Post on 08-Nov-2014
description
SERII SIMPLE (date negrupate) 0 ULPL�PHGLL 0HGLD�DULWPHWLF 0HGLD� DULWPHWLF HVWH� UH]XOWDWXO� VLQWHWL] ULL� vQWU-R� VLQJXU � H[SUHVLH� QXPHULF � D� WXWXURU�
QLYHOXULORU� LQGLYLGXDOH� REVHUYDWH�� RE LQXW � SULQ� UDSRUWDUHD� YDORULL� WRWDOL]DWH� D� FDUDFteristicii la QXP UXO�WRWDO�DO�XQLW LORU�
n
x
x
n
ii∑
== 1
unde:
ix - UHSUH]LQW �QLYHOXULOH�LQGLYLGXDOH�DOH�YDULDELOHL�
∑=
n
iix
1
- UHSUH]LQW �YROXPXO�FHQWUDOL]DW�DO�YDULDELOHL�
n - UHSUH]LQW �QXP UXO�XQLW LORU observate.
3URSULHW L a) GDF �x1=x2=...=xi=...=xn=xc , atunci cxx =
b) maxmin xxx <<
c) 0)(1
=−∑=
n
ii xx
d) GDF � axx i #=’ , atunci axx #=′ , de unde axx ±= ’
6HULL�GH�GLVWULEX LH�D�IUHFYHQ HORU
e) GDF �h
xx i=" , atunci
h
xx =′′ , de unde hxx ⋅′′= ,
respectiv :
GDF � hxx i ⋅=" , atunci hxx ⋅=′′ , de unde h
xx
′′=
Formule de calcul simplificat al mediei aritmetice:
( )a
n
ax
x
n
ii
±=∑
=1
#
hn
h
x
x
n
i
i
⋅
=∑
=1 sau ( )
hn
hxx
n
ii
:1∑
=
⋅=
ahn
h
ax
x
n
i
i
+⋅
−
=∑
=1
0HGLD�DUPRQLF
0HGLD� DUPRQLF � � hx �� VH� GHILQHúWH� FD� ILLQG� LQYHUVD�PHGLHL�DULWPHWLFH� FDOFXODW �GLQ�YDORULOH�
inverse ale termenilor aceleLDúL�VHULL�
∑=
=n
i i
h
x
nx
1
1
0HGLD�S WUDWLF
0HGLD� S WUDWLF � )( px � HVWH� DFHD� YDORDUH� FDUH� vQORFXLQG� WHUPHQLL� VHULHL� ULGLFD L� OD� S WUDW� QX�
PRGLILF �VXPD�S WUDWHORU�ORU�
n
xx
n
ii
p
∑== 1
2
0HGLD�JHRPHWULF
0HGLD�JHRPHWULF �� gx ��UHSUH]LQW �DFHD�YDORDUH�FX�FDUH��GDF �VH�vQORFXLHVF�WR L�WHUPHQLL�VHULHL�
úL�VH�IDFH�SURGXVXO�ORU��YDORDUHD�OD�FDUH�VH�DMXQJH�HVWH�HJDO �FX�SURGXVXO�WHUPHQLORU�UHDOL�
n
n
iig xx ∏
=
=1
3ULQ�ORJDULWPDUH�VH�RE LQH�
n
xx
n
ii
g
∑== 1
lglg
ÌQWUH�PHGLLOH�SUH]HQWDWH�H[LVW �XUP WRDUHD�UHOD LH�GH�RUGLQH�
pagh xxxx <<<
9DORUL�PHGLL�GH�SR]L LH�VDX�GH�VWUXFWXU Mediana (Me)� UHSUH]LQW � YDORDUHD� FHQWUDO � D� XQHL� VHULL� VWDWLVWLFH� RUGRQDWH� FUHVF WRU� VDX�
