Chap 5 - Thong Ke Mo Ta (Handout)

8/13/2019 Chap 5 - Thong Ke Mo Ta (Handout)

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Thng k m t

Hong Vn H

[email protected]

Ngy 27 thng 10 nm 2012

Mt s khi nim c bn 2Bin v d liu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Tng th v mu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Chn mu ngu nhin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Thng k m t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Thng k suy lun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

M t d liu bng th 11Gii thiu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Phn phi tn s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Xy dng mt phn phi tn s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Xy dng mt phn phi tn s - v d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 th Stem-and-Leaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 th Stem-and-Leaf - v d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 th phn tn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

M t d liu s 29Gii thiu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Cc o xu hng trung tm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Trung bnh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Trung v. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Trung v. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 o no l tt nht? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 o s bin thin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Min gi tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Nhc im ca min gi tr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Min phn v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 th Boxplot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Cng thc tm phn v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Phng sai. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 lch tiu chun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 o s bin thin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46So snh cc lch chun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47nh l Chebyshev. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Quy tc thc nghim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49H s bin thin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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So snh h s bin thin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Cc phn phi thng gp trong thng k 53Phn phi chun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Phn phi chun ha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Phn phi chi bnh phng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Phn phi Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Phn phi mu 72Phn phi mu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Phn phi mu ca trung bnh v phng sai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Sai s chun ca trung bnh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Phn phi mu ca t l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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Mt s khi nim c bn 2

Bin v d liu

Bin (variable):mt c trng m thay i t ngi hay vt, hin tng ny sang ngi hayvt, hin tng khc. Bin gm hai loi: bin nh tnh(qualitative variable) vbin nh lng

(quantitative variable). Bin nh tnh:biu din tnh cht ca c trng m n th hin, c tc dng phn loi; v d:

nhm mu (A, B, AB, O), gii tnh (nam, n), mu mt (en, nu, xanh), . . .

Bin nh lng:biu din ln ca c trng m n th hin; v d: chiu cao, cn nng,thi gian, ...

Bin nh lng bao gmbin ri rc(discrete variable) vbin lin tc(continuous variable).

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Bin v d liu

Thng thng, bin ri rc lin quan n bi ton m s cc phn t ca mt tng th; v d:s sn phm hng trong 1 l hng, s con trong 1 gia nh, s cuc in thoi n tng itrong 1 gi, . . . trong khi bin lin tc lin quan n s o c; v d: cn nng ca 1 sn phm,chiu cao ca 1 ci cy, cng dng in, nhit , ...

D liu (data):cc gi tr ca mt bin. Tp hp tt c nhng quan trc cho mt bin c thc gi l mt tp d liu (Data set).

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Tng th v mu Tng th (population):Tp hp tt c nhng phn t mang c trng quan tm hay cn

nghin cu.

Mu (sample):l mt tp con c chn ra t tng th. Ta thng k hiu N ch s phnt ca tng th v n ch c mu.

Tham s (parameter):l mt c trng c th ca mt tng th. Thng k (statistic):l mt c trng c th ca mt mu.

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Tng th v mu

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V d v tng th

S c tri ng k i bu c, Thu nhp ca cc h gia inh trong thnh ph, im trung bnh ca tt c cc sinh vin trong mt trng i hc, Trng lng ca cc sn phm trong mt nh my.

Thng thng, ta khng th chn ht c tt c cc phn t ca tng th nghin cu bi v: S phn t ca tng th rt ln,

Thi gian v kinh ph khng cho php,

C th lm h hi cc phn t ca tng th.

Do , ta ch thc hin nghin cu trn cc mu c chn ra t tng th.

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Chn mu ngu nhin

Mtmu ngu nhin (random sample)gm nphn t c chn ra t mt tng th phi tha cciu kin sau:

Mi phn t trong tng th phi c chn ngu nhin v c lp,

Mi phn t trong tng th c kh nng c chn nh nhau (xc sut c chn bng nhau),

Mi mu c ncng c cng kh nng c chn t tng th.

Phng phpchn mu ngu nhin n gin (simple random sampling):

+ nh s cc phn t ca tng th t 1 n N. Lp cc phiu cng nh s nh vy.

+ Trn u cc phiu, sau chn c hon li nphiu. Cc phn t ca tng th c s th ttrong phiu ly ra s c chn lm mu.

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Thng k m t

Thng k m t (Descriptive statistics): l qu trnh thu thp, tng hp v x l d liu bin i d liu thnh thng tin.

+ Thu thp d liu: kho st, o c, ...

