ~~n we seek too un~:~. 81~=S do ~~ 6~ ~~$ ~~ed It ~ tO =~. ThCy P¢~, ~t ~~' :n tmng to e~n v~e =~s when -an ex =~ t=k ;s ca~ out by s - ~~! ~~ in ~~ of inch cham=~= of ~ah, Mmb~, =d Ru~. The "~am;~- wint wew ~s them ~~s ~ rep~ of ~ ia=~ Bum of ~~ ~ ~ ~= rioter ~~. ln :~al - ~~S that MAY cxpla:~ the Awe of Saran' Mature, =d :IR~. ~e transltIcn ~~m d~ an~S ~ i: ~,~s ~ p~l tn ~~l ~~.~ion. ~e sample ~n ~ is no Ion.= ) - a~ Of 10~. ~r ~~se ~a. T! is ~ =~on of ~ ~~ vantage ~ be ~~r~ - ~~ the ba£k~ound of ~ d~:ion of ~ ~~m Yanked' ~ mu~ be v~-~d agates Cat would he=en elf we repe~ed thea p~=io:n pmce~ ma~ na: =. The ~~- of thee ha- ideas =~t be ~~i:~. ~-~ely, the :~e connection of des:~d Pa pmdu~n ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ . · - _ w1th the 1~ of pmbab~litv and the IO~ of Here need n~ ~ Th=e Is mu~ valuable m~t into ~= t~o 5~4 ;t it At. ~~' i. ~4~ 0~-, t~ ~~ ~~ 8'4 ~~:6~0 based on ~ ~w indi ~~ 0~5 ~~ to us mCuences our thinking in SAYS ~~t -A Ad cxamin~ion and therefore ~~st ~ Amps I:~vidua! ~$ catch 0~4 attention ~~e thev ~ u:~.~! in so~e w~ ~ ~~e they ~~r in our ~~ I, :. ~~s an~ 6~84 -~ ~ ~ :~* ~ 0~ct thes0 08~5 to 6e ,~ 3~-V4 ~ twirl i=~: ~~g =~ - :~> ~4 ~~e sampies I'- t~0 :'*~.~5 (~ t50~` =0 3~0 ~: all ~:~= · an e~. at o:~* ~~:~e column1~ Ann ~~= condu~s ~ volunta) re 5~ - ~ - x '{~# ~~ 5\y 35~ t0t :4~5 to re pmvo=~ question. The ~~ts ~ ~~ Cad fOr news =- ticies 34~6 =~0 ~~ - rim th8t p~CiZO 6~r column :~r 6:~t 5~7 i~ t~6 =~ i~0 6~0 ~ 00~:~ is ~~#8i:~. I~ 1975 A~n Lan~= asked `~*If vou had )~t to ~ Over min. ~4~6 >~ ~*~0 0tii670~- ~~t 7'~ 0{ t60 ~~ i07000 =~5 33~6 ~04 M3*=r I their tCSpOOSCS bar ~~4~g t8~5 0: t~0 0~.~$ l~3i0~6 0~ them b~ the:r
I:~6 N~ =~£Y : ~ V . ~ ~~ * ~ ~} A ~ ~ ~ ~ ~ into ~n question. ~ ~:onvMe random ~~le caged ~~= to ~ anention p~ I.- ~ ~~m ~ =~ ~ th~ 91:% den behave thirds ~a:~- is ~e n~e of wi~ -~*~e tO at=~t pem ~~ =~ :~ ~~ pm~= 70yD "~- when the t=~ ^ :~ " - ~ S~ d~ c~ no =~l in~ - ~t *A ex e pe~le who sm~ Forward. Yet ~ n~ media not Aid: ~d ' w~e ~ as ~ ~ descnbed ~ ~~ .~' m ~~e ~~:n =*d wnte~:n p~s t~ pm~e mow ~,(b ~a,- Me students - ] -fib fled ~~+ all Of an - ~~ :~e arced volunt~ :~e makes clear ~ n=d ~ ~ systemic ~~d Aft ~ec6~g s~ - s" ~e smlim~,s =~ method is to Tet im~rson~ chance sele~ the =~+ film ~~ -~i-~*~:~s the ~~s of pe~ chol=, - ~~* ~ ~e ~~r ~ ~ the ~~* ~e -I ~ fib . :~ . A use :~f - ~~e is the mO~ :impo=~t Id pnnmple fol* If* I: ~~. l: 5~5 3: ~~ ~~ ~ ~~ 5~ j~7 ~t ~ ~ ~ tragecus when set again ane~ evidence and volu~* :: ~e u~ ~ chance is lllu~d by sImple random: ~mp]~5K' - ~~ Eve ~! po~e samples of ~e stated s~e t~ =~e chancre to be ~ ~~le ~~y :. hit ~~= ~~$ ~ 0~7 tO ~~ =~0 i~ ~ ~ C]~= 6~t by d~g n.~s - m ~ bat or van-~-~d ~~S from ~ sampling bowl ~e of ~ random number table ~~0W5$ -aim 6~7 00~: aim:: Do :~11 the wa~ that too rapid I of :~e comp~:~er w:~} obscure ~e naive ~f random ~~. The me elak me ~~m mmpl:~g de~s used in n=~: Anile m - ~~s need not 8~t in tDl=~*~) is The aim: Id =~=i~ exper:~ts =e closely mIated to s3~e mn~m,* ~~. once ~i:n the :~d ~r ~ design can be -I appa:*~:~t ball In* of some uncontMIe~d or all Id Ax;:.. He~ ts an =e,:~.