basic statistics session 1

7/25/2019 Basic Statistics Session 1

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Basic StatisticsIntroductory Workshop MSBAPM

John R Wilson

[email protected](ote that !uch o" the !aterial and de#nitions are "ro! A $irst %ourse in Statistics& Mc%la'e and Sinci

edition)
mailto:[email protected]:[email protected]


2/78

Section *ne + *'er'ie, o" ter!


3/78

So!e de#nitions

escripti'e Statistics - Methods "or describin a as it is.

In"erential Statistics + Reach conclusions beyondthe i!!ediate data

Sa!ple data insiht -----In"ers population insiht

Para!eter + su!!ary !easure "or population Statistic + su!!ary !easure "or a sa!ple


4/78

More de#nitions//

Population + A unit that ,e are interested in stud(ie& reistered 'oters& collee students& !lb playetc.)

Sa!ple + subset o" the population 0ariables + %haracteristics o" the population or sa!pl

Measures + Assin!ent o" a 'alue "or each character

%ould be continuous (ie ae& ,eiht) or cateorical (A&B&% et


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Why in"erential statistics1

Population 's Sa!ple Population !ay not be practical or is too e2pensi'e

32. We ,ant to !easure likely 'oter outco!es4polli'oter is not practical so ,e poll a rando! sa!ple o" 'and then in"er ,hat ,e ,ould et i" ,e had polled A6

7here is a potential "or error


6/78

32ercise *ne

A sur'ey sho,s that the a'erae ae o" 70 'ie,8. $o2 belie'es the a'erae ae o" their 'ie,eless than 8. 7o test this& they sa!ple 9:: 'ie,

escribe the population

escribe the 'ariable o" interest

escribe the sa!ple escribe the in"erence


7/78

32ercise *ne Ans,ers

A sur'ey sho,s that the a'erae ae o" 70 'ie,8. $o2 belie'es the a'erae ae o" their 'ie,eless than 8. 7o test this& they sa!ple 9:: 'ie,

escribe the population (All $o2 'ie,ers)

escribe the 'ariable o" interest (Ae)

escribe the sa!ple (9:: o" the $o2 'ie,ers) escribe the in"erence (7he in"erence is that the a'er

o" the SAMP63 appro2i!ates the a'erae ae o" the epopulation)


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Reliability

As noted earlier& there could be error in a sa!plestatistic correctly appro2i!atin a population pa(ie $o2 sa!ple ae correctly appro2i!ates ae o$o2 'ie,ers)

We use !easures o" reliability to re;ect the der

uncertainty about statistical in"erence7his is o"ten re"erred to as the !arin o" error or

con#dence inter'al We ,ill discuss these in !ore detail later


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ata types


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32ercise 7,o

7he Ar!y %orps o" 3nineers is !easurin to2ic#sh. 7he captured a total o" #sh and noted "ollo,in 'ariables

Ri'erCcreek ,here #sh ,as captured

Species

6enth Weiht

7o2ic concentration in pp!

Which are =uantitati'eC=ualitati'e


11/78

32ercise 7,o

7he Ar!y %orps o" 3nineers is !easurin to2ic#sh. 7he captured a total o" #sh and noted "ollo,in 'ariables

Ri'erCcreek ,here #sh ,as captured (


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Section 7,o + escripti'e Statis


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escribin


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32ercise 7hreePatient # Insured (1), Medicare/Medicaid (2),

Uninsured (3)

9

9

8

G

H

9

What is the "re=ueach o" the ins

types1

What type o" 'arKPatient L


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6ets assu!e the "ollo,in colu!n o" 'alues459

8G

99G8

9

What is n1

What 'alue ,ould correspond to 21

What is the 'alue o" 1


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6ets assu!e the "ollo,in colu!n o" 'alues459

8G

99G8

9

What is n1 n is

What 'alue ,ould correspond to 21 8G

What is the 'alue o" 1 989


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32ercise $our

Assu!e a dataset o" &&&8&&&G

$ind

$ind 9

$ind


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32ercise $i'e5

G

8

H

9

8

G

8

:

What is the !ean1Median1

Mode1


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32cel "or central !easures

*pen up ne, e2cel ,orkbook 3nter nu!bers in "ro! 32ercise 8

Qse e2cel "unctions to calculate !ean& !edian a!ode

SA03 this ,orkbook..,e ,ill use this later


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Sa!ple !ean

$or!ula "or sa!ple !ean ,hich is denoted as

(Cn ,here n is the nu!ber o" obser'ations

is usually re"erred to as K2 bar (Population !eK!u)

7he roundin o" is subect to the deree o" accnecessary


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Median

When n is odd& the !edian is si!ply the !iddle a"ter arranin all obser'ations either in ascenddescendin order

When n is e'en& then ,e use the a'erae o" the t,o nu!bers

32D &8&G&&& 7he !edian ,ould be H (A'erae o" G&)


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Mean or Median1

I" the !ean and !edian or close& ,e typically us!ean

To,e'er& the !ean is sensiti'e to e2tre!e 'alue(outliers) ,hereas the !edian is not. In that cas!edian is o"ten used.

*"ten& the !edian is used ,hen considerin househoinco!es& ,hich can ha'e e2tre!e outliers (0irinia s!aors)


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Sy!!etric 's Ske,ed

When the !ean is to the riht o" !edian& it is riske,ed (Ske,ness nu!ber ,ould be neati'e)

When the !ean is to the le"t o" !edian& it is le"t (Ske,ness nu!ber ,ould be posti'e)


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So!e other !easures o"=uantitati'e 'alues

Ma2 and Min + the hihest and lo,est 'alue Rane - the diNerence "ro! the hihest 'alue an

lo,est 'alue

6o,er hal"-all 'alues to le"t o" !edian

Qpper hal"- all 'alues to the riht o" the !edian


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32ercise Si2

%onsider U&H&&8&9&&9&&&V %alculateD

Min

Ma2

Rane

st

=uartile rd=uartile


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Section + 0ariance and Standae'iation


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0ariance

%entral !easures are only part o" the story We are also interested in the Kspread o" the dis

We earlier considered rane

o, ,e ,ant to consider the 'ariance and standard d


30/78

0ariance

7he distance "ro! an obser'ation to the !ean oobser'ations is called a de'iation

3ach obser'ation ,ould be noted as 2- (ote thcould be positi'e or neati'e

32.

I!aine a plot ,ith three points U&&8V7he !ean is & so the de'iation "or is neati'e one

de'iation "or 8 is plus one (7he de'iation "or is :)


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32a!ple o" 'ariance

o, consider these t,o data sets U&9&&&8V and U9&&&&V

To, ,ould you describe these datasets1

What are the ra, de'iations "or each nu!ber in each

Why can>t ,e si!ply Ka'erae out the de'iations1


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32a!ple o" 'ariance solution

U&9&&&8V and U9&&&&V

To, ,ould you describe these datasets1 7hey both ha'e a !ean o"

7he ranes are and 9& respecti'ely

7hey both ha'e 8 !e!bers& but there is reater 'ariance inset

What are the ra, de'iations "or each nu!ber in each U-9&-&:&&9V and U-&:&:&:&V

Why can>t ,e si!ply Ka'erae out the de'iations to a sinle !easure o" 'ariablility1

It ,ould al,ays e=ual ero

7here are t,o ,ays to handle this4use absolute 'alues (,h

cause proble!s) or s=uare each de'iation


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%alculatin 'ariance

Re!e!ber our last set U&&8V We had de'iations o" -&:& I" ,e su!!ed the! up& they ,ould e=ual ero& but th

clearly not the case as there is a distribution

7o resol'e that& ,e s=uare each o" the de'iations& su!up and then di'ide by - (In this case& 9)

7here"ore our 'ariance is E:EC9

*ur "or!ula looks like thisD


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Standard de'iation o" sa!ple

*ur standard de'iation is the s=uare root o" our'ariance

In our last case& our 'ariance ,as & so the s=uare roo


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Si2 si!a


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Si2 si!a

St d d i ti d 0 i


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Standard e'iation and 0arianc32cel Reopen our pre'ious ,orkbook

We can use the stde' and 'ar "unctions


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32ercise se'en

Qsin e2cel

3nter the "ollo,in nu!bers into a colu!n

98&9&9&9G&9&9H&&9:&9&9

Qsin e2cel "unctions& return the !ean& !edian&

'ariance and standard de'iation


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7he need to 'isualie

7he !easures discussed (!ean& std de'iation& edon>t al,ays tell the ,hole story

Ansco!be>s =uartet


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6unch Break


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Probability

Why do pro"essional poker players al,ays see! t

6Q%X11

So!e& but they also ha'e a reat understandin oprobability and odds


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Si!ple probability + the coin tos