+ Biu din d liu: dng bng v th,

+ Tng hp d liu: tnh cc tham s mu nh trung bnh mu, phng sai mu, trung v, ...

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Thng k suy lun

Suy lun l mt qu trnh rt ra cc kt lun hoc a ra cc quyt nh v mt tng th davo cc kt qu nghin cu t mu.

Thng k suy lun (Inferential statistics): x l cc thng tin c c t thng k m t, t a ra cc c s d on (predictions), d bo (forecasts) v c lng (estimations).

+ c lng: v d c lng t l sn phm km cht lng trong 1 nh my; c lngtrng lng trung bnh s dng trung bnh mu,

+ Kim nh gi thuyt: v d cn kim nh khng nh trng lng trung bnh ca 1 snphm l 20 kg.

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M t d liu bng th 11

Gii thiu

a ra quyt nh t d liu dng th thng rt kh. Do vy, cn phi t chc li d liu.

Cc dng t chc d liu: Bng

th

Cc loi th c s dng s ph thuc vo bin c tng hp.

Cc dng th quan trng thng dng: biu ng (line chart), th t chc tn s(histogram), th stem-and-leaf, th phn tn (scatter plot).

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Phn phi tn s

Phn phi tn s (Frequency distribution) l g?

l mt danh snh hoc bng, cha cc khong c phn nhm theo d liu quan trc, v cc tn s tng ng ca d liu nm bn trong tng khong.

Ti sao s dng phn phi tn s?

phn phi tn s dng tng hp d liu, bin i d liu th thnh cc thng tin hu ch hn, s dng mt ci nhn trc quan gii thch d liu.

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Xy dng mt phn phi tn s

Trong bng phn phi tn s:

Mi nhm c b rng bng nhau,

B rng ca mi nhm c xc nh bi

Gi tr ln nht Gi tr b nhtS khong cn chia

Ti thiu l 5 khong, nhng khng nhiu hn 20 khong,

Cc khong khng trng nhau,

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Xy dng mt phn phi tn s - v d

V d1.Chn ngu nhin 20 ngy ma ng c nhit cao v o nhit (v: F) c s liunh sau

24 35 17 21 24 37 26 46 58 3032 13 12 38 41 43 44 27 53 27

Hy lp bng phn phi tn s cho s liu ny.

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Cc bc thc hin:

Sp xp d liu theo th t tng dn12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

Xc nh min d liu (range): 58 12 = 46 Chn s khong cn chia: 5(thng thng t 5 n 15)

Xc nh rng ca khong: 10 (lm trn 46/5)

Xc nh bin ca cc khong: t 10 n di 20, t 20 n di 30, . . . , t 50 n di 60

m s gi tr d liu nm trong mi khong

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D liu c sp tng:12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

Khong Tn s Tn s quan h Phn trm[10,20) 3 0.15 15[20,30) 6 0.30 30[30,40) 5 0.25 25[40,50) 4 0.20 20

[50,60) 2 0.10 10Tng 20 1.00 100

th biu din bng phn phi tn s gi l th t chc tn s (histogram).

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Xy dng mt phn phi tn s

Cu hi:chia d liu thnh bao nhiu khong l tt? Chn cc im ct th no cho ph hp?

L qu trnh "th" v "sai",

Mc tiu l to c 1 phn phi khng qu "lm chm" (jagged), c nhiu nh v khng c

dng "khi" (blocky), Mc tiu l ch ra c s bin thin trong d liu.

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Hnh dng ca phn phi

Hnh dng ca phn phi (shape of the distribution) gi l i xng (symmetric) nu cc gi trquan trc cn bng xung quanh trung tm.

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Hnh dng ca phn phi

Hnh dng ca phn phi gi l bt i xng (skewed) nu d liu quan trc khng phn b ixng xung quanh trung tm.

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Hnh dng ca phn phi

S dng th histogram nhn bit phn phi xc sut ca mt i lng ngu nhin.

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th Stem-and-Leaf

Mt phng php n gin nhn bit cc chi tit ca phn phi trong mt tp d liu

Phng php:sp xp d liu theo th t tng dn, chia cc gi tr sp xp thnh hai phn:phn th nht gm cc ch s dn u (stem) v phn t hai l ch s ui (leaf).

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th Stem-and-Leaf - v d

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V d2.V th stem-and-leaf cho tp d liu sau:

61 63 70 71 71 81 83 84 64 6565 66 84 87 73 75 92 93 77 7878 88 88 95 79

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th phn tn

th phn tn (Scatter plot)c s dng xc nh mi lin h gia hai bin Xv Y.

