chap 5 - thong ke mo ta (handout)
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Thng k m t
Hong Vn H
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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Xy dng mt phn phi tn s - v d
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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Xy dng mt phn phi tn s - v d
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 - v d
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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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th Stem-and-Leaf - v d
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th Stem-and-Leaf - v d
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th Stem-and-Leaf - v d
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.
58
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)."
61
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).
63
Phn phi chi bnh phng
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Phn phi chi bnh phng
65
Phn phi chi bnh phng
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.
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Phn phi student
69
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Phn phi student
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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
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Phn phi mu ca trung bnh v phng sai
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
Phn phi mu ca trung bnh v phng sai
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