GHVFUHVF WRU�FDUH�vPSDUWH�WHUPHQLL�VHULHL�vQ�GRX �S U L�HJDOH�
Locul 2
1+= nMe unde n�UHSUH]LQW �QXP UXO�WHUPHQLORU�VHULHL�
Valoarea medianei: • GDF �QXP UXO�WHUPHQLORU�VHULHL�HVWH�LPSDU��n=2p+1): 1+= pxMe
• GDF �QXP UXO�WHUPHQLORU�HVWH�SDU��n=2p): 2
1++= pp xx
Me
Modul (Mo)�HVWH�YDORDUHD�FDUH�VH�UHSHW �GH�FHOH�PDL�PXOWH�RUL��PRWLY�SHQWUX�FDUH�PDL�HVWH�
FXQRVFXW�vQ�OLWHUDWXUD�GH�VSHFLDOLWDWH�úL�VXE�GHQXPLUHD�GH�dominanta seriei. ,QGLFDWRULL�YDULD LHL
a) ,QGLFDWRULL�VLPSOL�DL�YDULD LHL
Amplitudinea abVROXW (Aa��VH�FDOFXOHD] �FD�GLIHUHQ �vQWUH�QLYHOXO�PD[LP��xmax��úL�QLYHOXO�
minim (xmin) al caracteristicii: Aa = xmax - xmin
$PSOLWXGLQHD�UHODWLY �D�YDULD LHL��A%��VH�FDOFXOHD] �FD�UDSRUW�vQWUH�DPSOLWXGLQHD�DEVROXW �D�
YDULD LHL�úL�QLYHOXO�PHGLX�DO�FDUDFteristicii:
100% ⋅=x
AA a
Abaterile individuale absolute (di)� VH� FDOFXOHD] � FD� GLIHUHQ � vQWUH� ILHFDUH� YDULDQW �
vQUHJLVWUDW �úL�PHGLD�DULWPHWLF �D�DFHVWRUD�
nixxd ii ,1 =−= Abaterile individuale relative (di%)� VH� FDOFXOHD] � UDSRUWkQG� DEaterile absolute la nivelul
mediu al caracteristicii:
nix
xx
x
dd ii
i ,1 100100% =⋅−
=⋅=
ÌQ�DQDOL]D�YDULD LHL��GH�PXOWH�RUL��QH�OLPLW P�OD�D�FDOFXOD�DEDWHULOH�PD[LPH�vQWU-un sens sau altul.
b) ,QGLFDWRULL�VLQWHWLFL�DL�YDULD LHL
$EDWHUHD� PHGLH� OLQLDU )(d � VH� FDOFXOHD] � FD� R� PHGLH� DULWPHWLF � VLPSO � GLQ� DEDWHULOH�WHUPHQLORU�VHULHL�GH�OD�PHGLD�ORU��OXDWH�vQ�YDORDUH�DEVROXW �
n
xx
d
n
ii∑
=−
= 1
$EDWHUHD�PHGLH�S WUDWLF sau abaterea standard )(σ �VH�FDOFXOHD] �FD�R�PHGLH�S WUDWLF �din DEDWHULOH�WXWXURU�YDULDQWHORU�VHULHL�GH�OD�PHGLD�ORU�DULWPHWLF �
n
xxn
ii∑
=
−= 1
2)(σ
&DOFXOD L�SHQWUX�DFHHDúL�VHULH��FHL�GRL�LQGLFDWRUL�YHULILF �UHOD LD�
d>σ ÌQ�OLWHUDWXUD�GH�VSHFLDOLWDWH�VH�DSUHFLD] �F �SHQWUX�R�VHULH�GH�GLVWULEX LH�FX�WHQGLQ �FODU �GH�
QRUPDOLWDWH��DEDWHUHD�PHGLH�OLQLDU �HVWH�HJDO �FX�����GLQ�YDORDUHD�DEDWHULL�PHGLL�S WUDWLFH� Dispersia� �YDULDQ D�� XQHL� FDUDFWHULVWLFL� )( 2σ � VH� FDOFXOHD] � FD� PHGLH� DULWPHWLF � VLPSO � D�
S WUDWHORU�DEDWHULORU�WHUPHQLORU�ID �GH�PHGLD�ORU�� Formulele de calcul sunt:
n
xxn
ii∑
=
−= 1
2
2
)(σ sau
2
11
2
2
−=∑∑
==
n
x
n
xn
ii
n
ii
σ
&RHILFLHQWXO�GH�YDULD LH (v��VH�FDOFXOHD] �FD�UDSRUW�vQWUH�DEDWHUHD�PHGLH�S WUDWLF �úL�QLYHOXO�
mediu al seriei:
100⋅=x
vσ
Coeficientul GH�YDULD LH�VH�PDL�SRDWH�FDOFXOD�úL�GXS �UHOD LD�
100⋅=′x
dv
6(5,,�'(�',675,%8 ,(�81,',0(16,21$/(��GDWH�JUXSDWH� ,QGLFDWRUL�GH�IUHFYHQ H
)UHFYHQ HOH� UHODWLYH ( *in �� VH� RE LQ� UDSRUWkQG� IUHFYHQ D� ILHF UHL� JUXSH� �ni) la totalul
IUHFYHQ HORU� )(1
∑=
k
iin :
∑=
=k
ii
ii
n
nn
1
* sau 100
1
*(%) ⋅=
∑=
k
ii
ii
n
nn
)UHFYHQ HOH�FXPXODWH�VH�QRWHD] �FX� iF sau *iF �vQ�IXQF LH�GH�IHOXO�IUHFYHQ HORU�LQFOXVH�vQ�
calcul (absolute sau relative):
;,...,3,2,1 ;1
kinFi
jji == ∑
=
respectiv ;,...3,2,1 ; 1
** kinFi
jji == ∑
=
0 ULPL�PHGLL 0HGLD�DULWPHWLF &DOFXOXO�FX�IUHFYHQ H�DEVROXWH�
∑
∑
=
==k
ii
k
iii
n
nx
x
1
1
3URSULHW L� a) PHGLD� VH� vQFDGUHD] � vQ� LQWHUYDOXO� FX� IUHFYHQ D� PD[LP � VDX� vQ� XQXO� GLQ� FHOH� GRX �
intervale învecinate.
b) 0)(1
=−∑=
k
iii nxx
c) GDF � axx i #=′ , atunci ax
n
nax
xk
ii
i
k
ii
#
#
==′
∑
∑
=
=
1
1
)(
d) GDFh
xx i=′′ , atunci
1
1
h
x
n
nh
x
xk
ii
n
ii
i
=⋅
=′′
∑
∑
=
=
UHVSHFWLY�GDF � hxx i ⋅=′′ , atunci xhn
nhx
xk
ii
n
iii
⋅=⋅⋅
=′′
∑
∑
=
=
1
1
)(
3URSULHW LOH� GH� OD� SXQFWHOH� F�� úL� G�� VHUYHVF� OD� calculul simplificat al mediei aritmetice. 6H�XWLOL]HD] �vQ�DFHVW�VFRS�IRUPXOHOH�
( )a
n
nax
xk
ii
i
k
ii
±=
∑
∑
=
=
1
1
#
h
n
nh
x
xk
ii
i
k
i
i
⋅
=
∑
∑
=
=
1
1 , respectiv
( )h
n
nhx
xk
ii
i
k
ii
÷⋅
=
∑
∑
=
=
1
1
ah
n
nh
ax
xk
ii
i
k
i
i
+⋅
−
=
∑
