GAUSSOVA KRIVULJAuc.fmf.uni-lj.si/mi/arhivpoletih/gradiva/1920/mr_g.pdf · 2019-09-25 · GAUSSOVA...
Transcript of GAUSSOVA KRIVULJAuc.fmf.uni-lj.si/mi/arhivpoletih/gradiva/1920/mr_g.pdf · 2019-09-25 · GAUSSOVA...
GAUSSOVA KRIVULJA
Martin Raic
Izzivi poucevanja matematikeUL FMF
21. september 2019
Martin Raic GAUSSOVA KRIVULJA
GAUSSOVA KRIVULJA (TRANSFORMIRANA)
-3 -2 -1 0 1 2 3
01
23
y = a e−b(x−µ)2
x
y
a=1, b=1, µ=0 a=1, b=1,µ=2
a=2, b=4, µ=0
a= 12 , b=
14 , µ=0
Martin Raic GAUSSOVA KRIVULJA
TELESNA VISINA 1382 BAVARSKIH VOJASKIHOBVEZNIKOV IZ 19. STOLETJA
visina
frekv
enca
150 160 170 180
050
100
150
Vir podatkov (9. 8. 2015): http://www.uni-tuebingen.de/fakultaeten/wirtschafts-und-sozialwissenschaftliche-fakultaet/faecher/wirtschaftswissenschaft/lehrstuehle/volkswirtschaftslehre/wirtschaftsgeschichte/data-hub-height.html
Martin Raic GAUSSOVA KRIVULJA
OBSEG PRSNEGA KOSA PRI 5738 SKOTSKIHVOJAKIH
velikost
frekv
enca
35 40 45
020
040
060
080
010
00
Vir podatkov (9. 8. 2015): https://vincentarelbundock.github.io/Rdatasets/datasets.html
Martin Raic GAUSSOVA KRIVULJA
ZRACNI TLAK PO SVETU (761286) MERITEV
tlak [mb]
frekv
enca
9600 9800 10000 10200 10400 10600
010
000
2000
030
000
4000
0
Vir podatkov (9. 8. 2015): http://cdiac.ornl.gov/epubs/ndp/ndp026c/ndp026c.html
Martin Raic GAUSSOVA KRIVULJA
DEM : USD (1980–1987)
0 500 1000 1500
0.30
0.35
0.40
0.45
0.50
0.55
dan
DE
M/U
SD
Vir podatkov (9. 8. 2015): https://vincentarelbundock.github.io/Rdatasets/datasets.html
Martin Raic GAUSSOVA KRIVULJA
DEM : USD (1980–1987)
DEM/USD
frekv
enca
0.30 0.35 0.40 0.45 0.50 0.55 0.60
050
100
150
Vir podatkov (9. 8. 2015): https://vincentarelbundock.github.io/Rdatasets/datasets.html
Martin Raic GAUSSOVA KRIVULJA
DEM : USD (1980–1987) – SPREMEMBE
sprememba
frekv
enca
-0.02 0.00 0.02 0.04
050
100
150
200
250
Vir podatkov (9. 8. 2015): https://vincentarelbundock.github.io/Rdatasets/datasets.html
Martin Raic GAUSSOVA KRIVULJA
TOCKE V SVETOVNEM POKALU PRI ZENSKEMSMUCANJU (2012/13)
tocke
frekv
enca
0 500 1000 1500 2000 2500
010
2030
4050
Vir podatkov (27. 3. 2013):http://www.fis-ski.com/uk/disciplines/alpine-skiing/cupstandings.html
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 1. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 2. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 3. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 4. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 5. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 6. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 7. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 8. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 9. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 10. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 30. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 100. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 300. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 10000. izvedba
0 5 10 15 20 25 30
010
2030
4050
60
stevilo grbov
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: 10000. izvedba
0 5 10 15 20 25 30
0.00
0.05
0.10
0.15
0.20
stevilo grbov
rela
tivna
frekv
enca
Martin Raic GAUSSOVA KRIVULJA
MET 30 POSTENIH KOVANCEV: VERJETNOSTI
0 5 10 15 20 25 30
0.00
0.05
0.10
0.15
0.20
stevilo grbov
verje
tnos
t
Martin Raic GAUSSOVA KRIVULJA
100 METOV POSTENEGA KOVANCA
n = 100, p = 0.5
pik
verje
tnos
t
0 20 40 60 80 100
0.00
0.02
0.04
0.06
0.08
Martin Raic GAUSSOVA KRIVULJA
100 METOV POSTENEGA KOVANCA
n = 100, p = 0.5
pik
verje
tnos
t
30 40 50 60 70
0.00
0.02
0.04
0.06
0.08
Martin Raic GAUSSOVA KRIVULJA
Abraham de Moivre(1667–1754)
francoski matematik, deloval v Angliji
Vir: Vikipedija
Martin Raic GAUSSOVA KRIVULJA
IZPELJAVA ZA POSTEN KOVANEC (1)
n-krat vrzemo posten kovanecX := stevilo grbov, ki padejoVerjetnost, da pade natanko g grbov, oznacimo sPn(X = k).
