statistik gia politikoÔc mhqanikoÔcusers.auth.gr/dkugiu/teach/civilengineer/oh_1.pdf · 2012. 11....

Eisagwg PERIGRAFIKH STATISTIKH

Statistik gia PolitikoÔc MhqanikoÔc

Dhm trhc Kougioumtz c

26 NoembrÐou 2012

Dhm trhc Kougioumtz c Statistik gia PolitikoÔc MhqanikoÔc


Perieqìmena

Perigrafik statistik (Shmei¸seic: Kef. 1)

EktÐmhsh paramètrwn (Shmei¸seic: Kef. 2)

StatistikoÐ èlegqoi (Shmei¸seic: Kef. 3)

Susqètish, Palindrìmhsh (Shmei¸seic: Kef. 4)

IstoselÐda maj matoc:http://users.auth.gr/dkugiu/Teaching/CivilEngineer/

http://blackboard.lib.auth.gr/

kwdikìc maj matoc 10U001



Eisagwgik

Bebaiìthta

bèbaia fainìmena −→ qr sh kajoristik¸n mejìdwn / montèlwn• Pìsh ¸ra knei ènac podhlthc apì to èna sto llo krothc Nèac ParalÐac JessalonÐkhc (3km), ìtan phgaÐnei mestajer taqÔthta 20km/h?• Poia ja eÐnai h jèsh thc ghc wc proc ton lio thnprwtoqroni 2013?

Abebaiìthta / tuqaiìthta

abèbaia tuqaÐa fainìmena −→ qr sh statistik¸n mejìdwn /montèlwn• Pìsh ¸ra knei kpoioc th diadrom xxxxxx � PoluteqneÐo mepod lato / autokÐnhto?• EÐnai akribèc to dromolìgio tou OSE gia Aj na �JessalonÐkh? (p.q. InterCity 12.18 – 17.41)



ParadeÐgmata

Poia eÐnai h antoq jraÔshc enìc tÔpou skurodèmatoc?

Poia eÐnai h mèsh ro kÐnhshc se mia odik arthrÐa tomeshmèri? EÐnai megalÔterh apì 1000 oq mata thn ¸ra?

Pìsoi q¸roi prkingk an diamèrisma èqei kat mèso ìromia polukatoikÐa sthn ToÔmpa? Sthn Kalamari?Uprqei diafor stic dÔo sunoikÐec?

Pwc metablletai h katanlwsh hlektrik c enèrgeiac meto mègejoc tou diamerÐsmatoc? MporoÔme na problèyoumegia kpoio diamèrisma gnwstoÔ megèjouc thn katanlwshenèrgeiac?

Pwc ja apant soume? =⇒ Statistik



DiadikasÐa Statistik c1 Δειγματοληψία (sullog dedomènwn)2 Perigrafik Statistik (perigraf / parousÐash

dedomènwn kai sunoptik¸n mètrwn)3 Statistik SumperasmatologÐa Statistik

(anlush statistik¸n dedomènwn kai ermhneÐaapotelesmtwn)

Pardeigma: Q¸roi prkigk

1 Sullègoume dedomèna apì 20 polukatoikÐec sthn ToÔmpa

kai 19 sthn Kalamari. X

2 �Blèpoume� tic parathr seic (pÐnakec, graf mata) kai

upologÐzoume kpoia mètra (p.q. mèso ìro). V

3 Efarmìzoume statistikèc mejìdouc gia na ektim soume tomèso q¸ro prkigk stic dÔo sunoikÐec kai gia na elègxoume

an uprqei shmantik diafor. V



Qr simoi ìroi Statistik ctuqaÐa metablht (t.m.): opoiod pote qarakthristikìtou opoÐou h tim allzei sta difora stoiqeÐa touplhjusmoÔ, π.χ. η αντοχή θραύσης σκυροδέματος.

dedomèna: èna sÔnolo tim¸n miac t.m. pou èqoume sthdijesh mac, π.χ. μετρήσεις αντοχής θραύσης σκυροδέματος.

plhjusmìc: mia omda mia kathgorÐa sthn opoÐaanafèretai h t.m., π.χ. σκυρόδεμα γενικά, ή ένας τύποςσκυροδέματος.

deÐgma: èna uposÔnolo tou plhjusmoÔ pou meletme, π.χ.25 δοκίμια σκυροδέματος γενικά ή ενός τύπου.

parmetroc: èna mègejoc pou sunoyÐzei me kpoio trìpotic timèc thc t.m. ston plhjusmì, π.χ. η μέση τιμή αντοχήςθραύσης σκυροδέματος.

statistikì: èna mègejoc pou sunoyÐzei me kpoio trìpotic timèc thc t.m. sto deÐgma, π.χ. ο μέσος όρος αντοχήςθραύσης από τα 25 δοκίμια σκυροδέματος που μετρήσαμε.



