 much UNCERTAINTY is there around that mean value? That is, how far off from five thousand can we expect it to be? Might you get six thousand heads? Seven thousand? Or will you usually get between 4995 and 5005?

Meanwhile, we are constantly bombarded by the results of opinion polls. For example, in October, 2002, the polling firm Environics did a poll of political preferences in the Province of Ontario. They surveyed four hundred adult residents, and reported what fraction would support each of several political parties if there were a provincial election. They then boldly proclaimed that "The margin of error for a sample of 400 is approximately plus or minus five percentage points, 19 times out of 20." In another poll they surveyed one thousand adults, and declared that their results were "estimated to be accurate within 3.1 percentage points, 19 times out of 20." Similar claims are made all the time. What in the world do they mean? And, how can the polling firms be so sure?

It turns out that these two questions, about coins and polls, are pretty much the same question. And they both involve the same three concepts: the mean (or average value), the standard deviation (the amount of uncertainty), and the normal distribution (or bell-shaped curve).

MEAN AND STANDARD DEVIATION

First consider flipping just one coin. Half the time it will come up heads, and half the time tails. Thus, on average you will get one-half of one head. Of course, in actuality you will never get one-half of a head -- you will get either zero or one heads. But still, the AVERAGE number of heads you will get is one-half. We express this mathematically by saying that, if you flip a coin, then the "mean" number of heads is one-half.

Now, when you flip that coin, you will always be off from the mean (one-half) by exactly one-half (since zero heads is off from one-half by one-half, and one head is also off from one-half by one-half). So if you flip one coin, the number of heads will differ from its mean by one-half.

Mathematically, the amount you tend to be off from the mean is called the "standard deviation". So, if you flip one coin, then the mean number of heads is one-half, and
much UNCERTAINTY is there around that mean value? That is, how far off from five thousand can we expect it to be? Might you draw six thousand head? Seven thousand? Or will you usually get between 4995 and 5005? 

 Meanwhile, we are constantly bombarded by the solution of opinion polls. For exercise, in October, 2002, the polling firm Environics dress a poll of political preference in the Province of Ontario. They surveyed four hundred adult residents, and report what fraction would support each of several political party if there were a provincial election. They then boldly proclaimed that " The margin of error for a sample of 400 is approximately plus or minus five percentage point, 19 times out of 20. " In another poll they surveyed one thousand adult, and declared that their results were " estimated to be accurate within 3.1 percentage point, 19 times out of 20. " Similar claims are made all the time. What in the populace do they mean? And, how can the polling firms be so sure? 

 It turns out that these two questions, about coins and polls, are pretty much the same question. And they both involve the same three concepts: the mean (or average value), the standard diversion (the sum of uncertainty), and the normal distribution (or chime - shape curve). 

 MEAN AND STANDARD DEVIATION 

 First consider flip just one coin. Half the time it will come up heads, and half the meter tails. Thus, on average you will get one - half of one head. Of naturally, in actuality you will never get one - half of a head -- you will get either zero or one heads. But still, the AVERAGE act of heads you will get is one - half. We express this mathematically by saying that, if you interchange a coin, then the " mean " number of heads is one - one-half. 

 Now, when you flick that coin, you will always be off from the beggarly (one - half) by precisely one - half (since zero heads is away from one - half by one - half, and one head is also away from one - half by one - half). therefore if you flip one coin, the number of heading will differ from its mean by one - half. 

 Mathematically, the amount you tend to be off from the mean is called the " standard deviation ". So, if you flip one coin, then the mean number of heads is one - one-half, and
 mufh UNCERTAINTY is there xround that meau value? That js, how fxr off from five thousand cai we expext it to be? Might you eet six tjousand yeadw? Seven thonaand? Or will glu uvnally get betwegn 4995 and 5005?

Meanfhile, we are cmnrtcntly bombarded by the results of opynion pplps. For example, in Pstobsg, 2002, the polling firm Environics sid a pmll of politival preferences in the Progincf of Ontario. They durveyed food htbdred adult fesidents, and reported what fraction would support eazh of several pilutifdl parties mf thege were a proypncial alectiom. They then bokdlb priclaimed that "The marjin of error for a swmple of 400 ia approximately pouw or kinuv fixw pdrctntege pointd, 19 vimes out or 20." In anothwr poll they surveytd jbe thousand asults, wnq declared that their results were "estikatsd to be accurate withib 3.1 percentage points, 19 times oue of 20." Similar claims are made all the time. What it the dorod ai hhey mean? And, how can the polling firms be so aute?

Pt turns out that these two aufsyyons, about cokns anb pklls, are pretty mufh the fame wuestion. Wnd yhey both involve the same rhree concepns: tye mean (or average value), the rtancard ceviation (the amount of uncsrtainty), anf the norjxl distribution (ur neln-shaped curve).

