STATYSTYKA OPISOWA

Analiza struktury

∑=

=n

i

ixn

x1

1 ∑

=

⋅=k

i

ii nxn

x1

1 ∑

=

⋅=k

i

ii nxn

x1

ˆ1

∑=

=n

i i

H

x

nx

1

1

∑=

=n

i i

i

H

x

a

nx

1

∑=

=n

i i

i

H

x

a

nx

1 ˆ

N

nx

x

k

j

jj

og

∑=

⋅

= 1

( )N

nxx

N

nxS

S

k

j

jj

k

j

jj

og

∑∑==

−

+= 1

2

1

2

2

)(

( )∑=

−=n

i

i xxn

xS1

22 1)( ( ) i

k

i

i nxxn

xS ⋅−= ∑=1

22 1)( ( ) i

k

i

i nxxn

xS ⋅−= ∑=1

22 ˆ1

)(

%100)(

)( ⋅=x

xSxV )()( 2 xSxS =

∑=

−=n

i

i xxn

xd1

1)( i

n

i

i nxxn

xd ⋅−= ∑=1

1)( i

n

i

i nxxn

xd ⋅−= ∑=1

ˆ1

)(

)()( xSxxxSx typ +<<−

{ } { }ii xxR minmax −=

( )∑=

−=n

i

i xxn

xM1

3

3

1)( ( ) i

k

i

i nxxn

xM ⋅−= ∑=1

3

3

1)( ( ) i

k

i

i nxxn

xM ⋅−= ∑=1

3

3ˆ

1)(

)(

)(3

3

3xS

xM=γ

)(xS

DxAs

−=

( ) ( ) D

DDDD

DDD x

nnnn

nnxD ∆⋅

−+−−

+=+−

−

11

1 β

β

βββ Q

Q

i xn

ncumnxQ ∆⋅

−⋅+= − )( 1

2

13 QQQ

−= %100⋅=

Me

QVQ

)()(

)()(

13

13

QMeMeQ

QMeMeQAQ −+−

−−−=

Analiza korelacji

( )( )

( ) ( )∑ ∑

∑

= =

=

−⋅−

−−=

⋅=

n

i

n

i

ii

n

i

ii

xy

yyxx

yyxx

ySxS

yxr

1 1

22

1

)()(

),cov(

r

( )( )1

6

12

1

2

−⋅

−⋅−=

∑=

nn

rr

R

t

i

yx

S %10022 ⋅= xyrR

( )∑∑= =

−=

r

i

k

j ij

ijij

n

nn

1 1

2

2

ˆ

ˆχ ,

n

nnn

ji

ij

•• ⋅=ˆ { }1,1min

2

−−⋅=

krnV

χ

2

2

χχ+

=n

C ( )( )11

2

−−=

krnT

χ )()()(( 21211122211211

21122211

nnnnnnn

nnnn

+++

−=ϕ

baxy +=ˆ 2

11

2

1 11

2 )(

),cov(

)(

)(

−

−===

∑∑

∑ ∑∑

==

= ==

n

i

i

n

i

i

n

i

n

i

i

n

i

iii

xy

xxn

yxyxn

xS

yx

xS

ySra xayb −=

dcyx +=ˆ 2

11

2

1 11

2 )(

),cov(

)(

)(

−

−===

∑∑

∑ ∑∑

==

= ==

n

i

i

n

i

i

n

i

n

i

i

n

i

iii

xy

yyn

yxyxn

yS

yx

yS

xSrc ycxd −=

( )

22

ˆ

)( 1

2

1

2

2

−=

−

−=

∑∑==

n

u

n

yy

uS

n

i

i

n

i

ii

( )