I!aine tossin a coin

What is the probability o" the toss resultin in KT

7he ans,erD .8 or 8:F


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Si!ple probability

6et>s take our last e2a!ple4,e ;ipped a coin anca!e up heads

We no, ,ant to ;ip the coin aain What is the probability that it ,ill co!e up heads aa

I" ,e ;ip it : ti!es and it is heads all ten ti!es& ,haprobability that it ,ill co!e up heads the thti!e.


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32peri!ents

7hese coin tosses are acts or obser'ations that lsinle outco!e& but that cannot be predicted wicertainty.

3ach obser'ation is called a sa!ple point or si!e'ent


45/78

Sa!ple points

In the case o" the coin& there ,ere t,o sa!ple pheads and tails

What are the sa!ple points on a sinle die (ice

We ha'e to be care"ul ,hen considerin the sa!

points that are possible (See ne2t slide)


46/78

Sa!ple points

Proble!D 6ist all o" the sa!ple points i" ,e ;ip 7W

We !iht think there are three sa!ple points ,he;ippin these coins

7ailsC7ails

TeadsCTeads

TeadsC7ails

444


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Sa!ple points ---sa!ple spac

But there are really sa!ple points

TT

77

T7

7T

7he set o" sa!ple points is re"erred to as a sa!pspace and& as obser'ed abo'e& is denoted as SD

77& T7& 7TV


48/78

32ercise eiht

enote the sa!ple space o" a sinle dice

enote the sa!ple space o" the "ace cards (J&


49/78

Probability de#ned

7he probability o" an e'ent is the likelihood that outco!e ,ill occur ,hen the e2peri!ent is per"o

Probability is usually denoted as KP

What is the likelihood that ,hen ,e roll a standathe roll ,ill result in a K1

It is in G& or appro2. G.HF

b bili d d


50/78

Probability e2panded

Re!e!ber the : consecuti'e coin ;ips that res

heads1

I" ,e ,ere to conduct that e2peri!ent ($lippin thone !illion ti!es& ,e ,ould 'ery likely see a resusuests rouhly hal" the ;ips ,ere heads and ha;ips ,ere tails.

7his is the la, o" lare nu!bers ,hich states threlati'e "re=uency o" an outco!e approaches its ttheoretical probability the !ore ti!es you repeat

Q l l


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Qnclear sa!ple spaces

5ou are startin a business. What is the probability th

succeed1 We could si!ply state 8:F...it ,ill either succeed or "a

To,e'er& ,e kno, that is not true& so ho, ,ould ,e aprobability1

We could consider e2perience in runnin a si!ilar business

We could look at success rates o" si!ilar businesses We could apply statistical techni=ues usin 'ariables such a

location& etc.

Whiche'er techni=ues ,e use& the #nal assess!ent o"probability is still subecti'e

P b bilit l


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Probability rules

A66 probabilities& ,hether subecti'e or not& !us

9 basic rules.7he probability o" a sa!ple point MQS7 lie bet,een :

7he probabilities o" all sa!ple points ,ithin a sa!ple!ust e=ual

3 i i %l di i


53/78

32ercise nine + %lass discussion

Suppose you are tra'elin to *rane county %al

and are interested in stayin at a hotel that has conser'ation prora!.

What are the sa!ple points "or this e2peri!ent1

To, ,ould you o about assinin the probabiliyour sa!ple points1

3 t


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3'ents

$irst& ,hat are the sa!ple points "or a sinle die

SD U&9&&&8&GV

Suppose that instead o" the probability o" a sinsa!ple point& ,e ,ere interested in so!ethin l

probability that the nu!ber ,ill be e'en (or oddcalled an event

An e'ent is typically noted as A.