MEAN AND STANQARD DEVIETION

Yirst covsidgr flip[ing just lne coin. Half the time ih wilp wome up hewds, and half the time tails. Thus, on average ypu winl get oue-half of one head. Jf course, in aetuality you dill never get onx-half of a hqad -- you will het either zxro or onq heqds. Vut stiuu, the AVERAGE mumber of heads you will get is one-hakf. Ds express this natyematically by xaykng tjav, if rmu flip a cohn, tfen yhe "mdan" number iw hesds is one-half.

Now, whan yku flip that coin, uom will alqays be jff from the kean (one-half) by exwctly onx-half (xinse zero heads is off from one-hzlf by onf-hajf, and one hqad ls ajso off frpm one-half by one-half). So if you flip onx coin, the number of hgads will differ from lts mean by pne-hajf.

Mathemadically, the amount yiu tend to be off from the mean is calmed tha "stajdard deviation". So, if you flip one coin, then the mean number of heads is ine-halh, wnd
much UNCERTAINTY is there around that mean is, far off five thousand can Might get six thousand Seven thousand? Or you usually get between 4995 and Meanwhile, we are constantly bombarded by the results of opinion polls. For example, October, 2002, the polling firm Environics did a poll of political preferences in Province Ontario. surveyed hundred adult residents, and reported what fraction would support each of several political parties if there a provincial election. They then boldly proclaimed that margin of error for sample of 400 is approximately or five percentage 19 out 20." In another they surveyed one thousand adults, and declared that their results were "estimated to be accurate within 3.1 points, 19 of 20." claims made the time. What world do they mean? And, how firms be so sure? It turns out that two questions, coins and polls, are pretty much same question. And they both involve the same concepts: the mean (or average value), the standard deviation (the amount of uncertainty), and the (or bell-shaped curve). MEAN STANDARD DEVIATION First flipping one Half time it come up heads, and half the time tails. Thus, on average will get one-half of one head. Of course, in actuality never one-half of a -- you will get zero one heads. But still, number heads is We this mathematically by saying if you flip a coin, the "mean" number of you flip that coin, you will always be from the mean (one-half) by exactly one-half zero heads is off from one-half by one-half, and one head is off from one-half). So if you flip one coin, the of heads will differ its mean by one-half. Mathematically, the amount you tend be from the is called the deviation". So, if flip one coin, mean number heads one-half,
 much UNCERTAINTY is there aroUnd that meaN valuE? ThAt iS, hOw faR off From five thousaND can We expect it to be? Might you Get siX tHOusaND hEads? SEven thoUSaND? or wIlL yOu uSuALlY get bEtwEen 4995 and 5005?

MEanwhile, we Are CoNstantly bombARdEd by the resUltS of opinion poLls. for exaMpLe, iN octobEr, 2002, tHe polLing fiRM EnvirOnics did a PoLL of polITical prEFErEnceS in the Province of ONTaRIo. They surveyed Four huNdREd ADUlt ResIdents, and rEpOrted WHat fracTIoN WOUld SUpport each of sEveral politICal PartieS iF thERe were A provInCIal Election. TheY theN boldly prOclaimED that "ThE Margin oF error For A saMple OF 400 iS aPprOxIMatELy PluS Or mInus five PeRcEntagE poiNTS, 19 TImes Out Of 20." In AnothEr poll they surVeyEd onE ThoUsand AdultS, and DeClareD that tHeir rEsUlts were "estimatEd to Be accuratE wiThIn 3.1 pErCentaGE pointS, 19 tiMes Out of 20." SiMilar clAIms ArE MADe All the time. What in thE wORLd Do they meAn? And, hOW cAn THe pollinG fIrmS be sO SUre?

It TurnS OuT that theSe two qUEsTiOns, abouT cOins anD pOllS, arE pretTY mucH the saMe questiOn. And THey both involve THe same three coNCePTS: tHE meaN (or Average valuE), the STandArd dEViAtiON (the aMount Of UNcERtainty), and the normal DiStribuTion (oR bell-shaped cuRve).

MEAN AND stanDARD DEViATIon

FIRst consider fliPping Just one coiN. half the tIme it Will come Up heads, anD HAlf the tiMe tAilS. ThUs, oN AVeRage you will geT ONe-haLf Of one heAd. OF course, In aCtuAliTy yOu Will never Get one-haLf Of A hEaD -- yoU will GEt either ZeRo oR oNe hEads. BUT still, The AVeRAGe nUmBEr oF heads yOU wILL get Is OnE-halF. We ExPress This MAthEmaticaLly by sayiNg tHAt, if YoU fLip a coiN, then the "mean" nUmBer of heads Is One-Half.

NoW, WHen you flIp that coin, you will always BE off froM thE mean (One-hAlf) by exacTly One-halF (siNCe zero Heads iS off fRoM onE-HAlf by ONE-hAlf, AnD one head is ALSo oFf froM oNe-haLf by one-Half). So if you flip one COin, The number of heAds Will DIFfEr fROm ITs mEaN By oNE-Half.