~. ~ ~~ science ink ln the e~:~s of pr gan~ in chan~ =~.ions Conducted an expenment wIth ~~: d=t subjects. The :: took ~ test of their att:~de tow~rd occupy, then read oerman prop~:~-~ ~:~.~v ~r ~~! months, after which their attitude was Chin measured. The ~~r was ~ 940 Belween te~ and retest' Oe~ imbued and con~ nce. The st~, Phi: towa~ :~ =~Y 053~d -I but we shaH never know how -much of this :. was Me to -~:~g Oe~n prop - ~~a
The d~ ~ ~ th~s expenme~ had ~ f o~ Ail t~ in {~ ex pe~s in ~ =~M sciences. 0~e the ~60 0~t ~ t50 i~7 =~ ~ =,(5 5~: ~~$ ~~n ~: ~ K i~ ~ tr=~= =~ be distinguishedm the e~= of I- Blest I 8= ~ ~~ K s - ~~V - ~~: eXpenmen~ into ~ b=~e ~~ - 'come ion-: TO s:~st Ado zed p~e .~ =~s two t~s o~ of whl~ ~ ~ ~ ~ ~ con~ t:~:~t such as no =~.~g ~~.~* H~ is ~e USA . 0~ K .~ \ 7 ~ ~ ~ ]~t ~ ~ ~~ The ~~m allOcatmn Visits ~ ~~e mn~m ~~e of ~e ~~s t~ ~~t :7 t5-8 ~~g ~5 =~ Tr=~t 2. Recomb i:za~on amp that there \s no blaS ln assigning subjects to Amp:. The =~s aw there~e Sim:~r (:~n ~'0 aVe=~) before ~e tr~s ~~-~ 355~$ tt8t O~:~Si6C ~~S 30: CQ~ - OO 870 imps ~^~^~°~K-K~ _ ~ b''0~ =~. If care ts We'd ~ weat ~l 016~S similarly exc~ ~r ~e expenm-~:~! t=~m-~, a~ ~~e d~e in ~~w ~~ w- ~~Ct the e~ct ~ the t=atmentsK The i~c of -=mpam:ve randomized ex=~:~s aliowS -I ~~t caus~:~nthe reSwn~ is not :~ Amp with the treatment but :s actu~v caused b~ it4 As in the -ape of samp:ling~ :mox el~me designs are common in pt30ti00 6~t ~6 ~~ 8~t i~ - .. ink CIas~m expe i,, K . ~ 60~ ~ 63~ =6 ~~. ~~$i667, ~ ~ t0~5 ~ 38 ~~p ~*~5 t5~ =~m 5~0~:3 ~ ~ ~ two com=~g twatm~-~s ~r sewre I.. Stu*~*~$ cam-~ out the Sodom ass~. Some of the tokens bear ~ marl; on the bottom, ~~vis:~e when the =~om:~:~n is done* ~~e s~, -I to ekes ~7 Is ~ If* t~{ t~t him Isis ;at07 Augment in c~t ye- How ~+ \ ]~d 3nc~L0mi7st:-~n divide t35~0 subjects be the `o ~9 Do the ran~-miza:~ ~~v and display -~e dies I:* of counts" Repeated -I pmv:~es e:xperie-~ce ~r:Llb -~:~m van.~-~n that imps toward prob~litV and infer-~-
I ,\ ~ Be- ~~s With ~ fi~n=:~s of both d~a =~i:s and Ma ~~n :n hand, older ~~ ~ =~em0e senous ~~ studies" ~ am~= - = ~ cum=~m pm~s i~de it Tommie of st~t Opera: ~~t t~ :~5 86~ ~ ~¢ ~~ - ~ - ~ ~ vehicle= at :a ~ ~ion, cia~ifed by type =;d h~-~W as- [~41~ tv ~ bCeI186 pint =6 ~ e~enmem on the e~= of dis- t~ md aged on ~~s ~ shommg ~ =d ~~as~ad. her Flesh of su~ ~~- Movies vacate expene~ :~n a~g stat~s~al please An~s ~ r~ ~~ {Q arrive at said ~s~ is m~n~ But p~ p~e'ms ~ ~~uci~ t~ d~a must be amm mthin acceptable Ilmi~. Here is an exce~t ~om ~ w~ of ~ chew ~~! of new shames m~ or secon~t sc35~* Some of the ~m pmdu=~n a=~'i1;ies W0= Dim 0~$ i~. 