∑
=
=
1
1 ,
unde: a - mijlocul unui intHUYDO�GH�RELFHL�FHQWUXO�LQWHUYDOXOXL�FX�IUHFYHQ D�FHD�PDL�PDUH� h -�P ULPHD�LQWHUYDOXOXL�GDF �VHULD�DUH�LQWHUYDOH�GH�YDULD LH�HJDOH� e) GDF �vQWU-R�VHULH�VH�UHGXF�SURSRU LRQDO�WRDWH�IUHFYHQ HOH��PHGLD�FDOFXODW �SH�ED]D�QRLORU�
IUHFYHQ H�U PkQH�QHVFKLPEDW �
x
c
nc
nx
i
ii
=∑
∑
$FHDVW � SURSULHWDWH� VHUYHúWH� OD� FDOFXOXO� PHGLHL� FX� DMXWRUXO� IUHFYHQ HORU� UHODWLYH. În acest caz ∑= inc :
∑= *ii nxx sau
100
*(%)∑= ii nx
x
0HGLD�DUPRQLF
• &DOFXOXO�FX�IUHFYHQ H�DEVROXWH�
∑
∑
=
==k
ii
i
k
ii
h
nx
nx
1
1
1
• &DOFXOXO�FX�IUHFYHQ H�UHODWLYH�
∑=
=k
ii
i
h
nx
x
1
*11
sau
∑=
=k
ii
i
h
nx
x
1
*(%)
1100
)RUP �WUDQVIRUPDW �D�PHGLHL�DULWPHWLFH • &DOFXOXO�FX�IUHFYHQ H�DEVROXWH�
xnx
x
nxx
k
iii
i
k
iii
h ==∑
∑
=
=
1
1
1
• &DOFXOXO�FX�IUHFYHQ H�UHODWLYH�
∑=
=k
iii
i
h
nxx
x
1
*11
sau
∑=
=k
iii
i
h
nxx
x
1
*(%)
1100
0HGLD�S WUDWLF
• &DOFXOXO�FX�IUHFYHQ H�DEVROXWH�
∑
∑
=
==k
ii
i
k
ii
p
n
nxx
1
1
2
• &DOFXOXO�FX�IUHFYHQ H�UHODWLYH�
*
1
2i
k
iip nxx ∑
=
= sau 100
*(%)
1
2i
k
ii
p
nxx
∑==
0HGLD�JHRPHWULF
1
1
∑= = ∏=
k
ii
in k
i
nig xx
Prin lRJDULWPDUH�VH�YD�RE LQH�
∑
∑
=
=⋅
=k
ii
k
iii
g
n
xnx
1
1
lglg
ÌQWUH�PHGLLOH�SUH]HQWDWH�H[LVW �XUP WRDUH�UHOD LH�GH�RUGLQH�
pagh xxxx <<<
9DORUL�PHGLL�GH�SR]L LH�VDX�GH�VWUXFWXU Modul (modulul, dominanta)
ÌQ�FD]XO�XQHL�VHULL�GH�GLVWULEX LH�SH�YDULDQWH��PRGXO�HVWH�YDULDQWD�FX�IUHFYHQ D�PD[LP � ÌQ�FD]XO�JUXS ULL�SH�LQWHUYDOH��ORFXO�PRGXOXL�HVWH�LQWHUYDOXO�FX�IUHFYHQ D�PD[LP �LDU�YDORDUHD�
VH�FDOFXOHD] �DVWIHO�
21
10 ∆+∆
∆+= hxMo
în care:
x0 - UHSUH]LQW �OLPLWD�LQIHULRDU �D�LQWHUYDOXOXL�PRGDO� h - P Uimea intervalului modal;
1∆ - GLIHUHQ D�GLQWUH�IUHFYHQ D�LQWHUYDOXOXL�PRGDO�úL�D�FHOXL�SUHFHGHQW�
2∆ - GLIHUHQ D�GLQWUH�IUHFYHQ D�LQWHUYDOXOXL�PRGDO�úL�D�FHOXL�XUP WRU�
Mediana (Me) a) În cazul datelor grupate pH� YDULDQWH�� ORFXO� PHGLDQHL� HVWH� YDULDQWD� D� F UHL� IUHFYHQ �
FXPXODW � HVWH� SULPD� PDL� PDUH� GHFkW�2
1+n iar valoarea medianei este chiar varianta
UHVSHFWLY �
b) ÌQ� FD]XO� GDWHORU� JUXSDWH� SH� LQWHUYDOH�� ORFXO�PHGLDQHL� HVWH� LQWHUYDOXO� D� F UXL� IUHFYHQ �
FXPXODW �HVWH�SULPD�PDL�PDUH�GHFkW�2
11
∑=
+k
iin
��LDU�YDORDUHD�PHGLDQHL�VH�FDOFXOHD] �GXS �
formula:
m
m
ii
k
ii
n
nn
hxMe∑
∑ −
=
= −+
⋅+=
1
1
1
0
2
1
,
unde: h - P ULPHD�LQWHUYDOXOXL�PHGLDQ� m - indexul intervalului median;
∑−
=
1
1
m
iin
- VXPD� IUHFYHQ HORU� SUHFHGHQWH� LQWHUYDOXOXL� PHGLDQ� �IUHFYHQ D� FXPXODW �a intervalului precedent celui median);
nm - IUHFYHQ D�DEVROXW �D�LQWHUYDOXOXL�PHGLDQ� Cuartilele VXQW�DFHOH�YDORUL�DOH�FDUDFWHULVWLFLL��FDUH�VHSDU �VHULD�vQ�SDWUX�S U L�HJDOH�
u
u
ii
k
ii
n
nn
hxQ∑∑
−
==
−++=
1
1101
)1(4
1
,
unde u�HVWH�LQGH[XO�LQWHUYDOXOXL�FDUH�FRQ LQH�Q1
MeQ =2
v
v
ii
k
ii
n
nnhxQ
∑∑−
==
−++=
1
1103
)1(4
3
,
unde v�HVWH�LQGH[XO�LQWHUYDOXOXL�FDUH�FRQ LQH�Q3
Decilele GLYLG�VHULD�vQ�]HFH�S U L�HJDOH�IRORVLQG�vQ�DFHVW�VFRS�QRX �GHFLOH
1
11
1101
)1(10
1
d
d
ii
k
ii
n
nnhxD
∑∑−
==−+
+= ,
unde d1�HVWH�LQGH[XO�LQWHUYDOXOXL�FDUH�FRQ LQH�D1
eMD =5
9
19
1109
)1(10
9
d
d
ii
k
ii
n
nnhxD
∑∑−
==
−++= ,
unde d9 HVWH�LQGH[XO�LQWHUYDOXOXL�FDUH�FRQ LQH�D9.
,QGLFDWRULL�YDULD LHL
a) ,QGLFDWRULL�VLPSOL�DL�YDULD LHL
$PSOLWXGLQHD�DEVROXW (Aa) Aa=xL-xl ,
unde: xL – liPLWD�VXSHULRDU �D�XOWLPXOXL�LQWHUYDO� xl –�OLPLWD�LQIHULRDU �D�SULPXOXL�LQWHUYDO� $PSOLWXGLQHD�UHODWLY �D�YDULD LHL��A%)
100% ⋅=x
AA a
b) ,QGLFDWRULL�VLQWHWLFL�DL�YDULD LHL
$EDWHUHD�PHGLH�OLQLDU )(d • &DOFXOXO�FX�IUHFYHQ H�DEVRlute:
∑
∑
=
=
−=
k
ii
i
k
ii
n
nxx
d
1
1
• &DOFXOXO�FX�IUHFYHQ H�UHODWLYH�
*
1i
k
ii nxxd ∑
=−= sau
100
(%)*
1
i
k
ii nxx
d∑
=
−=
$EDWHUHD�PHGLH�S WUDWLF sau abaterea standard )(σ :