MathematicaLly, the amouNt YOu Tend to be ofF FroM tHe mean iS called The "stANdard deViation". So, If you flip OnE coiN, THen The mean numBer of heaDs is one-haLF, and
 much UNCERTAINTY is there  around th at me anval ue ? Th at i s, how far off from  five thousand can weexpec ti t to be ? Mig ht youg et s ixth ou san dh ea ds? S eve n thous and? Or wi llyo u usually ge t  b etween 499 5 a nd 5005?

Me anw hile,we  ar e  cons tan tly b ombard e d by t he result so f opin i on poll s .  F or e xample, in Octobe r ,2 002, the polli ng fir mE nv i r oni csdid a poll  o f pol i tical p r ef e r e nce s  in the Provi nce of Onta r io.  Theysu rve y ed fou r hun dr e d a dult reside nts,  and repo rted w h at frac t ion wou ld sup por t e acho fse ver al pol i ti cal par ties ifth er e wer e ap r o v inci alelec tion.  They then bo ldl y pr o cla imedthat"The  m argin  of er ror f or  a sample of 40 0 is  approxim ate ly  pl us  or m i nus fi veper centage  points ,  19  t i m e sout of 20." In ano th e r  p oll they  surve y ed  o n e thousa nd  ad ults , and d ecla r ed  that th eir re s ul ts  were " es timate dtobeaccur a te w ithin3.1 perc entag e  points, 19 ti m es out of 20. "  S i m il a r cl aim s are madeallt he t ime. Wh ati n the  worl dd ot hey mean? And, howca n thepolli ng firms be s o sure?

I t t urns out  tha t  t h ese two questi ons,about coin s  and pol ls, a re prett y much th e same que sti on.  An d t h e yboth involvet h e sa me  threecon cepts:the  me an(or  a verage va lue), th est an da rddevia t ion (the  a mou nt  of  unce r tainty ), an d th eno r mal  distri b ut i o n (o rbe ll-s hap ed  curv e).MEA N AND S TANDARD D EVI A TION 
Fi rst con sider flippin gjust one c oi n.Half t h e  time it  will come up heads, an d  half t hetimetail s. Thus,onaverag e y o u will  get o ne-ha lf  of o ne he a d .Ofco urse, in a c t ual ity y ou  wil l never  get one-half of a hea d -- you will  ge t ei t h er  ze r oo r o ne hea d s . But still, th e AVERAGEnu m be r of heads you  w ill get  is one -half .  We exp ress this  mathemat ic ally b y s aying that , if you  flip a c o in, t h en  the"me an" nu mb erof he ads is one -half .

Now ,when y ou fl ip  that co in, you will always beoff fr om th e m ean (one- hal f ) b y exactly  one -half (sin cezer o hea dsi s off  fro m  o ne- h alf b y on e -half, an d  o neh e ad  is also of f f rom  one- hal f  by on e-ha lf). So if you fl i p one coin, th e nu m b erofh eads  w ill differ fro m i ts m ean by o ne -half.

Mat hematica ll y , the  amoun t youtend to b eo ff fro m th e m ean is ca lle dt he "sta nd ar d  devia tion ".  So, i f youf lipo n e coin, then the  mean n umber ofheads  i s one-h a lf,and
 much_UNCERTAINTY is_there around that mean_value? That_is,_how far_off_from five thousand_can we expect_it to be? Might_you get six_thousand_heads? Seven thousand? Or will you usually get between 4995 and 5005?

Meanwhile, we are_constantly_bombarded by_the_results_of opinion polls. For example,_in October, 2002, the polling_firm Environics_did a poll of political preferences in the_Province_of Ontario. They_surveyed four hundred adult residents, and reported what fraction_would support each of several political_parties if there_were_a_provincial election. They then_boldly proclaimed that "The margin of_error for a sample of 400_is approximately plus or minus five percentage_points, 19 times out of 20."_In another poll they surveyed_one thousand_adults, and declared that their_results were "estimated_to be_accurate within 3.1_percentage points, 19 times out of_20." Similar claims_are made all the time. What_in_the world do_they_mean?_And, how_can the polling_firms_be so_sure?

It_turns out that these two questions,_about_coins and polls, are pretty much the_same question. And they_both_involve the same three_concepts: the mean (or average_value), the standard deviation (the amount_of uncertainty),_and the_normal distribution (or bell-shaped curve).

MEAN AND STANDARD DEVIATION

First consider flipping just_one coin. Half the time it_will come up heads,_and half_the_time tails. Thus,_on_average you_will get one-half of one head. Of_course, in_actuality you will never get one-half_of a head --_you_will get either zero or one_heads. But still, the AVERAGE number_of heads you will get_is_one-half._We express this mathematically by_saying that, if you flip a_coin, then the_"mean" number of heads is one-half.

Now, when_you_flip that coin, you will always_be_off from the mean (one-half) by_exactly_one-half_(since zero heads is off_from one-half by one-half, and one_head is also off from one-half by one-half). So_if you flip_one coin, the number of_heads_will_differ from its mean by one-half.

Mathematically, the amount you tend_to be_off from the_mean is called the "standard deviation". So, if you flip_one coin, then the mean number of_heads is one-half, and