50~6 ~ [~0 t~0 ~~ 3~ t50 ~~ ba~ ~~:~. Thmr eXpenence is cautI:~. Our 5~d te~ eXpenenc~ h~:~d us ~~t ~~a =~n I:s an Im~t c~:~t of ~~:~.s ~~:~n ~r at I=~ ~ ~~~ nr~ t=~:~:~g bow to~~ and =~ ~ta c~on =~:~= (~, dem~ng l~nt ~ ~ ~ ~ x~= and ~~e sIze) Is ~~! ~ :. S~, da:a `:~:~n ~s ~ mm:~:~g ~~= :~: m~s ~1 anat~s mom ~~ - and lnter=~ to :* Out eXpen=~= alm cO~ =^ how - ~r th~ data cOllec~n ~ p=~: some fO:~e chaneng~ In the cta~m Fm =~e our 501d test teaC =~; ~~ they they ~~nt 3,~ tO,0~t,N,A,'t0 8,=~UD't Of Ci~.~SS 't 1~! = d~ ~ ~ K ~ ~ ~ ~g 6~8 0~ t~ ~ ~6 t~3 ~ 5~ t~ ~8 ~8 at_ co*~*~e of* {~. These en: - proved to ~ 5'0 dismpuve ~ =~c press that the As gm~ r~nt to =~ct S=~ ,~ns that ~~ on da~ c~:~- K ~ _ ~ ~ I. ~ .. _. .~ . . . ~ ~BAB~ILI1~* Cha:~= wnat~n c~ ~ inve~i~d e*~Pi:~' Solving the tools of dam an~s to display the regularity in :~om outco~* Prob~-ilu s ~ baby o~f mathematics that descnbes chance in much more deli} than observation can hope to d~. :~abilitv theory is an i: -I d~:~*tion of the p0~4 0t =~*~ti05 t~ 606000 0~5i~ :~s ~m ~~:~e assumptions. :n toss:~' tor eXampie, is de=~ simply as ~ sequence of ins : t:~s each yielding ~ hem mth ~rob~i1:ity ~ ~ ~ K 6~ dais unassuming ~~*tion follow such beautify} msults as the Iaw of the iterated Io~m, which gives ~ precise bounda~ ~r the fluctuations in the count of 503465 35 t085~g :~ti~. The d:~-n of the count
hi: : - ~= ci: 20: 1:D / f ~ /~' ~~ - -hi -' - - D ~ - - ~- - -' :0 10 20 ~-8 0= T05$~ ~~ ~ id: ~'~6 (K ~~ i3~ 0~¢ ).~6 ~~= 605£~= ~C ~~O ~ :~ns 1n (~n ~~ ~e cent= I1ne lS the mean ni2 50~ OD ~= $~0 5~ CU - ~ - ~ ~~C ~~= ttC =~: :~:~e }:s :~. : ~~: of heads ~~r ~ to-~s of ~ fa:r co-in has ~ m=n of ni9, catch ~= plotted acme ~ a~ ~ ~ ~~;~1~ne (~ Figure 6~. ~e $~da~ deviation of the cOunt of ~~s 1n ~ tosses is O.~. ~ I~ of the :~ Io~thm ~~s ~~t fluctuations in the count o~f :h=~ extend j21~ogn I dev~s on e:~r s:~ of We mean. The I; of h=~s Mot~ maims ~ will approach w:~n a~v ~~ d:~e of ~~$ 50~ i~:i~V ~~ 8:5 t0$~g C0~ti~S, tUt ~~it C=~ 1t 0~)' If o~^ Dam an~' - ~n aided b~ comp:~r It =~- n~r dimmer the Iaw of the itemted Iogam~m# As m\h ot~er beautify! ~d uwN! areas Of mathemat10s> proba:bil itY has in practice oni~ ~ limited plate in even s~' - oc! ins stmction. B~e the ~~s ~ p~abili~ are m~hematical~- rather Limp-: it is easy A;- o~YerIook the extent :o which the =~ts of pmb~.~.ity conflict wall Intuitwe ldea.s that aw 6~Y set and di~.! to dislodge b~ the time students =~h secondary whorl St:~^~^ tio~s open persist even when ~~:ts =n answer typical te~ 4~ns I-:. The c.~) Fifth of pm~-~:~ty ideas is afford by both the c:~e of teachers and by resea - .~:
120 =~ To ~:~Y O~d -~e mth ~~s in e~r ~~= is an ::mp~t p~10 tO 5~l tC306= 0; ~~ p~]it~. it t~ ~0 = .~hemad~ ~~t It. tn ~e ~~v of ~~s of ~~=, =e ~ the £iw ~:tn~slmple :' = phenomena a~ ObSe~ omen e= - ~ ~~ cl~ ~~ =~* ~~.. 34.~.g =;n a~ to ;~ 1;~.s 3~.~c~ develo=.~t ~ re= dam 0m ~an= Is =d i=er ~ ~~m sawing am mm puter s~:~lat~. But no ~~= wh - = wch ex~e ce wars or Me in ~ Is ~~.' it ~~s Ills ti~ ~ ~~n ~ ~ 8~C i.~gh1; l.O.~O typo bFCh&~r~,~ Of ~~= ~ ~ ~ -a I'm ~ 6m ~ ~ ~ ~~ ~ · ~ - ~ K ~ ~ ~ ~ O , ~ 8~O devices 1.