3'ent A (Ke'en in this case)contains sa!ple pall ,ith probabilities o" CG4,e si!ply add the!

et P o" A o" C9

St " l l ti P " t


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Steps "or calculatin P o" e'ent

e#ne the e2perie!ent..that is& describe the pro

used to !ake an obser'ation and the type o"obser'ation that ,ill be recorded

6ist the sa!ple points

Assin probabilities to the sa!ple points

eter!ine the collection o" sa!ple points contathe e'ent

Su! the sa!ple point probabilities to et the pro" the e'ent

32ercise ten


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32ercise ten7his is a study o" di'orced peop

Group Description Proportion

PP Joint custody and et alon ,ell .9%% *ccasional con;ict .

AA %ooperate on children& con;ict other,ise .98

$$ Tostile to each other& in con;ict on e'ery issue .98

Suppose that :: couples are selected at rando!D 6ist the sa!ple points Assin probabilities to the sa!ple points What is the probability that the spouses "all into the $$ cateory1 What is the probability o" at least so!e con;ict1

Identi"yin sa!ple points "ro!


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de y sa p e po s oroups So "ar& ,e ha'e seen s!all co!binations& so ide

the sa!ple points ,as easy But let>s assu!e ,e ,anted to select any sa!pl

ite!s "ro! a ar o" :: ite!s. 7o understand theprobability o" any *3 roup o" 8 ite!s& ,e needkno, ho, !any diNerent roups o" 8 there are.

We could try and list the! all& but that ,ould be tediodoesn>t scale ,ell

So ,hat do ,e do1

%o!binatorial !ath


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%o!binatorial !ath

We utilie a co!bination "or!ula

$ro! our last scenario& lets assin :: (7otal nu!ite!s in the ar) and n8 (7otal nu!ber o" ite!s ,e select)

7he "or!ula ,e use isYD

/ C n/(-n)/

7he e2cla!ation point is called a K"actorial (see ne

Y7his assu!es once an ite! is selected it is *7 replaced). I" it ,as replaced& the "or!ula ,::8& but it ,ouldn>t !ake sense in this conte2t because ,e need 8 I$$3R37 ite!s

$actorial


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$actorial

$actorial si!ply !eans that ,e !ultiply a nu!b

each nu!ber be"ore it

32 8/7his !eans 8 2 2 2 9 2 (:/ is by de#nition)

So 8/ 9:

Back to co!binations


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Back to co!binations

So in our e2a!ple& :: and n8

So ,e ,ould ha'e ::/C8/(:: + 8)/

7his e=uals a R3A665 bi nu!ber4let>s open up and use the K"act "unction to calculate it

5ou should et H89H89:7hat !eans there are o'er H8 !illion co!binatio

7he probability o" selectin any *3 o" theco!binations is CH89H89:

32ercise ele'en


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32ercise ele'en

%alculate ho, !any sa!ples o" 8 ite!s out o" a

possible 9: there are. (Ans,er is 8&8:)

6et>s calculate 6*77*//

Assu!e 8 nu!bers and you need the rihtco!bination o" G to ,in. What is the probability

Multiplicati'e probability o"


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p p yindependent e'ents What i" ,e ha'e !ultiple e'ents occurrin1

$or e2a!ple& the P o" sur'in heart surery is .the p o" sur'i'in the reco'ery is .. What is thprobability that i" you ha'e heart surery& you ,actually o ho!e1

We si!ply !ultiply the probabilities& so the p o" ho!e is . Y ..G (So the hospital ,ould saprocedures ha'e an 8F success rate)

32ercise (6otto)


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32ercise (6otto)

8/C(G/)(H/)

7his e=uals 99&8H&:

7he probability is C99&8H&:

7he ticket ,ould state that the odds o" ,innin arouhly in 9!illion

iscrete probability distribution


64/78

In our sa!ple point proble!s& ,e ,ere able to li

either !anually or usin co!binatorial !ath& thesa!ple points.

7he "act that ,e can Klist these !akes the! ,hre"er to as discrete rando! 'ariables. 7his ter!that there is a fnite nu!ber o" distinct possible

(or sa!ple points)7his diNers "ro! continuous 'alues that are in#n

lead to the continuous probability distribution todiscussed shortly

iscrete probability distribution


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iscrete probability distributionQsin our e2a!ple o" t,o coins

TT T& 7 77:

:.8

.8

9

9.8

7oss o" t,o coins

7oss o" t,o coins

iscrete Probability istribution


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TT T& 7 77-:.98

:

:.98

:.8:.H8

7oss o" t,o coins

7oss o" t,o coins

Re!e!ber "ro! our earlier e2a!ple that there ,ere sa!ple points UTT& T7& 7T& 77V

I" ,e treat T7 and 7T as a sinle 'alue& ,e cancalculate the p o" that 'alue as Z E Z or [

7he raph abo'e depicts our discrete probabilitydistribution

We can also depict this ,ith a "or!ula



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7he probability distribution o" a discrete rando! 'ariable is a or "or!ula that speci#es the PR*BABI6I75 associated ,ith each'alue that the rando! 'ariable can assu!e

7here are t,o conditions that !ust be !et P(2) \ or to : "or A66 'alues o" 2.