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M t d liu s 29

Gii thiu

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Cc o xu hng trung tm

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Trung bnh

Trung bnh (mean)l i lng thng c s dng nht o gi tr trung tm ca d liu.

Vi mt tng th c Nphn t, trung bnh tng th tnh bi

=Ni=1 xi

N =

x1+ x2+ . . . + xN

N (1)

Vi mt mu c n, trung bnh mu

x=

ni=1 xi

n =

x1+ x2+ . . .+ xnn

(2)

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Trung bnh

Trung bnh b nh hng bi cc gi tr ngoi lai (ouliers).

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Trung v

Trong mt tp d liu c sp xp (tng),trung v (median)l gi tr "chnh gia" ca d liu(50%bn trn, 50% bn di).

Trung v khng b nh hng bi cc im oulier.

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Trung v

V tr ca trung v: sp xp d liu theo th t tng dn, gi il v tr ca trung v

i=n

+ 12

+ Nu ichn, trung v = Xi,

+ Nu il, trung v =X[i]+ X[i]+1

2 , vi[i]l phn nguyn ca i.

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Mode

Model mt i lng o xu hng trung tm ca d liu,

L gi tr thng xy ra nht,

Khng b nh hng bi cc im outlier,

C th s dng cho c d liu s v d liu phn loi,

C th c nhiu mode hoc khng tn ti mode.

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o no l tt nht?

Trung bnhlun lun c s dng, nu cc im outlier khng tn ti.

Trung vthng c dng, bi v trung v khng b nh hng bi cc im outlier.

V tr ca trung v v trung bnh nh hng bi hnh dng ca phn phi:

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o s bin thin

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Min gi tr

Min gi tr (range)l o s bin thin n gin nht,

L chnh lch gia gi tr ln nht v b nht ca d liu quan trc

Min gi tr= XMax XMin

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Nhc im ca min gi tr

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Min phn v

Ta c th loi b cc im outlier bng cch s dng min phn v (interquartile range).

Cng thc tnh min phn v:IQR= Q3 Q1 (3)

vi Q1l phn v th 1 (mc 25%) v Q3l phn v th 3 (mc 75%) ca d liu.

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th Boxplot

biu din min phn v v cc im outlier ta dng th boxplot,

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Cng thc tm phn v

Sp xp d liu theo th t tng dn, gi Q1, Q2(trung v), Q3ln lt l phn v th 1, 2 v 3 cad liu; v tr ca Q1, Q2v Q3c xc nh nh sau

V tr phn v th nht = 0.25(n+ 1)

V tr phn v th hai= 0.5(n+ 1)

V tr phn v th ba= 0.75(n+ 1)

vi nl s gi tr quan trc.

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Phng sai

Phng sai (Variance)l trung bnh ca bnh phng lch cc gi tr so vi trung bnh.

Phng sai phn nh phn tn hay s bin thin ca d liu.

Phng sai tng th

2 = Ni=1(xi )2

N (4)

vi Nl s phn t ca tng th, l trung bnh tng th, xil gi tr th ica bin x.

Phng sai mu

S2 =

ni=1(Xi X)2

n 1 (5)

vi Xl trung bnh mu, nl c mu, Xi l gi tr th ica bin X.

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lch tiu chun

S dng o s bin thin, biu din s bin thin xung quanh trung bnh,

C cng n v o vi d liu gc.

lch chun ca tng th, k hiu l :

=

Ni=1(xi )2

N (6)

lch chun ca mu,

S=

ni=1(Xi X)2

n 1 (7)

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o s bin thin

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So snh cc lch chun

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nh l Chebyshev

nh l 1(Chebyshev). Vi mt tng th bt k c trung bnh v lch chun , v k >1, phntrm cc gi tr quan trc nm trong khong

[ + k]

bngt nht100

1 1

k2

%

V d:

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Quy tc thc nghim

Quy tc thc nghim (The Empirical Rule):nu d liu c phn phi chun (hoc tim cnchun), th khong

+ 1cha khong 68% gi tr d liu ca mu hoc tng th.

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Quy tc thc nghim

+ 2cha khong 95% gi tr d liu ca mu hoc tng th.+ 3cha khong 99.7%gi tr d liu ca mu hoc tng th.

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H s bin thin

H s bin thin (Coefficient of Variation)c s dng so snh s bin thin ca hai haynhiu tp d liu, c th o cc n v khc nhau,

o mi lin h gia s bin thin v trung bnh.

n v tnh bng %.