Pijanìthtec

PijanojewrÐa

TuqaÐa metablht t.m. X

Sunrthsh puknìthtac pijanìthtac sunrthsh mzacpijanìthtac fX (x)

Ajroistik sunrthsh katanom c FX (x)

KÔriec parmetroi katanom c:� Mèsh (prosdok¸menh) tim : E(X ) = µ� Diaspor (diakÔmansh): Var(X ) = σ2

Skopìc PijanojewrÐac

gnwrÐzoume katanom (paramètrouc) −→ meletme thn t.m.

Skopìc Statistik c

gnwrÐzoume timèc thc t.m. −→ sumpersmata gia thn t.m.Dhm trhc Kougioumtz c Statistik gia PolitikoÔc MhqanikoÔc


Me bsh ta dedomèna jèloume na bgloume sumpersmata:apì deÐgma −→ gia plhjusmì

apì statistikì −→ gia parmetro

Parmetroc: stajer ki gnwsth

Statistikì: metablhtì kai gnwstì

DeÐgma: tuqaÐo kai antiproswpeutikì tou plhjusmoÔ



PeÐrama: �tuqaÐo kai antiproswpeutikì deÐgma�?

1 Pre èna qartki me thneikìna sto diplanì sq ma.

2 Dilexe tuqaÐo kaiantiproswpeutikì deÐgmapènte kÔklwn apì touc 60tou plhjusmoÔ.

3 Upolìgise to mèso ìro twnaktÐnwn twn 5 kÔklwn kaigrye to sto pÐsw mèroc,p.q.

1 + 0.5 + 1.5 + 1 + 2.5

5= 1.3



Perigraf Statistik¸n dedomènwnPerigrafik Mètra Statistik¸n Dedomènwn

PERIGRAFIKH STATISTIKH

Perigraf Statistik¸n dedomènwn1. Diakekrimènec timèc (k kathgorÐec lÐgec arijmhtikèctimèc xi , i = 1, ..., k, deÐgma megèjouc n)

Suqnìthtec

suqnìthta emfnishc thc tim c xi , fi

sqetik suqnìthta / posostì, pi =fin

ajroistik suqnìthta Fi =∑i

j=1 fj ìpou xj ≤ xiajroistik sqetik suqnìthta Pi =

∑ij=1 pj ìpou xj ≤ xi

ParousÐash suqnot twn gia ta xi :

pÐnaka suqnot twn: mia gramm gia kje tim , kjest lh eÐnai ènac tÔpoc suqnìthtacrabdìgramma: mia rbdoc gia th suqnìthta kje tim ckuklikì digramma (�pÐta�), ...




Pardeigma: αριθμός δωματίων διαμερίσματος

DÐnetai o arijmìc dwmatÐwn se èna deÐgma 120 diamerismtwnmiac sunoikÐac thc pìlhc.

1 4 2 2 2 3 4 3 1 1 3 31 2 1 2 1 1 2 3 3 5 1 22 3 2 1 1 4 3 4 1 1 6 21 3 2 1 2 2 3 2 4 3 3 51 3 5 3 1 2 2 3 1 2 6 41 2 5 4 3 1 2 4 2 1 3 42 2 2 3 2 1 3 3 4 2 1 52 2 3 3 2 4 6 3 2 3 1 32 1 5 1 1 4 4 2 5 4 2 24 2 1 2 2 2 3 2 3 2 1 4




Pardeigma (sunèqeia)

Πίνακας συχνοτήτωνxi fi pi Fi Pi

1 28 0.23 28 0.232 39 0.33 67 0.563 27 0.23 94 0.784 16 0.13 110 0.925 7 0.06 117 0.976 3 0.03 120 1.00

