The document provides statistical formulas and analyses for descriptive statistics, correlation analysis, dynamic analysis, probability calculations, interval estimation, and parametric tests. It includes formulas for measures of central tendency, variance, standard deviation, correlation coefficients, trend lines, probability distributions, confidence intervals, and hypothesis tests regarding means and variances.

1. STATYSTYKA OPISOWA
Analiza struktury
∑=
=
n
i
ix
n
x
1
1
∑=
⋅=
k
i
ii nx
n
x
1
1
∑=
⋅=
k
i
ii nx
n
x
1
ˆ
1
∑=
= n
i i
H
x
n
x
1
1
∑=
= n
i i
i
H
x
a
n
x
1
∑=
= n
i i
i
H
x
a
n
x
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 xx
n
xS
1
22 1
)( ( ) i
k
i
i nxx
n
xS ⋅−= ∑=1
22 1
)( ( ) i
k
i
i nxx
n
xS ⋅−= ∑=1
22
ˆ
1
)(
%100
)(
)( ⋅=
x
xS
xV )()( 2
xSxS =
∑=
−=
n
i
i xx
n
xd
1
1
)( i
n
i
i nxx
n
xd ⋅−= ∑=1
1
)( i
n
i
i nxx
n
xd ⋅−= ∑=1
ˆ
1
)(
)()( xSxxxSx typ +<<−
{ } { }ii xxR minmax −=
( )∑=
−=
n
i
i xx
n
xM
1
3
3
1
)( ( ) i
k
i
i nxx
n
xM ⋅−= ∑=1
3
3
1
)( ( ) i
k
i
i nxx
n
xM ⋅−= ∑=1
3
3 ˆ
1
)(
)(
)(
3
3
3
xS
xM
=γ
)(xS
Dx
As
−
=
( ) ( ) D
DDDD
DD
D x
nnnn
nn
xD ∆⋅
−+−
−
+=
+−
−
11
1
β
β
β
ββ Q
Q
i
x
n
ncumn
xQ ∆⋅
−⋅
+= − )( 1
2
13 QQ
Q
−
= %100⋅=
Me
Q
VQ
)()(
)()(
13
13
QMeMeQ
QMeMeQ
AQ
−+−
−−−
=
Analiza korelacji
( )( )
( ) ( )∑ ∑
∑
= =
=
−⋅−
−−
=
⋅
=
n
i
n
i
ii
n
i
ii
xy
yyxx
yyxx
ySxS
yx
r
1 1
22
1
)()(
),cov(
r
( )
( )1
6
1 2
1
2
−⋅
−⋅
−=
∑=
nn
rr
R
t
i
yx
S
%10022
⋅= xyrR
( )
∑∑= =
−
=
r
i
k
j ij
ijij
n
nn
1 1
2
2
ˆ
ˆ
χ ,
n
nn
n ji
ij
•• ⋅
=ˆ
{ }1,1min
2
−−⋅
=
krn
V
χ
2
2
χ
χ
+
=
n
C
( )( )11
2
−−
=
krn
T
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
yS
ra 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
xS
rc ycxd −=

2. ( )
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
ct
ct
y
yy −
=∆ /
1
1
1/
−
−
−
−
=∆
t
tt
tt
y
yy
c
t
ct
y
y
i =/
1
1/
−
− =
t
t
tt
y
y
i
1
1
1
1/
1
1/2/31/2 ... −−−
− ==⋅⋅⋅= n
nn
n
n
nng
y
y
iiiii
∑
∑
∑
∑
=
=
=
=
⋅
⋅
=== n
i
ii
n
i
itit
n
i
i
n
i
it
t
w
qp
qp
w
w
w
w
I
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
iit
L
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
itit
P
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
iti
L
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
itit
P
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
n
uSyS
1
2
2
1
1)()(
( )∑=
−=
in
i
tt
i
i yy
n
O
1
ˆ
1
0
1
=∑= i
d
i
O ∑=
=
in
i t
t
i
i
y
y
n
S
1 ˆ
1
dS
i
d
i
=∑=1
)ˆ( ittt Oyyz +−= ittt Syyz ⋅−= ˆ
ip ObaTy ++= )( ip SbaTy ⋅+= )(
( )
( )∑=
−
−
++⋅= n
t
i
tp
tt
tT
n
zSyS
1
2
2
1
1)()( %100
ˆ
)(
⋅=
p
p
w
y
yS
b
2
)( 1
2
2
−
=
∑=
n
z
zS
n
t
t
t