Su! o" p(2)



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So!eti!es& the distribution is disco'ered a"ter !any obser'at

is not kno,n a priori $or e2a!ple& Q o" A] researchers used historical records o" dro7e2as to sho, that the distribution o" 2& ,here 2 nu!ber o" !ust be sa!pled until a dry year is obser'ed& could be sho,n"or!ulaD p(2) (.)(.H)(2-) 2 &9&4

What is the p that a"ter any consecuti'e years& a drouht ,o

P() (.)(.H)(-) (.)(.H)9 (.)(.) .H

7hus there is a 8F chance that "or any years& a drouht !i

Mean and std de'iation o" discrd i bl


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rando! 'ariable

TT (:) T& 7 () 77 (9)-:.98

:

:.98

:.8:.H8

7oss o" t,o coins

7oss o" t,o coins

6ets assin 'alues o" :& & 9 to the

possible outco!es. While ,e canlook at the raph to see that the!ean appears to be & ,e cancon#r! that by !ultiplyin our'alues by their respecti'e p

So :(C) E (.8) E 9(.98) 7hus the !ean is one.

7his !ean is o"ten re"erred to as theKe2pected 'alue and denoted 3(2)-- 5ou ,ill see this later in the BAPMprora!

32ercise t,el'e


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32ercise t,el'e

6et>s say you ,ork "or an insurance co!pany an

sell a one year ^:&::: policy ,ith a pre!iu! o7he policy pays out i" the custo!er dies& but thethat happenin is .::. What is the e2pected athis transaction1

32ercise t,el'e


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32ercise t,el'e

6et>s say you ,ork "or an insurance co!pany an

sell a one year ^:&::: policy ,ith a pre!iu! o7he policy pays out i" the custo!er dies& but thethat happenin is .::. What is the e2pected athis transaction1Gain x Sap!e point p

9: %usto!er li'es .

-&H: (9: +

:&:::)

%usto!er dies .::

7he e2pected ain is 9:(.) E (-H:)(.::) ^9:

I" the co!pany ,ere to sell a 'ery lare nu!ber o" policies it ,ould neon a'erae& 9: per sale.

7he 'ariance (and std de'iationd i bl


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a rando! 'ariable7he 'ariance o" a rando! 'ariable 2 isD

_9 3` (2-)9 (2-)9)Y(p(2))

7his is re"erred to as the expected value o the sq

distance rom the mean

7he std de'iation is si!ply the s=uare root o" _9

32ercise thirteen


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32ercise thirteen

A certain type o" che!otherapy is success"ul H:

ti!e. 6et 2 e=ual the nu!ber o" success"ul cure8.

So this !eans that "or any #'e treated patients& 9sur'i'e .9F o" ti!e& ,ill sur'i'e GF o" ti!e&

$ind the !ean& 'ariance and standard de'iation distribution

" 1 2 3 4 5

P(2) .::9 .:9 .9 .: .G: .G

32ercise thirteen


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e c se ee

3(2) :(.::9) E (.:9) E 9(.9) E (.:) E (.G:) E 8(.G) .87his !eans that the nu!ber o" success"ul cures& on a'erae& "or #'e patien,ill be .8 (H:F success rate)

0ariance _9 3` (2-)9 (2-)9)Y(p(2)) (:-.8)9 Y(.::9) E (-.8)9 Y (.:9) 4(8-.8)9 Y (.G) .:8

Std de'iation is s=uare root o" .:8 .:9

%onclusion o" Session


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pp

Website ,ith statistical sy!bols

httpDCC,,,.rapidtables.co!C!athCsy!bolsCStatis!bols.ht!

reek letters

] score table


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7-table


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basic statistics session 1

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