Pn(X = k) =
(nk
)2n(
nk
)=
n(n − 1)(n − 2) · · · (n − k + 1)
k !
Martin Raic GAUSSOVA KRIVULJA
IZPELJAVA ZA POSTEN KOVANEC (2)
Aproksimacija temelji na kvocientu zaporednih verjetnosti:
Pn(X = k)
Pn(X = k − 1)=
n!2n k! (n−k)!
n!2n (k−1)! (n−k+1)!
=(k − 1)! (n − k + 1)!
k ! (n − k)!=
=n − k + 1
k.
Zaradi enostavnosti vzemimo, da je n sodo stevilo, t. j. n = 2m.Naj bo k = m + d . Tedaj lahko zgornjo enakost prepisemo vobliki:
P2m(X = m + d)
P2m(X = m + d − 1)=
m − d + 1m + d
=1− d−1
m
1 + dm
.
V naslednjem koraku bomo to aproksimirali za primer, ko jed � m.
Martin Raic GAUSSOVA KRIVULJA
IZPELJAVA ZA POSTEN KOVANEC (3)
Za majhne x je 1 + x ≈ ex :
x
y
1 2
1
2
y = ex
y = 1 + x
Martin Raic GAUSSOVA KRIVULJA
IZPELJAVA ZA POSTEN KOVANEC (4)
Ce d ni prevelik, je torej:
P2m(X = m + d)
P2m(X = m + d − 1)=
m − d + 1m + d
≈ e−(d−1)/m
ed/m = e−(2d−1)/m .
Z mnozenjem teh priblizkov dobimo (za d ≥ 0):
P2m(X = m + d)
P2m(X = m)≈ e−(1+3+5+···+(2d−1))/m = e−d2/m .
Sledi:P2m(X = m + d) ≈ P2m(X = m) e−d2/m .
Zaradi simetrije to velja tudi za negativne d .Verjetnosti torej sledijo Gaussovi krivulji.
Martin Raic GAUSSOVA KRIVULJA
IZPELJAVA ZA POSTEN KOVANEC (5)
Priblizek za P2m(X = m) dobimo iz dejstva, da je vsota vsehverjetnosti enaka 1:
m∑d=−m
P2m(X = m + d) = 1 .
Ker se vse dogaja v obmocju d = O(√
n)
(sicer je faktor e−d2/m
zanemarljiv), smemo aproksimirati:
∞∑d=−∞
P2m(X = m) e−d2/m ≈ 1 .
Martin Raic GAUSSOVA KRIVULJA
IZPELJAVA ZA POSTEN KOVANEC (6)
Prejsnjo formulo prepisemo v obliki:
√m P2m(X = m)
∞∑d=−∞
1√m
e−(d/√
m)2 ≈ 1 .
Posamezen sumand je enak ploscini pravokotnika s sirino1/√
m in visino e−(d/√
m)2, ta pa je priblizno enaka ploscini pod
krivuljo y = e−x2na intervalu okoli d/
√m s sirino 1/
√m:
x
y
d/√
m
Φ(c)
y = e−x2
Martin Raic GAUSSOVA KRIVULJA
IZPELJAVA ZA POSTEN KOVANEC (7)
Celotna vsota pa je priblizno enaka ploscini pod celotno krivuljoy = e−x2
, za katero je znano, da je enaka√π:
∞∑d=−∞
1√m
e−(d/√
m)2 ≈∫ ∞−∞
e−x2dx =
√π .