~. ~C =,~>r grew tam I. emi1 pa=m and not attempt ~ ca=~ =~:~on of =ch outcome ("Site -I pu~ the I- :~. ~~s ~ion: is ~e eas~r oecause mo~ t or: me overall p=~m of ~~ is one of the cow 5~= 0[ 68~ =~# Next wC~e 'that, ~~ =~s of outco~S increase With amend tn~, the pmpon,10~: (~r relative - ~.'enCIe's) of tn~s On Warm each 0~-~6 =~5 Ibis i~ tt~ i0~g =~K PRelines am the ma~ -~:~! ~dealiz~ion ~ ~~e ~~e Ion~ wiat~.~ ~~.~-~s,. A,.s, Students Ieam the mathematics of propo~i~, ^~ ~ pmb~tv =n be~n mth ass~s of pmb~ilitl~ to finite sets of o-~s and companion of obsewetL pmpod~s to the~ Ii- comparison ~ outcomes to p0~illt:= can bes~ if not =~v planned~ Computer sim~:ion lS w~ help~1 in ~~di.~ ;b0 I~~ DOCt Of t=~iS =gOi~d if -id t61811Nr<' I -my to 1:~ rel~y clow to pr~:~:~ties. T:n sho~ seq~ -of tnals, the dev:iat~ns of ~~ Czechs mom :~:ities mU o~en seem 1~*~ = ~~ ~~ ~~: Id ~~ -- that the nW described b~ pr-~iln~ applies ~n to shod sequences of mn~m outcomes* Th:s belief :in an ::~= **I of small nu~*~- ~ ~~ ~ .~ all ~ _ £ ~1: _ - . .~ . -I ~ ~ Aim Tne AT -ok ~~l,0= w~o see ~ =*n Of w1~^~g thrnws w11h dice as w1~ce that the M~er is "^h01~- ~ mu~ explanation o~ bee we ~~v un~:~e Oh's p;~-~il:~v of Mets in random ~~A Ask s - ~~! :~1-e to wnte dow~ ~ seque~ Is that imitates :D tosws ~ ~ b~*:~ed co:n H~° :~ng was the Io:~t mn of =~:ti~ heads or consecutive tai:~9 Most pecple wall wn:~e ~ sequence with Oo =~s of :~e than KAYO cons-~tive heads or talls But ~~t the p='~'6ility of ~ =n
~~ 'T:~:~y of th=e or -arm he~s ~n ~en.~:t, tosses of ~ :~r ~n 1s Onto%, and ~ p - I ~ e~r ~ mn of at I=~ three h=~ or ~ my. of at I~t th~ =~s i.s ~~r ~= ON 3Pr~ =~= -~ mns ~ quite did is ~ -~a ~r m~e:r sib* The mns of =~:~e hays or ~~ve ~~:]s In: anne~ ::n ~:[ coin Motif (~ ~ :~ed pi ~,=~) ~~ 5~.~ t0' Ad:. bent ~ don,! ex~ to ~ 00~ t05505 3~t ~~ :~dent thst ~< ~ in4~ce is ~~ng ~e :~ do: of ~ min" The =~e mlw=~-~. ~ on tM ~~n cou~. If ~ - ~r s"~} conseo31~ve shots, bmb ~s and -atl~ ~8t he or ~e h~ ~ "*IBM b.~" =d i:s'mor'0 'l - ~y ~ make ~e :~t - ~. Yet examin=on of shooting ~~, sh~ ~~ m::aS of bats made or missed are no more :~t the w~d be =~d :n ~ sequence of i. mndom m.-~^ ~~ ~ ba$~h is I::ke throwing Aces th~ of cou~ ~e palof ma~ ~ shot dies ~ pl:* ~ ~~+ As ~~e cx~s su=~t' ~ :~e idea of pmbabili~ as :~-~erm Motive fact -is -~-e sophist.~d and :~s -A empirics backing. So- I=~r ~ Thomson unde=~g of p~l,0~s the ma~ m~: for pmb~il;~- a* s~e =~e (~= of ~! pos~ s151e oute-~) and an a~nt of p~li~ untidying ~ few b=:c Iaws or =.~s th~ 1.~e the addition mIe P<A or B) ~ P(~) ~ p(~) ~r disto:~t e~. ~~r Fictive laws for simple combinations of events c~ be d-~ed An ttese or, more simpIy^* mot:~ dir~v - m the behavior of pm~~ions. ~~e ~~e taws am the mat mat)=! content of e.~~ prOb~ilitV" *At this-~-~nt in the d~nt of mathem~ pm~dity~ T~ ~0 ~ ~ $~= ~ ~~ =~ ~ ~5 tt8t ~ l~ ~~ ~ ~ A ~ ng w1th =~er as=ct of math~ati~ thinking that ls ~t I ~ -in students*" caw{~l =d :lit-~ ads of I~: ~~. Psy~ oh.. ~~vi~ pm~ilitv Id* o~r ma~ eXewi~ ~.~t ~1 misconceptions and =n help to co~ct them. For ex;:m:p~' Tve~ky am Kahneman'~: p~d -~ ~~:~s w:th ~ personality s~h of ~ vouch: *I and In asked ~~:~h -of t:~-~e ~~-~s was more pmbab.~- ~ Lln~a is ~ bank teller" ~ :~a :s ~ banJk teller and :s a-~e in the ~~:ni~ movement. A~t S5~ of the st-~s chose tbe ~~d statement, ewn tho~ Axis -~-~:t ls ~ s4set of the Crst. This -~or =~ed despite venous at:~s at a:~*~w p:*~:~:~s ~~t mart make the issue more pNtOlb~billt71~. O~`v'= ~ 6~ Of