Cng thc

CV = S

X100% (8)

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So snh h s bin thin

D liu A:

Trung bnh xA= 50

lch chun sA= 5

CVA= SAxA

100% = 550 100% = 10% D liu B:

Trung bnh xB = 100

lch chun sB = 5

CVB = SB

xB100% =

5

100100% = 5%

C hai tp d liu c cng lch chun, nhng d liu B bin thin t hn so vi gi tr ca n.

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Cc phn phi thng gp trong thng k 53

Phn phi chun

nh ngha 2(Phn phi chun). Bin ngu nhin lin tcXnhn gi tr trong khong(, +)c gi l c phn phi chun tham s, nu hm mt xc sut c dng

f(x) = 1

2exp

(x )222 < x 0, <


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Phn phi chun

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Phn phi chun

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Phn phi chun

Nh vo nh l sau, nn nu bin ngu nhin Xc phn phi chun th bin i tuyn tnh ca Xcng c phn phi chun.

nh l 3(Tnh "tuyn tnh" ca phn phi chun). Nu bin ngu nhin Xc phn phi chun vi

k vng, phng sai2

v nuY =aX+ b, (a, bl hng s v a = 0), thYc phn phi chunvi k vnga + bv phng saia22.nh l 4. Nu cc bin ngu nhin X1, . . . , X n l c lp v nuXi c phn phi chun vi kvngiv phng sai2i,(i= 1, 2, . . . , n), th tngX1+ + Xn c phn phi chun vi k vngl 1+ + n v phng sai l 21+ + 2n.

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Phn phi chun

Mnh 1. Nu cc bin ngu nhin X1, . . . , X n l c lp v Xi c phn phi chun vi k vngiv phng sai2i,(i= 1, . . . , n). ai, . . . , an v bl cc hng s sao cho c t nht mtai= 0, thbin ngu nhin a1X1+ + anXn+ bc phn phi chun vi k vnga11+ + annv phngsaia21

21+ + a2n2n.

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Phn phi chun ha

nh ngha 5(Standard normal distribution). Bin ngu nhin Xc gi l c phn phi chunha nu n c phn phi chun vi tham s= 0v 2 = 1, k hiuX

N(0;1).

Theo quy c, hm phn phi ca bin ngu nhin chun ha c k hiu l (x), tc

(x) = 1

2

x

ey2

2 dy

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Phn phi chun ha

Theo nh l v tnh tuyn tnh ca phn phi chun, nu X N; 2th X

c phn phi

chun ha hayX

N(0; 1)

Da vo tnh cht ny ta c th tnh xc sut ca bin ngu nhin X N; 2.P (X b) = P

X

b

=

b

Tng t, vi a bth

P (a < X b) = P (X b) P (X a) =

b

a

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Phn phi chun ha

Nu X N; 2thP (|X | k) = P

k X

k

= 2(k) 1

ngi ta hay gi ng thc trn l "Quy tc k-sigma(k)".

Vik= 3ta c quy tc 3-sigma:

P (|X | 3) = Pk X

k

= 2(3) 1 0.9973

"Sai s gia Xv khng qu 3 l gn chc chn (xc sut gn bng 1)."

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Phn phi chun ha

nh ngha 6(Phn v chun ha, Normal quartile). Cho bin ngu nhin X N

; 2, phn v

chun ha mc, k hiux, l gi tr ca bin ngu nhin Xtha mn iu kin P (X x) = V d3.ng knh ca mt chi tit my do mt my tin sn xut c phn phi chun vi k vng20mm, phng sai (0.2mm)2. Tnh xc sut ly ngu nhin mt chi tit

a) c ng knh trong khong 19.9mm n 20.3mm.

b) c ng knh sai khc vi k vng khng qu 0.3mm.

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Phn phi chi bnh phng

nh ngha 7(Chi-squared distribution). Bin ngu nhin lin tcXnhn gi tr trong khong(0, +)c gi l c phn phi chi bnh phng vinbc t do, k hiuX 2(n), nu hm mt xc sut c dng

f(x) =

0 vix

0,

12n2(n

2)

xn21ex2 vix >0.

trong (x) =0 t

x1etdtl hm Gamma .

Lu : thy phn phi chi bnh phng xut pht t phn phi chun ngi ta cn nh nghaX 2(n)nu X= ni=1 X2i vi Xi l cc bin ngu nhin c lp v Xi N(0, 1).

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64

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27/32


65


nh l 8(Cc c trng ca bin ngu nhin c phn phi chi bnh phng) . ChoXl bin ngunhin c phn phi chi bnh phng vinbc t do th

i) K vngE (X) =n,

ii) Phng saiVar (X) = 2n,iii) NuX 2(n), Y 2(m)v X, Yl hai bin ngu nhin c lp thX+ Y 2(m + n).