( ) )(

)()(ˆ

2

2

1

2

1

2

2

yS

uS

n

kn

yy

yy

n

i

ii

n

i

ii

⋅−

=−

−=

∑

∑

=

=ϕ 22 1 ϕ−=R

Analiza dynamiki

ctct yyd −=/ 1/ −−= ctct yyd

c

ctct

y

yy −=∆ /

1

11/

−

−−

−=∆

t

tttt

y

yy

c

tct

y

yi =/

1

1/

−− =

t

ttt

y

yi

1

1

11/

11/2/31/2 ... −−−− ==⋅⋅⋅= n

nnn

nnng

y

yiiiii

∑

∑

∑

∑

=

=

=

=

⋅

⋅===

n

i

ii

n

i

itit

n

i

i

n

i

it

tw

qp

qp

w

w

w

wI

1

00

1

1

0

1

0

F

q

F

p

L

q

P

p

P

q

L

pw IIIIIII ⋅=⋅=⋅=

∑

∑

∑

∑

=

=

=

= =⋅

⋅=

n

i

n

i

p

n

i

ii

n

i

iitL

p

w

iw

qp

qp

I

1

0

1

0

1

00

1

0

∑

∑

∑

∑

=

=

=

= =⋅

⋅=

n

i p

t

n

i

t

n

i

iti

n

i

ititP

p

i

w

w

qp

qp

I

1

1

1

0

1

P

p

L

p

F

p III ⋅=

∑

∑

∑

∑

=

=

=

= =⋅

⋅=

n

i

n

i p

t

n

i

ii

n

i

itiL

q

w

i

w

qp

qp

I

1

0

1

1

00

1

0

∑

∑

∑

∑

=

=

=

= =⋅

⋅=

n

i

p

n

i

t

n

i

iit

n

i

ititP

q

iw

w

qp

qp

I

1

0

1

1

0

1

P

q

L

q

F

q III ⋅=

Trend

baty +=ˆ ( )∑ ∑

∑∑ ∑

= =

== =

−⋅

⋅−⋅=

n

t

n

t

n

t

t

n

t

n

t

t

ttn

yttyn

a

1 1

22

11 1

tayb −=

( )

22

ˆ

)( 1

2

1

2

2

−=

−

−=

∑∑==

n

u

n

yy

uS

n

i

t

n

t

tt

( )

( )∑

∑

=

=

−

−=

n

t

tt

n

t

tt

yy

yy

1

2

1

2

2

ˆ

ϕ

22 1 ϕ−=R

( )

( )∑=

−

−++⋅=

n

t

p

tt

tT

nuSyS

1

2

21

1)()(

( )∑=

−=in

i

tt

i

i yyn

O1

ˆ1

01

=∑= i

d

i

O ∑=

=in

i t

t

i

iy

y

nS

1 ˆ

1 dS

i

d

i

=∑=1

)ˆ( ittt Oyyz +−= ittt Syyz ⋅−= ˆ

ip ObaTy ++= )( ip SbaTy ⋅+= )(

( )