Pi SCt:~e ~~e ~ ~ c=~ ~e the ~ =~r tn ~ ~~r W~. ~ . ~ # I, TO N~? ~ An- c~ to. thelr ~~= is ~.~ so~e {~: ~~ n~ps ~ deco: we r~ of :~th~^ - ~t If :n - Away Ants N~e~ et ~7 ~~ ~~ =~£~;: companions ~ pr:~ ~~= ~~- of ~ ED of Cad sawn Empha~s on: the Anal am quadut:ve =~s of prosaic ~~ bo~ pn~ to- =d :n currant win ~~ of ~e Ice 0[ ~~, it ~~t ~0~# Few ~ e development of If; appl~le s~11~^ as oppo=d to ~ bas~c ~~ ~~` ~f Cab Air t~~qU,1~ me detailed Shy* ~~t tti~ # ~ we ie~e ~e com domain of Awns concepts tOin at:! ~ ~~ ~ expo~ ~~e =e ~~l A. pMbs :~o tnm mediate Ill ~e choice of ~~] win ~67 ~: exhume' on wI:~er pr~abi:li~ will ~ pu~d as an :~at topic :n its own right or ~~r ~ is I Ail* t:o Ie~ to =~i.~l In~:~-~. lFim' ~ vegans :~* do not dwell! on Allis m~otLs for cal¢~-ing p~WH:ities :in Die sampan sparse gambit namncs is ~ I harderBy than ~iluy* Students at all: - els f~d =~! promos mussing and ~ffcult. ~e s:~ or como~:~= coes not advance ~ concepm~ understanding o~f chance and Libra i655 t0~.= t~ 0~t t0pi05 it ~~i~ the ~~:litv to use pmbabi~l:~* mode~* I:n m~-~t c~ ~T but the simplest count:~g problems should be abided. ~ more - ~~! step fo~d - m the basics of lid :s to cola sider Additional p~litv:~ i~' =d multi:~.i~ :~.~. Be of ~e occu=~ce of an e~t ~ often modifies the I* :~ assigned to- another ~:m B. W: ex~1~, 1~g ~~t ~ randLomIv selected university professor is femur red== the prob~il:tsr that the Dro~r ~ bald is I The conditional pmbabil: - of ~ ~~ A, denoted ~ P(~, need not ~ equal to Pat, if the two aw equal, COtS ~ 8:~6 ~ 8~ id l50$C ~~:~S ilk tO~ ~0W i068S and basic skills i;~t are i: in construing probability models i ~ the net ~ ~ and social sc:~s . lt is gu:te ~~le to pmsent the iota of :~:e a~d the mules tipllc~ion Moe P(/ and B)P(~)P(~) ~r i: e\~s arid lime if Beer 8:~:~tiO~ tO CO~dLitiOD~ p:Ob~illt? in ~~. Chili p81h is att=~:~w if the ~~! is to r=ch statistics inference most e~icientiv and also words the =~siderable co;~1 d::~ties associated with conditional pr~-ility. ~e binomial dist~but;~s ~r the count of suc .4k.- HE ~ ~ · , A ,* ~ ~ ~ h~d number of inde}~nden: :~s aide q~ r Aphid. :~h
~ a:' Ace: ~ aw other inte:~w wpl:~ti:~:s such as mliabill - ~. :~f com~X sW Tf m~! probability ls ~~ =~ ~e I meaning of ire a~ t~ d=' of ~~V ~~.~g th~ id s - e ~~ ~ ~9 Ads exa~31~s ad Lions on cawal assumption of ln~en~=' wig emphasIs on in~nt tmimo~v ~ AL m~. ~~= mI=~d ~ if to b:~ dim dI:~' and to- tM ~.tip];i~io;n Mae ~r .in~en~t ~~- shou]d be shames of u:~ade secon ~ ~1 ~~y of con~ion~ ~:il:~y 18- a. when the ~~ is to enable students to- =~*= and use mall descr:~s 0[ P=~$ ~ ~ ~~v 3~ I~* :~^ of m:uki=~ proce~s my are n~ Oete~.