'Ajroisma 120 1.00

Rabdìgramma

1 2 3 4 5 60

5

10

15

20

25

30

35

40

αριθµoς δωµατιων

συχν

oτητ

α

ραβδoγραµµα δωµατιων




2. Arijmhtikèc timèc (πολλές διακεκριμένες τιμές, τιμές σε διάστημα)

OmadopoÐhsh

QwrÐzoume ta dedomèna se k omdec −→ pÐnakec / graf mataìpwc prin gia tic k timèc (omdec)

Qwrismìc se omdec: (Ðdio eÔroc tim¸n r se kje omda)

EÔroc dedomènwn: R = xmax − xmin

R/k ' r

To pr¸to disthma prèpei na perièqei to xminTo teleutaÐo disthma prèpei na perièqei to xmax

Rabdìgramma (en¸nontac tic pleurèc) −→ istìgramma

'Alla graf mata: fullogrfhma, shmeiogrfhma, ...




Jèma 1

Prosdiorismìc arijmoÔ omdwn eÔroc diast matocistogrmmatoc (number of bins or bin width): Mèjodoi kaiperiorismoÐ.

Jèma 2

Fullogrfhma (stem and leaf plot): ParousÐash,pleonekt mata kai pardeigma.




Pardeigma: αντοχή θραύσης σκυροδέματος

Metr jhke h antoq jraÔshcskurodèmatoc se dÔo deÐgmataapì dÔo tÔpouc skurodèmatoc Akai B.

A/A tÔpoc A tÔpoc B

1 5.3 5.02 4.5 4.23 5.7 5.44 5.8 5.55 4.8 4.66 6.4 6.17 6.4 6.18 5.6 5.39 5.8 5.510 5.7 5.411 5.5 5.212 6.1 5.813 5.2 4.914 7.0 6.715 5.5 5.216 5.7 5.417 6.3 6.018 5.6 5.319 5.5 5.220 5.0 4.821 5.822 4.723 6.124 6.725 5.1

SÔnolo 141.8 107.6





X antoq jraÔshc tou skurodèmatoc tÔpou A (se ksi)Qwrismìc se omdec

xmin = 4.5 ksi xmax = 7.0 ksiR = xmax − xmin = 7.0− 4.5 = 2.5 ksi.Dialègoume na qwrÐsoume ta dedomèna se 10 omdec (k = 10)

r =R

k=

2.5

10= 0.25 ksi,

Dialègoume h pr¸th omda (disthma) na arqÐzei apì thn tim 4.5 ksiomda 1: 4.50− 4.75omda 2: 4.75− 5.00...omda 10: 6.75− 7.00

H teleutaÐa omda perilambnei to xmax = 7.0 ksiDhm trhc Kougioumtz c Statistik gia PolitikoÔc MhqanikoÔc




PÐnakac suqnot twn

Disthma tim¸n fi pi Fi Pi

4.50 � 4.75 2 0.08 2 0.084.75 � 5.00 2 0.08 4 0.165.00 � 5.25 2 0.08 6 0.245.25 � 5.50 4 0.16 10 0.405.50 � 5.75 5 0.20 15 0.605.75 � 6.00 3 0.12 18 0.726.00 � 6.25 2 0.08 20 0.806.25 � 6.50 3 0.12 23 0.926.50 � 6.75 1 0.04 24 0.966.75 � 7.00 1 0.04 25 1.00'Ajroisma 25 1.00





Istìgramma

4.5 5 5.5 6 6.5 70

1

2

3

4

5Histogram of beton data

bins

MporoÔme na deqtoÔme ìti h X akoloujeÐ kpoia gnwst katanom ?EÐnai shmantikì gia thn statistik anlush na eÐnai hkatanom kanonik




Perigrafik Mètra Statistik¸n Dedomènwn

• mètra jèshc: prosdiorÐzoun qarakthristikèc jèseic mèsasto eÔroc twn dedomènwn

• mètra metablhtìthtac: dÐnoun perilhptik th diaskìrpishkai metablhtìthta twn dedomènwn

