National Academies Press: OpenBook
« Previous: Quantity
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 95
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 96
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 97
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 98
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 99
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 100
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 101
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 102
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 103
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 104
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 105
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 106
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 107
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 108
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 109
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 110
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 111
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 112
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 113
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 114
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 115
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 116
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 117
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 118
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 119
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 120
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 121
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 122
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 123
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 124
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 125
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 126
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 127
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 128
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 129
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 130
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 131
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 132
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 133
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 134
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 135
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 136
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 137
Suggested Citation:"Uncertainty." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 138

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

~ ~ ~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, how—er- 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 ~ -to—0 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 ~m—m bow I: ~~es and cIln:~] tn~s =d Ws~ and finance ~. M~v c~s must d~ m~ ~a in more—se) 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 predic—le outcomes. drop ~ co:n—m ~ 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=a—m 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 died—m 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 to—b- ~ 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=~es—can ~~ 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, how—e~ ~ 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 a—ys~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 d—I- The—chic ~~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~'s—st 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~ ~ simile—aims ~ 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~ she—how 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:dual—se~. 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~.~n—of 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 d—cI~d in the setting of models for r—or pattems in data. ~~ di~:bmi-~-s in ~e m~:~! =~= =~e We p gm$sion of descn.~.~ m~s ~r s:~ewada~e dat,3..~ t.~.~y mmt ,8W ~ar :~:~er ~~e even when it is undem~d that distributions can appear be~m ~ ~~: introduction to pr~v M~r,~e :~C With seve-~e dam w-~Id flaw ~en ~lvancing as ~~ts d—elop the ne~ mathematics! co:~s =d skids. The begonia study of Able dam comes tater than examination of ~ sledge van~-~' in ac*~e with our hm principle, but usable mathematical ~~s are more accessible in the Ace c=~. The bas:c `~ph ~r two van~le ~ta is the 50~17 ~5 p:= Vides ~ Setting ~r fling coordinates ln the p;~.~" C:~s (~e and male StudentS9) and outlic~ in ~ watte~ot proVoke din cu$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~s—is 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 ~^ hiding—m 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$~=s and measure to; pmduce ~~s data bmb amid the I: -of :~^ ~~ " me=~re ~ ch~:e means to r~nt -it by ~ ~~. This c notion ~-~-~ mtm~s an I-- Thinking ~~t -my w:~:t i=~ at Acre to ~ mature =~ of w~ smog nu - ~~s are Chum and others ~ in -or n-~- Aims—~ iS ~ palm (~e or I w=~, to measum ~ pm:~r charm tensti~ Bmn mth mn~le physics ch=~stics. ~~h is -~we 8~ t~ ~ =~: ~~ 60 it" ~~= is 5~77 560~0 ~¢ ~~ ~0 hi- - ice that 7~0 can p~ be~ the :~ sh~es possible ln t~ dimensions as ~ -aft ~ mIer besl~ a~ Ien~. we man co.~m ou:~S with undeml=ding the -~k=~stic to be mea=:~d, w1~th d—I; ~ satin I i~$ 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 tte—endtin~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? No—there 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 the—a 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 distinguished—m 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~:~n—the 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 P—Relines 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 be—s~ if not =~v planned~ Computer sim~:ion lS w~ help~1 in ~~di.~ ;b0 I~~ DO—Ct 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 -at—l~ ~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 ba—ts 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 pal—of 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 tO—in 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 harder—By 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~es—prece~ 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 p—a~ 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~es—m 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 B—s:= school. ~es :ts Came ~ ~e con~ns of in—cnce an =pr~d :n w~s ~~ - abbey statements ~~t the u~ p= :~f t~ p—ability :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 v—n 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 p—nullity is quite natu~, but it dlve~s attention—m 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~r—en Pi inference iS the ~^ Two t>~s of :~:~, con:fi~-~-e id and signifiable te~' figs ure in intr~o—tnst:~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 p—uces 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 ~ 9—K 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 ~~s—o 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 ~r—m ~ 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{ p—Dim 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 in—Idren ::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.

Next: Shape »
On the Shoulders of Giants: New Approaches to Numeracy Get This Book
×
Buy Paperback | $34.95
MyNAP members save 10% online.
Login or Register to save!
Download Free PDF

What mathematics should be learned by today's young people as well as tomorrow's workforce? On the Shoulders of Giants is a vision of richness of mathematics expressed in essays on change, dimension, quantity, shape, and uncertainty, each of which illustrate fundamental strands for school mathematics. These essays expand on the idea of mathematics as the language and science of patterns, allowing us to realize the importance of providing hands-on experience and the development of a curriculum that will enable students to apply their knowledge to diverse numerical problems.

  1. ×

    Welcome to OpenBook!

    You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

    Do you want to take a quick tour of the OpenBook's features?

    No Thanks Take a Tour »
  2. ×

    Show this book's table of contents, where you can jump to any chapter by name.

    « Back Next »
  3. ×

    ...or use these buttons to go back to the previous chapter or skip to the next one.

    « Back Next »
  4. ×

    Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

    « Back Next »
  5. ×

    To search the entire text of this book, type in your search term here and press Enter.

    « Back Next »
  6. ×

    Share a link to this book page on your preferred social network or via email.

    « Back Next »
  7. ×

    View our suggested citation for this chapter.

    « Back Next »
  8. ×

    Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

    « Back Next »
Stay Connected!