~*:~c requtmS con~ti=~ p - -I To Eve onI~ ~ single example, the :~*e of ~w po-sitwes in testing ~r mre conditions applies Bill Ill to- que~i'0~s as cu=~t as Bins ~r d=~- ~ use of :~:~e ~~' and sc=~ng ~r A:~:~S w~ 1q ~~ ~~P h~.~d An ~ .< repon,6 wtew ~ Odin ^~^~ ^~ ~**. ~^ ~~ Hi ~~ =:~s can ~ ~~. We EMSA t~t was intmduced in the I* ~ screen donat~ blood ~r t~ pr=~ce of AlDs antibody. ~~en =- tib~s are p~, Eu.~A is p~i.ti.w m~ ~ pmb~itt.V of aims 0~-$ - ~~ t50 53~6 t=~6 is ~~t contami~*~d mt:h an- (~^t w~ 8 jpm6~i3~v 0[ t O:~. ~:~e num~ am :~tio:~] pr~:ilitIes:. If once in ~ tho-~d Of the units of blmd s~d bv EusA contain ALES Blip then 98.~ of all Reside respo:~s mI! 'be ~~e poM!:~ - The c~:~n of: ~e Id: scree~i:~:g te~s ~ s::~e tree dia~am such as that d~d in bile 7. Sm~s a~d mth an un~ndi~g of cond~ition~ prob~ilit\t ~ tr= diam:ms can easily problem c<~r simu*~n of w~ too c.~X to stu~- a-~ticallV. Conditional probability bnn~ ~ new s~ of co-~! difficulties that~ like those In the ca~v Shiv of pmb~ilitV*, can ~ easily and u~V ove~i<~:lked if ion is ove~) dir=~d ~ teaching de6ni1;io-~s and mies. :~ts 6~d the distinctions among P(~B , Pow and P(A and B~ hard to- ~~" D:~ ~ photograph of an att:~:ive and well- dressed ~~an and a~ the probability that she is ~ fashion model- The Andover sho~w that the question is i- as asking the conditional prob~li:~v that ~ wOma~ k.~.~n to be ~ ~~n mode) is archive and pre - ~~ce Of ~~se positives among EuSA M~:~S antibodies ~ be =~d Wt m~ ~ - 67085~-, l~t i57 t08~8 00~*-~6 ~/AB) and P(~-
~4 oh-: / ~ \ · ACES a:. :~:1 ":~S \ 'I No O ~8 1 8~^ it} - - C' 'A . ~ . PRO A103 & ELISA +) 'P(~A +) i i ~ i :~: , ~ _1 ~ ~ 0709:1 :~' ~ [0 \7~c x ~'A EUSA ~ Am. A 0:2 ~ ~ :.~? ~ sU ~-~s X A ~ ~ 3 ~92~ 7 ;~ ~ .; 0~ ~ ~ It'd =~0 0t ~= ~= t~ ttt 6~^ t¢~t ~ Am 30~] t~ = =:~ 0~t ~ ~ ~ BACK ~ t3~= i~ ~6 {~6 t60 ~0 =~tAA0~ pa - t.) ~ tt05 3= Q ~: SC$ ~ ~ 100~: tYl~g t00 l~ ~ O~ ~ th8t ~ $ kOO~ =d the -~t ~ - ~= pr-~ility Is wanted am an essenti~ p=~.~:ina~ tO ~] work w~h PI Trans;:~n ~ {~e Random sampling and cXpenment~ randomization provide expend ence w~h random-~s that mot-~s not WHINY lLO S!~) O: p - ability
but TO ~ reasoning of prob~ :~. Repe~d s~ ~ A - ~~ Spa= =~ Id ~ ~ ~ + his car: :s -Sodom in ~e techn~ 50~7 ~Ct 'God uses an explicit chan~ mechanlSm ~ :; pores mn~n -I 61~% -I ~! ^~mer~¢an a~:~s ~~t n~ be~h invoice ~~ wg~s ~ ~~ -of e~r ~ w 0~s the awake d~ of random van~:~n ~ Molar "male servos. S-~, the Conclusion Bat ~ n - ~ -m - ~~ - ~~t ompe - ~s =~ ~~nt ~ ~ whined only if t~ ma~n of wper~ exceeds ~ p~e ~~m vai~adon ~ simile ~~:~- rhe mn~ outwmes ~-~d from Ma brow are skeptics such as sa=le proportions am =~le me=~. -ample syndics are ran~m awns Band :~omena having numenc~ val:~" The :*:e~r 1~e~ Behavior of wch nat~-~;~s in repeated ~~li~ or mpeated ex~=tal ~~:n :s d=¢rib~ by ~ ~~ din t - Lion. It is u~ to view "~ di~mions as W~+ dies tnbutlonS of mudom cant ~~om wn~' their di~r~rnion:~' a~ shed moments