Torej je:
P2m(X = m) ≈ 1√mπ
,
P2m(X = m + d) ≈ 1√mπ
e−d2/m .
Martin Raic GAUSSOVA KRIVULJA
IZPELJAVA ZA POSTEN KOVANEC (8)
A tudi ce je n lih, se da z nekoliko tehnicno zahtevnejsoizpeljavo dobiti:
Pn
(X =
n2
+ d)≈√
2nπ
e−2d2/n
oziroma:
Pn(X = k) ≈√
2nπ
e−2(k−n/2)2/n .
Gaussovi krivulji pa sledijo tudi verjetnosti za nepostenkovanec, na katerem grb pade z verjetnostjo p ∈ (0,1), cifra paz verjetnostjo q = 1− p: za |k − np| � (npq)2/3 veljaLaplaceova lokalna formula:
Pn,p(X = k) ≈ 1√2πnpq
e−2(k−np)2/(2npq) .
Martin Raic GAUSSOVA KRIVULJA
Pierre-Simon de Laplace1749–1827
francoski matematik, astronom, fizik in politik
Vir: Vikipedija
Martin Raic GAUSSOVA KRIVULJA
100 METOV NEPOSTENEGA KOVANCA
n = 100, p = 0.1
pik
verje
tnos
t
0 20 40 60 80 100
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Martin Raic GAUSSOVA KRIVULJA
100 METOV NEPOSTENEGA KOVANCA
n = 100, p = 0.9
pik
verje
tnos
t
0 20 40 60 80 100
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Martin Raic GAUSSOVA KRIVULJA
CENTRALNI LIMITNI IZREK
Velja X = Y1 + Y2 + · · ·+ Yn, kjer je:
Yj =
{1 ; pri j-tem metu pade grb,0 ; pri j-tem metu pade cifra.
Gaussovi krivulji pa pod razmeroma milimi pogoji sledi tudiporazdelitev vsote veliko splosnejsih neodvisnih slucajnihspremenljivk: porazdelitve posameznih sestevancev ne smejobiti pregrde in nobeden ne sme prevec izstopati. Precizniformulaciji tega dejstva pravimo centralni limitni izrek.
Gaussovi krivulji torej sledi porazdelitev slucajne kolicine, kinastane kot rezultanta veliko majhnih neodvisnih vplivov, ki semed seboj sestevajo.
Martin Raic GAUSSOVA KRIVULJA
NORMALNA (GAUSSOVA) PORAZDELITEV
µ
σ σ
x
f (x)
f (x) = 1σ√
2πe−(x−µ)
2/(2σ2)
X ∼ N(µ, σ)
µ = E(X ) (pricakovana vrednost ali matematicno upanje)σ2 = var(X ) = D(X ) = E
[(X − E(X ))2]
(varianca ali disperzija)
Martin Raic GAUSSOVA KRIVULJA
Paul Pierre Levy(1886–1971)
francoski matematik zidovskega rodu
Vir: Vikipedija
Martin Raic GAUSSOVA KRIVULJA
2 META POSTENE KOCKE
n = 2
pik
verje
tnos
t
2 4 6 8 10 12
0.00
0.05
0.10
0.15
Martin Raic GAUSSOVA KRIVULJA
30 METOV POSTENE KOCKE
n = 30
pik
verje
tnos
t
50 100 150
0.00
0.01
0.02
0.03
0.04
Martin Raic GAUSSOVA KRIVULJA
GRDA PORAZDELITEV
stevilo
verje
tnos
t
0 2 4 6 8
0.0
0.1
0.2
0.3
0.4
0.5
Martin Raic GAUSSOVA KRIVULJA
VSOTA 10 GRDIH PORAZDELITEV
vsota
verje
tnos
t
0 20 40 60 80
0.00
0.05
0.10
0.15
Martin Raic GAUSSOVA KRIVULJA
VSOTA 10 GRDIH PORAZDELITEV
vsota
verje
tnos
t
0 20 40 60 80
0.00
0.05
0.10
0.15
Martin Raic GAUSSOVA KRIVULJA
VSOTA 50 GRDIH PORAZDELITEV
vsota
verje
tnos
t
-60 -40 -20 0 20 40 60
0.00
00.