( )∑=

−

−++⋅=

n

t

i

tp

tt

tT

nzSyS

1

2

21

1)()( %100

ˆ

)(⋅=

p

p

wy

ySb

2)( 1

2

2

−=∑=

n

z

zS

n

t

t

t

Rachunek prawdopodobieństwa

)()( xXPxF <= ∑<

=xx

i

i

pxF )( ∫∞−

=x

dttfxF )()(

∑=

⋅=n

i

ii pxXE1

)( ∫∞

∞−

⋅= dxxfxXE )()(

[ ]∑=

⋅−=n

i

ii pXExXD1

22 )()( [ ]∫∞

∞−

⋅−= dxxfXExXD )()()(22

[ ]22 )()( XEXEXD −= ( ) [ ]222 )()( XEXEXD −=

∑=

⋅=n

i

ii pxXE1

22)( ∫∞

∞−

⋅= dxxfxXE )()( 22

)()( 2 XDXD =

ccE =)( 0)(2 =cD

)()( XEcXcE ⋅=⋅ )()( 222 XDcXcD ⋅=⋅

)()()( YEXEYXE +=+ )()()( 222 YDXDYXD +=+ dla zmiennych niezaleŜnych

)()()( YEXEYXE −=− )()()( 222 YDXDYXD +=− dla zmiennych niezaleŜnych

)()()( YEXEXYE ⋅=

dla zmiennych niezaleŜnych

)()(2)(2)()()( 222 YEXEXYEYDXDYXD −++=+

)()( xXPXF <=

)()()( aFbFbXaP −=≤≤

∑<≤

=≤≤bxa

i

i

pbXaP )( ∫=≤≤b

adxxfbXaP )()(

σmX

Z−

=

−==

p

pxXP

1)(

0

1

=

=

x

x

pXE =)(

)1()(2 ppXD −=

knkqpk

nkXP −

== )(

),1( pq −= nk ,...,2,1,0=

npXE =)(

npqXD =)(2

!)(

k

ekXP k

λ

λ−

==

pn ⋅=λ , ,...2,1,0=k

λ=)(XE

λ=)(2 XD

Estymacja przedziałowa

ασσ

αα −=

+<<− 1n

uxmn

uxP ααα −=

−+<<

−− 1

11 n

Stxm

n

StxP

ααα −=

+<<− 1n

Suxm

n

SuxP

αχ

σχ αα

−=

<<−−−

12

1,2

1

22

2

1,2

2

nn

nSnSP ασ αα −=

+<<− 122 n

SuS

n

SuSP

αρ αα −=

−

+<<−

− 111 22

n

rur

n

rurP

ααα −=

−

+<<−

− 1

)1()1(

n

n

m

n

m

un

mp

n

n

m

n

m

un

mP

2

22

d

un

σα≥ 2

22

d

Sun α≥

2

2 )1(

d

ppun

−≥ α

2

2

4d

un α≥

Testy parametryczne

00 : mmH =

00 : mmH ≠

<

>

00

00

:

:

mmH

mmH

nmx

uσ

0−= −σ znane

10 −−

= nS

mxt

,30≤n

1−n -stopni swobody

nS

mxu 0−= 30>n

2

0

2

0 : σσ =H

2

0

2

0 : σσ >H

2

0

22

σχ

nS=

,30≤n

1−n stopni swobody

122 2 −−= ku χ 1−= nk 30>n

00 : ppH =

00 : ppH ≠

<

>

00

00

:

:

ppH

ppH n

pp

pn

m

u)1( 00

0

−

−=

210 : mmH =

210 : mmH ≠

<

>

210

210

:

:

mmH

mmH

2

2

2

1

2

1

21

nn

xxu

σσ+

−=

−21,σσ znane

2

2

2

1

2

1

21

n

S

n

S

xxu

+

−=

12021 ≥+ nn

+

−++

−=

2121

2

22

2

11

21

11

2 nnnn

SnSn

xxt

12021 <+ nn

221 −+ nn stopni

swobody

2

2

2

10 : σσ =H 2

2

2

10 : σσ >H 2

2

2

1

ˆ

ˆ

S

SF =

2,1 21 −− nn - stopni

swobody

210 : ppH =

210 : ppH ≠

<

>

210

210

:

:

ppH

ppH n

qp

n

m

n

m

u 2

2

1

1 −= , gdzie:

21

21

nn

mmp

++

= , pq −=1 , 21

21

nn

nnn

+=

0:0 =ρH

0:0 ≠ρH

>

<

0:

0:

0

0

ρρ

H

H

21 2

−−

= nr

rt 122≤n

nr

ru

21−=

122>n

Testy nieparametryczne

( )∑=

−=

r

i i

ii

np

npn

1

2

2χ

)(xS

xxu

G

iG

i

−=

1−− rk stopni swobody )1()(

)2(2

12

21

2

21

212121

21

21

−++−−

−+

−=

nnnn

nnnnnn

nn

nnk

U

( )∑∑= =

−=

r

i

t

j ij

ijij

e

en

1 1

2

2χ

n

nne

ji

ij

•• ⋅= ,

)1)(1( −− kr stopni swobody