make up ano~r Ink of maten~ in inte~e p~f~b~ili - No Proportions if the di~mi-~n of ~ cwnt7 which is be: under slightly idealism a~ - lions. SO m=ns he ~ normal din tribmion if the population I :s no~. Genem! mies ~r manipulating means and I- of random Vance= ~~ to emu pie p~s and means. In I the I deviations sample Ills and mews ~~h decmase at the me 17~ as the =~le sue it; it ~ ~= ~~ leads to an undemanding of the advancers -of larger ~~. What 6.~5 85 t~.-0 ~~t ~ 0[ ~~s ~~s W4~ ~ ~ 69 e ma~r i:-~t thec~ms of or-~ilitv addre~ this oue~i=. The iaw of Ia~ num~ - s that ~~e pmpo~s and mews ap~b (~n venous senses) the -~ding pr-~ions and means in the under lying population. The cen~ limb theorem Ace t~t both portions and means become app:~ximate6? nor:mat~ distributed as the spew $~C ~~. Fight ~ i: the centm! limit theorem in ~aph:ical fo~. It be~s With the did of ~ bide obsession that is n~t s~d ante ~r from no~- Di~:ibutio-ns of this ~~ ~ often used to ~~ sc=~e the :~e in service of pa~s that do n~ wear out. ~e mean of this I. di~:tion is 't. ¢e other =.~s in the 5~e show the distribution of the mean ~ of samples of size ~ and of size ID drawn mom the on~ distribution. The char=~tic no~l sham is al- read~Y starting to em, when oW ID ob=~tio:~s are ~~. computer $~tion could show the e~ wen more dram?~ticaU:~.
126 ~~-~ ~ ~~Y _ I_ ~ by ~ my: an, ~ w i ~ :3 :5 ~~R ~ ~ I'd cent=l ~ Im'l! the~ \:n a:~~ ~e ~~= of ~~s O~f ~ K ~ ~ ~ <. ~ ~ ~ ~ t~ no~ ~stn6~mn ~ ~e =~¢ SI" :~. - 'm ~ (b') ~ 1c (~) ~ no is ~ suc5~.~ oom' o: m~ ~~ is quite Hi. if ~~> m~- pw=~. T~tio=1 Liege inst=~on :n stat:~i~ :: th substantial dow of ~~iT~t I=~t topics on ink and on mndom v~esprece~ the at: of ink ~me unde=~nding of independence and of distributions with their ~~s and standard ~ viat:~s :s ce~ainIv n~. But ~e degree of Akin ~~is:m th dice theSe topics aw trad~:tionaDy- ~ght is ~~* un:~$~i~ at the coll~ Ie~ a~ out of the q~;~n in wco:~dasy sake Both the length and the Bilk of the path to statistics via formal pa~ b.ilib-- a~e against ~~S tm~ition.~l appr.~+ ^~S Ca~d =d *I concInde'5 TeaChIng ~ (~;~! msp of p~~lV st\~! appears to be ~ w~ At task~ ~t width Emil and A. *:: we :~e the pm=~c I;.- -My two rcse-3=:h e~ns that would p~ed in Am::: on-e I;: 00~ t0- 0~ =~5 t~ I* Tam :~s o: p~v and =e that cx~= It'd I i~s of ~~l 1~e Can be taught .~N' o't teChn1~Y Co~ pmbabl.~. ~~- ~e cm:pincal emphasis of ~ a:~^ d~d Add dally beginning ~ the ea~y go o~s ~ setting Or teaching ~~h bat Si: pt066b: it) ~06 0~" :~" 5~? §~st Pepsi=] 806 then u~g m~' can demon~e the essent:~! =~s of prob~bi:~- id Offs :s panicuIarly suited to displaying ~~ng distnb~lons. OnIN -tulle ill probability is neeW to th)~k ~-~t sampling diStnbu~f tions. As the earlier d:~ssion of no~at distnb-utions India data ~ cscn:~o n: provides an ade quate conteXt ~ ~ ~~tic ns as i: i zed :~:~tica1 models for Donation. The core mat:~emati~ =~m tau~t to all stu~ts should include data analysis a~d an cm:pirica1 introduction to CO1NY basic pr~il~V ~~s and I~s at ~~.t the Iwe! of the Qua;~tive 3[:ite:~v matenal.~5