x1, x2, . . . , xn: parathr seic tou deÐgmatoc




Mètra jèshc

deigmatik mèsh tim arijmhtikìc mèsoc mèsoc ìroc

x̄ =x1 + x2 + · · ·+ xn

n=

1

n

n∑i=1

xi

H mèsh tim eÐnai to �kèntro isorropÐac� twn dedomènwn




Mètra jèshc

deigmatik dimesoc

EÐnai h kentrik tim ìtan diatxoume ta dedomèna se aÔxousaseir

x̃ =

{x(n+1)/2 n = 2k + 1

xn/2+xn/2+12 n = 2k

p.q. deÐgma x1 ≤ x2 ≤ x3 −→ x̃ = x2p.q. deÐgma x1 ≤ x2 ≤ x3 ≤ x4 −→ x̃ = x2+x32

deigmatik epikratoÔsa tim

EÐnai h tim pou emfanÐzetai me th megalÔterh suqnìthta




Mètra jèshc

Parathr seic:

x̄ eÐnai to pio shmantikì mètro kai apoteleÐ ektÐmhsh tou µ

gia ton upologismì tou x̄ qrhsimopoioÔntai ìlec oiparathr seic, gia th dimeso mìno h txh touc

to x̄ ephrezetai apì makrinèc timèc, h dimesoc ìqi

Jèma 3

To sunoptikì mètro tou perikommènou mèsou (trimmed mean):ParousÐash, idiìthtec kai pardeigma.




Mètra metablhtìthtac

H metablhtìthta diaspor twn parathr sewn eÐnai ènadeÔtero shmantikì qarakthristikì tou deÐgmatoc

Ta kuriìtera mètra diasporc eÐnai:

deigmatikì eÔroc R

Den eÐnai anjektikì mètro kai upologÐzetai mìno apì tic dÔoakraÐec timèc





deigmatik diaspor deigmatik diakÔmansh s2

Metrei th metablhtìthta twn parathr sewn gÔrw apì thmèsh tim .Apìklish miac xi apì th mèsh tim : xi − x̄To jroisma ìlwn twn apoklÐsewn eÐnai 0!n∑

i=1

(xi − x̄) =n∑

i=1

xi −n∑

i=1

x̄ = nx̄ − nx̄ = 0

Gi autì qrhsimopoioÔme ta tetrgwna twn apoklÐsewn

s2 =1

n − 1

n∑i=1

(xi − x̄)2 X

isodÔnama

s2 =1

n − 1

(n∑

i=1

x2i − nx̄2)

V





DÔskolo na ermhneÔsoume th deigmatik diaspor s2

deigmatik tupik apìklish s

H deigmatik tupik apìklish s eÐnai pio katllhlo mètrogiatÐ epidèqetai fusik ermhneÐa

Jèma 4

To sunoptikì mètro thc dimeshc apìluthc apìklishc (medianabsolute deviation, MAD): ParousÐash, idiìthtec kaipardeigma.

p-ekatostiaÐa shmeÐa

p-ekatostiaÐo shmeÐo: posostì parathr sewn to polÔ p% eÐnaimikrìterec ap' aut n thnparat rhsh (0 ≤ p < 1)H dimesoc eÐnai to 50-ekatostiaÐo shmeÐo





Qarakthristik ekatostiaÐa shmeÐa eÐnai ta tetartomìria

pr¸to kat¸tero tetartomìrio Q1: to 25-ekatostiaÐoshmeÐo

trÐto an¸tero tetartomìrio Q3: to 75-ekatostiaÐoshmeÐo

Q1 kai Q3 orÐzontai ìpwc h dimesoc all periorÐzontac tosÔnolo twn dedomènwn sta antÐstoiqa uposÔnola (kat¸tero an¸tero misì).





endotetartomoriakì eÔroc I

I = Q3 − Q1 eÐnai to eÔroc pou kalÔptoun ta mis apì tadedomèna pou eÐnai pio kont dimeso

sÔnoyh twn 5 arijm¸n - jhkìgramma

Sunj kec gia apodoq kanonik c katanom c apì jhkìgramma

x̃ ìqi kont sto Q1 sto Q3

to eÔroc twn tim¸n sta dÔo akraÐa tetartomìria na mhdiafèrei shmantik

na mhn uprqoun akraÐec timèc




Pardeigma: περιεκτικότητα σε ραδιενέργεια του χάλυβα

DÐnetai h periektikìthta se radienèrgeia tou qluba se 10dokÐmia apì èna ergostsio A

A/A ergostsio A ergostsio B1 0.40 0.112 0.51 0.133 0.51 0.264 0.54 0.275 0.55 0.336 0.59 0.377 0.63 0.528 0.67 0.659 0.7510 2.10