005
0.01
00.
015
0.02
00.
025
0.03
0
Martin Raic GAUSSOVA KRIVULJA
VSOTA 100 GRDIH PORAZDELITEV
vsota
verje
tnos
t
-60 -40 -20 0 20 40 60
0.00
00.
005
0.01
00.
015
Martin Raic GAUSSOVA KRIVULJA
VSOTA 500 GRDIH PORAZDELITEV
vsota
verje
tnos
t
-200 -150 -100 -50 0 50 100 150
0.00
00.
002
0.00
40.
006
0.00
8
Martin Raic GAUSSOVA KRIVULJA
DOHODKI 43886 DRUZIN V KIBERGRADU
dohodek
frekv
enca
0 50000 100000 150000 200000
050
010
0015
0020
0025
0030
0035
00
Vir podatkov: J. Rice: Mathematical Statistics and Data Analysis, Duxbury, 2007
Martin Raic GAUSSOVA KRIVULJA
1000 POVPRECIJ PO 20000 DOHODKOV
dohodek
frekv
enca
40800 41000 41200 41400 41600 41800
050
100
150
200
Martin Raic GAUSSOVA KRIVULJA
TOCKE V SVETOVNEM POKALU PRI ZENSKEMSMUCANJU (2012/13)
tocke
frekv
enca
0 500 1000 1500 2000 2500
010
2030
4050
Vir podatkov (27. 3. 2013):http://www.fis-ski.com/uk/disciplines/alpine-skiing/cupstandings.html
Martin Raic GAUSSOVA KRIVULJA
LOGARITEM TOCK
log10(tocke)
frekv
enca
0 1 2 3 4
010
2030
40
Vir podatkov (27. 3. 2013):http://www.fis-ski.com/uk/disciplines/alpine-skiing/cupstandings.html
Martin Raic GAUSSOVA KRIVULJA
STANDARDNA NORMALNA PORAZDELITEV
z
f (z)
N(0,1)
f (z) = 1√2π
e−x2/2
Z ∼ N(0,1) =⇒ a + bZ ∼ N(a, |b|)
Martin Raic GAUSSOVA KRIVULJA
PRIMERJALNI KVANTILNI (Q–Q) GRAFIKON:IDEJA 1
n razlicnih podatkov razdeli realno os na n + 1 delov:Primer: 3, 22, 7, −20, 11, −10, 15, 18, 25
y
−20
−10
0
10
20
Martin Raic GAUSSOVA KRIVULJA
PRIMERJALNI KVANTILNI (Q–Q) GRAFIKON:IDEJA 2
Realno os pa lahko razdelimo tudi glede na ploscine podGaussovo krivuljo:
x
Martin Raic GAUSSOVA KRIVULJA
PRIMERJALNI KVANTILNI (Q–Q) GRAFIKON:PRIMER 1
Podatki: 3, 22, 7, −20, 11, −10, 15, 18, 25
x
y
−20
−10
10
20
Martin Raic GAUSSOVA KRIVULJA
PRIMERJALNI KVANTILNI (Q–Q) GRAFIKON:PRIMER 2
Podatki: 5, 20, 5, −20, 10, −10, 10, 10, 25
x
y
−20
−10
10
20
Martin Raic GAUSSOVA KRIVULJA
OBSEG PRSNEGA KOSA PRI 5738 SKOTSKIHVOJAKIH
velikost
frekv
enca
35 40 45
020
040
060
080
010
00
Vir podatkov (9. 8. 2015): https://vincentarelbundock.github.io/Rdatasets/datasets.html
Martin Raic GAUSSOVA KRIVULJA
OBSEG PRSNEGA KOSA PRI 5738 SKOTSKIHVOJAKIH
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Vir podatkov (9. 8. 2015): https://vincentarelbundock.github.io/Rdatasets/datasets.html
Martin Raic GAUSSOVA KRIVULJA
TELESNA VISINA 1382 BAVARSKIH VOJASKIHOBVEZNIKOV IZ 19. STOLETJA
visina
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Vir podatkov (9. 8. 2015): http://www.uni-tuebingen.de/fakultaeten/wirtschafts-und-sozialwissenschaftliche-fakultaet/faecher/wirtschaftswissenschaft/lehrstuehle/volkswirtschaftslehre/wirtschaftsgeschichte/data-hub-height.html
Martin Raic GAUSSOVA KRIVULJA
TELESNA VISINA 1382 BAVARSKIH VOJASKIHOBVEZNIKOV IZ 19. STOLETJA