Air 127 3~5 is 00~6 ~~ ~ ~~, 0~:Q~7 ~ 8~^S)5 of ~a and m~ infe~esm d~ to t~ ~~ng wall.~- I: e=e - m ~a to wal:itv" ~s ~ kits top~c id ~h much room ~r di~eemem~s of ~ phito-~:bi=) nature. :~ its no ~~ng ~at t:. +~S ~e on the m:~t ~~ ~~h ~ ink Samet: = ~ mmparat:w ove - ~~ of the fig ~~- 3El~= or asss:~- ~ .. T~ :~ ~~.~t p3~ d:~e separates Bave~n mfe~ en:~e ~~ ci~! ~er=~-. ~~e un~r~a~i~ of the Id ls e~! to- mse =:~m d~. ~e queMi-on of induce in $~ - ~~ IS 50W tO ~ CO~:~- ~~t ~ ~~O ~~~ on ~e Isis ~f smI~= ~~ed-m ~ ~~. ~ I: is n:~r that ~~= the populate=' such as t~ m~ height ~ of ~ A;nen=:n women age :8 to )~. ~ I ~~tist:ic Tn this c~e is the acid m-=n height ~ of ~ random sample of -~ng women. W: pur~ poses of :~= ~ imagine haul ~ ~~d ~' in wpeated samples mom ~e ~e Amid The sampl~ dis~?u£ion of ~e statistic describes th:s V=ianonf The ~~ d~:~n ~fe0~;~s the id pa*ra~zmer~ tb:s caw ~ is the mean of ~e d~ributmn of x. It is be~w ~e dealing did depe~s on the unknown p~: th~ the I.: cam~ in:~rm~n about the }~:;~. CIa*ssi~ ;~ce :s rooted in the mn-~pt of pmb~ili~ as Ion~ Sari - ~~ =d :n the co=~;~ding idea that the concI:~s of ibid == are expre~d in te~s of - ~t ~~:~d h~n :n ~ pr~ionf To =y I; we oft -950~ con660nt ~ ~ lies Tweed 64~5 and 64~7 inch:- ~s sho~ha~ ~r "Wd ~t this inte - ~ by at m~d that is co=~t in 95~ of all possible samMes-- Pmbabili~ statements in classic inks amp to the m~ miller than to the specific = cl:~n ~ band:~- probability $~ements about ~ specific = s:~n :make no =~w - ~~e the population pam:~r is find, though I:.. The B=0f~ian a~b ~y1356s to bnng DnOr indention ~~t the value of the p=af:~:~:r irk play. This is done b~ re~g ~~e er ~ ~ ~ ~fndo~m quant:~ with ~ ~~wn prob~:litV distn~ tion that eXp:~s our imprecise indention ~~t its values The I= height ~ of all American ~~ng women is not w*:~om in the traditional Inset But :~.! :s u:~. T: am. quite sure that ~ iles ~~n 54 inches and 72 >~ches~ afnd ~ think it more li - A' that ~ lies near the center of this range. My su~-e 85508Y8~-~t of ~f:~naintV can be 0~3Y506 in ~ At,. ob~ ~ li ty ~~.~:ti ~ ~ ~r ~ ~
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
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
=~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*~:~.
:~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
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 <huh n:~s to bird dams, wee ~~d ~ the co~;~n to be close to O. The ob$~d Add- ~r ~
~ 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
:~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~+
: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
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.
:< -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. ~~ <o ~~ ace. Bed an. Ad: W.~. at, 1989. 14. Ados ~n~1 of ~~ ~ styli=. ^~~ ~_~^ a: ^~# Cane of ~~ of alkali=, 1989. 15. pan, Offs Cams, Baas; ha, Rigs. ~~ ~~> ~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.