SÔnolo 7.25 2.64





deigmatik mèsh tim :

x̄ =1

10

10∑i=1

xi =7.25

10= 0.725

deigmatik dimesoc:

x̃ =xn/2 + xn/2+1

2=

x5 + x62

=0.55 + 0.59

2= 0.57

eÔroc tim¸n deÐgmatoc:xmin = 0.40 xmax = 2.10 −→ R = 1.70

deigmatik diaspor (Pr¸ta to jroisma tetrag¸nwn)10∑i=1

x2i = 0.402 + 0.512 + · · ·+ 0.752 + 2.102 = 7.44

s2 =1

9

(10∑i=1

x2i − 10x̄2)

=1

9

(7.44− 10 · 0.7252

)= 0.243





deigmatik tupik apìklish s =√s2 =

√0.243 = 0.493

tetartomìria

endotetartomoriakì eÔrocI = Q3−Q1 = 0.67− 0.51 = 0.16

Jhkìgramma

1

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Θηκoγραμμα περιεκτικoτητας ραδιενεργειας





'Estw ìti h akraÐa tim ofeÐletai se sflma mètrhshc (deneÐnai pragmatik )Apaloif akraÐac tim c (−→ 9 parathr seic)δειγματική μέση τιμή x̄ = 5.159 = 0.572δειγματική διάμεσος x̃ = x(n+1)/2 = x5 = 0.55Διασπορά s2 = 18

(5.15− 9 · 0.5722

)= 0.010

Τυπική απόκλιση s =√

0.010 = 0.10Ελάχιστη τιμή xmin = 0.40Μέγιστη τιμή xmax = 0.75Εύρος R = 0.75− 0.40 = 0.35Πρώτο τεταρτομόριο (διάμεσος των {x1, . . . , x5}) Q1 = x3 = 0.51Τρίτο τεταρτομόριο (διάμεσος των {x5, . . . , x9}) Q3 = x7 = 0.63Ενδοτεταρτομοριακό εύρος I = 0.63− 0.51 = 0.12





1

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Θηκoγραμμα περιεκτικoτητας ραδιενεργειας





SÔgkrish periektikìthtac radienèrgeiac se qluba apì dÔoergostsia

Μέτρο X1 X2Μέση τιμή x̄1 = 0.572 x̄2 = 0.33Διάμεσος x̃1 = 0.55 x̃2 = 0.30Διασπορά s21 = 0.010 s

22 = 0.034

Τυπική απόκλιση s1 = 0.10 s2 = 0.18Ελάχιστη τιμή x1,min = 0.40 x2,min = 0.11Μέγιστη τιμή x1,max = 0.75 x1,max = 0.65Εύρος R1 = 0.35 R2 = 0.54Πρώτο τεταρτομόριο Q1,1 = 0.51 Q2,1 = 0.195Τρίτο τεταρτομόριο Q1,3 = 0.63 Q2,3 = 0.445Ενδοτεταρ. εύρος I1 = 0.12 I2 = 0.250





X_1 X_20.1

0.2

0.3

0.4

0.5

0.6

0.7

Θηκoγραμμα περιεκτικoτητας ραδιενεργειας, A και B

EÐnai h katanom Ðdia?

EÐnai h diaspor Ðdia?

EÐnai to kèntro Ðdio?




'Askhsh

'Eginan 15 metr seic thc sugkèntrwshc dialumènou oxugìnou(D.O.) s' èna potmi (se mg/l)

1.8 2.0 2.1 1.7 1.2 2.3 2.5 2.9 1.6 2.2 2.3 1.8 2.4 1.6 1.9

1 UpologÐste ta mètra jèshc kai metablhtìthtac gia tadedomèna tou deÐgmatoc kai sqhmatÐste to katllhlojhkìgramma.

2 Sqoliste an faÐnetai h sugkèntrwsh D.O. sto nerì toupotamoÔ na akoloujeÐ kanonik katanom .


Eisagwg'hPERIGRAFIKH STATISTIKHPerigraf'h Statistik'wn dedom'enwnPerigrafik'a M'etra Statistik'wn Dedom'enwn

statistik gia politikoÔc mhqanikoÔcusers.auth.gr/dkugiu/teach/civilengineer/oh_1.pdf · 2012. 11....

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