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−3 −2 −1 0 1 2 3
150
160
170
180
standardna normalna porazdelitev
visi
na v
ojak
a
Vir podatkov (9. 8. 2015): http://www.uni-tuebingen.de/fakultaeten/wirtschafts-und-sozialwissenschaftliche-fakultaet/faecher/wirtschaftswissenschaft/lehrstuehle/volkswirtschaftslehre/wirtschaftsgeschichte/data-hub-height.html
Martin Raic GAUSSOVA KRIVULJA
DEM : USD (1980–1987) – SPREMEMBE
sprememba
frekv
enca
-0.02 0.00 0.02 0.04
050
100
150
200
250
Vir podatkov (9. 8. 2015): https://vincentarelbundock.github.io/Rdatasets/datasets.html
Martin Raic GAUSSOVA KRIVULJA
DEM : USD (1980–1987) – SPREMEMBE
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−3 −2 −1 0 1 2 3
−0.
020.
000.
020.
04
standardna normalna porazdelitev
spre
mem
ba te
caja
Vir podatkov (9. 8. 2015): https://vincentarelbundock.github.io/Rdatasets/datasets.html
Martin Raic GAUSSOVA KRIVULJA
TOCKE V SVETOVNEM POKALU PRI ZENSKEMSMUCANJU (2012/13)
tocke
frekv
enca
0 500 1000 1500 2000 2500
010
2030
4050
Vir podatkov (27. 3. 2013):http://www.fis-ski.com/uk/disciplines/alpine-skiing/cupstandings.html
Martin Raic GAUSSOVA KRIVULJA
TOCKE V SVETOVNEM POKALU PRI ZENSKEMSMUCANJU (2012/13)
●
●
●
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●●
●●●●●●●●●●●
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−2 −1 0 1 2
050
010
0015
0020
0025
00
standardna normalna porazdelitev
tock
e
Vir podatkov (27. 3. 2013):http://www.fis-ski.com/uk/disciplines/alpine-skiing/cupstandings.html
Martin Raic GAUSSOVA KRIVULJA
LOGARITEM TOCK
log10(tocke)
frekv
enca
0 1 2 3 4
010
2030
40
Vir podatkov (27. 3. 2013):http://www.fis-ski.com/uk/disciplines/alpine-skiing/cupstandings.html
Martin Raic GAUSSOVA KRIVULJA
LOGARITEM TOCK
●
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● ●●
●
● ●
−2 −1 0 1 2
0.5
1.0
1.5
2.0
2.5
3.0
standardna normalna porazdelitev
log1
0(to
cke)
Martin Raic GAUSSOVA KRIVULJA
DOHODKI 43886 DRUZIN V KIBERGRADU
dohodek
frekv
enca
0 50000 100000 150000 200000
050
010
0015
0020
0025
0030
0035
00
Vir podatkov: J. Rice: Mathematical Statistics and Data Analysis, Duxbury, 2007
Martin Raic GAUSSOVA KRIVULJA
DOHODKI 43886 DRUZIN V KIBERGRADU
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−4 −2 0 2 4
0e+
001e
+05
2e+
053e
+05
4e+
05
standardna normalna porazdelitev
doho
dek
Vir podatkov: J. Rice: Mathematical Statistics and Data Analysis, Duxbury, 2007
Martin Raic GAUSSOVA KRIVULJA
1000 POVPRECIJ PO 20000 DOHODKOV
dohodek
frekv
enca
40800 41000 41200 41400 41600 41800
050
100
150
200
Martin Raic GAUSSOVA KRIVULJA
1000 POVPRECIJ PO 20000 DOHODKOV
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−3 −2 −1 0 1 2 3
4080
041
000
4120
041
400
4160
041
800
standardna normalna porazdelitev
doho
dek
Martin Raic GAUSSOVA KRIVULJA