66

Phn phi Student

nh ngha 9(Student distribution). Bin ngu nhin lin tcXnhn gi tr trong khong(, +)c gi l c phn phi Student vinbc t do, k hiuX t(n), nu hm mt xcsut c dng

f(x) = (n+12 )

n( n2 )

1 +

t2

n

n+12,

trong (x)l hm Gamma.

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Phn phi student

Lu

th ca hm mt phn phi Student c dng hnh chung nh th hm mt caphn phi chun, nhng c phn nh thp hn v hai phn ui cao hn so vi phn phi

chun. thy phn phi Student xut pht t phn phi chun v phn phi2(n)ngi ta cn nh

ngha X t(n)nu X= ZYn

vi Z N(0, 1), Y 2(n)v Z, Yl cc bin ngu nhin c

lp.

68

Phn phi student

69

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Phn phi student

70

Phn phi student

nh l 10(Cc c trng ca bin ngu nhin c phn phi Student). ChoX t(n)thi) K vngE (X) = 0nun >1, cc trng hp cn liE (X)khng c nh ngha.

ii) Phng saiVar (X) = nn2 nun >2; Var (X) =

nu1 < n

2cc trng hp cn li

Var (X)khng c nh ngha.

71

Phn phi mu 72

Phn phi mu

nh ngha 11. XtX1, X2, . . . , X nl mt mu ngu nhin chn t mt tng th v hm gi tr thc

(hay vc-t) T(x1, x2, . . . , xn). Th bin ngu nhin hay vc-t ngu nhin Y =T(X1, X2, . . . , X n)c coi l mt thng k. Phn phi xc sut ca thng kYc gi l phn phi mu ca Y.

Nhng phn phi mu c kho st:

Phn mi mu ca trung bnh, Phn phi mu ca phng sai, Phn phi mu ca t l.

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Phn phi mu ca trung bnh v phng sai

nh ngha 12. Trung bnh mu l trung bnh s hc ca cc gi tr trong mt mu ngu nhin, xcnh bi

X= X1+ X2+ + Xn

n =

1

n

n

i=1Xi

Phng sai mu l thng k xc nh bi

S2 = 1

n 1ni=1

(Xi X)2

74


nh l 13. Nu tng thXc phn phi chun X N(, 2)v (X1,...,Xn)l mt mu ngunhin t tng th trn th

X N(, 2

n).

(n1)

2 S2 =

ni=1

(XiX)22

2(n 1).

(X)

n

S t(n 1)

Xv S2 l hai bin ngu nhin c lp.

75


Trong trng hp tng th khng c phn phi chun, t nh l gii hn trung tm ta suy ra rng

(X )n

D N(0, 1)(X )n

S

D N(0, 1)

T kt qu ny, trong thc hnh, khi mu c kch thc, n, ln ta c cc phn phi xp x chun

sau(X )n

N(0, 1)

(X )nS

N(0, 1)

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Sai s chun ca trung bnh

nh ngha 14. XtX1, X2, . . . , X n l mt mu ngu nhin chn t mt tng th c trung bnh v phng sai2 < . Sai s chun (Standard Error) ca trung bnh, k hiuXc nh nghanh sau

X=

n (10)

ngha:

Xo bin thin ca Xxung quanh , Sai s chun cng nh, c lng tham s t tng th cng tt v tin cy cao.

77

Sai s chun ca trung bnh

Xb nh hng bi hai yu t:(1) C mu n: C mu cng lnsai s chun cng nh, ch rng khi n= 1th X=.(2) bin thin ca tng th: cng lnsai s chun cng ln.

78

Phn phi mu ca t l

Gi s cn kho st c trngAca mt tng th, kho st nphn t v t

Xi =

1, nu thaA0, nu khng tha A

thu c mu ngu nhin X1, . . . , X n vi Xi B(p), pl t l phn t tha c trngA. t X= ni=1l s phn t tha c trng Atrong mu kho st, th X B(n, p). T l mu Pl mt c lng ca t l pxc nh bi

P = X

n (11)

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Phn phi mu ca t l

K vng v phng sai ca Pbng

E

P

=p; Var

P

= p(1p)

n

Theo nh l gii hn trung tm ta c

P pp(1p)

p

N(0, 1)

V vy trong thc hnh, khi np 5, n(1p) 5, ta c P N

p,p(1p)

n

.

80

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Chap 5 - Thong Ke Mo Ta (Handout)