3. Rachunek prawdopodobieństwa
)()( xXPxF <= ∑<
=
xx
i
i
pxF )(
∫∞−
=
x
dttfxF )()(
∑=
⋅=
n
i
ii pxXE
1
)(
∫
∞
∞−
⋅= dxxfxXE )()(
[ ]∑=
⋅−=
n
i
ii pXExXD
1
22
)()( [ ]∫
∞
∞−
⋅−= dxxfXExXD )()()(
22
[ ]22
)()( XEXEXD −= ( ) [ ]222
)()( XEXEXD −=
∑=
⋅=
n
i
ii pxXE
1
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
a
dxxfbXaP )()(
σ
mX
Z
−
=



−
==
p
p
xXP
1
)(
0
1
=
=
x
x pXE =)(
)1()(2
ppXD −=
knk
qp
k
n
kXP −






== )(
),1( pq −= nk ,...,2,1,0=
npXE =)(
npqXD =)(2
!
)(
k
e
kXP k
λ
λ
−
==
pn ⋅=λ , ,...2,1,0=k
λ=)(XE
λ=)(2
XD
Estymacja przedziałowa
α
σσ
αα −=






+<<− 1
n
uxm
n
uxP ααα −=






−
+<<
−
− 1
11 n
S
txm
n
S
txP
ααα −=






+<<− 1
n
S
uxm
n
S
uxP
α
χ
σ
χ αα
−=










<<
−−−
12
1,
2
1
2
2
2
1,
2
2
nn
nSnS
P ασ αα −=






+<<− 1
22 n
S
uS
n
S
uSP
αρ αα −=





 −
+<<
−
− 1
11 22
n
r
ur
n
r
urP
ααα −=












−
+<<
−
− 1
)1()1(
n
n
m
n
m
u
n
m
p
n
n
m
n
m
u
n
m
P
2
22
d
u
n
σα
≥ 2
22
d
Su
n α
≥ 2
2
)1(
d
ppu
n
−
≥ α
2
2
4d
u
n α
≥

4. Testy parametryczne
00 : mmH =
00 : mmH ≠






<
>
00
00
:
:
mmH
mmH
n
mx
u
σ
0−
= −σ znane
10
−
−
= n
S
mx
t
,30≤n
1−n -stopni swobody
n
S
mx
u 0−
= 30>n
2
0
2
0 : σσ =H
2
0
2
0 : σσ >H
2
0
2
2
σ
χ
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
p
n
m
u
)1( 00
0
−
−
=
210 : mmH =
210 : mmH ≠






<
>
210
210
:
:
mmH
mmH 2
2
2
1
2
1
21
nn
xx
u
σσ
+
−
=
−21,σσ znane
2
2
2
1
2
1
21
n
S
n
S
xx
u
+
−
=
12021 ≥+ nn






+
−+
+
−
=
2121
2
22
2
11
21
11
2 nnnn
SnSn
xx
t
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
S
F =
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
mm
p
+
+
= , pq −=1 ,
21
21
nn
nn
n
+
=
0:0 =ρH
0:0 ≠ρH






>
<
0:
0:
0
0
ρ
ρ
H
H
2
1 2
−
−
= n
r
r
t 122≤n
n
r
r
u
2
1−
=
122>n
Testy nieparametryczne
( )
∑=
−
=
r
i i
ii
np
npn
1
2
2
χ
)(xS
xx
u
G
iG
i
−
=
1−− rk stopni swobody
)1()(
)2(2
1
2
21
2
21
212121
21
21
−++
−−
−
+
−
=
nnnn
nnnnnn
nn
nn
k
U
( )
∑∑= =
−
=
r
i
t
j ij
ijij
e
en
1 1
2
2
χ
n
nn
e
ji
ij
•• ⋅
= ,
)1)(1( −− kr stopni swobody