¾�&DOFXOXO�FX�IUHFYHQ H�DEVROXWH�
∑
∑
=
=
−=
k
ii
i
k
ii
n
nxx
1
1
2)(σ
¾�Calculul cu fUHFYHQ H�UHODWLYH�
i
k
ii nxx *
1
2)(∑=
−=σ sau 100
)( (%)1
2i
k
ii nxx∑
=
−=σ
&HL�GRL�LQGLFDWRUL�YHULILF �UHOD LD�
d>σ
&RHILFLHQWXO�GH�YDULD LH (v):
100⋅=x
vσ
, respectiv 100⋅=′x
dv
Dispersia (variaQ D� )( 2σ : • IRUPXOH�GHULYDWH�GLQ�GHILQL LH
¾�FDOFXOXO�FX�IUHFYHQ H�DEVROXWH�
∑
∑
=
=
−=
k
ii
i
k
ii
n
nxx
1
1
2
2
)(σ
¾�FDOFXOXO�FX�IUHFYHQ H�UHODWLYH�
i
k
ii nxx *
1
22 )(∑=
−=σ sau 100
)( (%)*
1
2
2i
k
ii nxx∑
=
−=σ
• ca moment centrat de ordinul doi ¾�calculul cu freFYHQ H�DEVROXWH�
2
1
1
1
1
2
2
−=∑
∑
∑
∑
=
=
=
=k
ii
k
iii
k
ii
i
k
ii
n
nx
n
nxσ
¾�FDOFXOXO�FX�IUHFYHQ H�UHODWLYH�
2
1
**
1
22
−= ∑∑
==
k
iiii
k
i
i nxnxσ sau
2
1
*(%)
*(%)
1
2
2
100100
−=∑∑
==
k
iiii
k
ii nxnx
σ
• prin formula de calcul simplificat ¾�SHQWUX�R�VHULH�GH�IUHFYHQ H�DEVROXWH�
22
1
1
2
2 )( axhn
nh
ax
k
ii
k
ii
i
−−⋅
−
=∑
∑
=
=σ
¾�pentru o serie cu frecven H�UHODWLYH�
22
1
*
2
2 )( axhnh
axk
ii
i −−⋅
−
= ∑=
σ sau
221
*(%)
2
2 )(100
axh
nh
axk
ii
i
−−⋅
−
=∑
=σ
1RW � a�úL�h�DX�DFHOHDúL�VHPQLILFD LL�FD�OD�FDOFXOXO�PHGLHL�DULWPHWLFH� &RUHF LD�OXL�6KHSSDUG�
12)(
222 h−=′ σσ ,
unde h �HVWH�P ULPHD�LQWHUYDOXOXL�GH�JUXSDUH� $FHDVW �FRUHF LH�VH�DSOLF �QXPDL�SHQWUX�VHULLOH�FDUH�SUH]LQW �XUP WRDUHOH�SURSULHW L� • UHSDUWL LD�HVWH�QRUPDO �VDX�XúRU�DVLPHWULF � • UHSDUWL LD�DUH�LQWHUYDOH�GH�JUXSDUH�HJDOH�
9DULD LD�LQWHUFXDUWLOLF �úL�LQWHUGHFLOLF $EDWHUHD�LQWHUFXDWLOLF ��Qd):
22
)()( 1331 QQMeQQMeQd
−=
−+−=
CoeILFLHQWXO�GH�YDULD LH�LQWHUFXDUWLOLF ��9T�:
Me
Me
Me
QV d
q 22 13
13
−=
−
==
$EDWHUHD�LQWHUGHFLOLF :
22
)()( 1991 DDMeDDMeDd
−=
−+−=
&RHILFLHQWXO�GH�YDULD LH�LQWHUGHFLOLF :
Me
DD
Me
DD
Me
DV d
d 22 19
19
−=
−
==
0HGLD�úL�GLVSHUVLD�FDUDFWHULVWLFLL�DOWHUQDWLYH 'LVWULEX LD�GH�IUHFYHQ H�D�FDUDFWHULVWLFLL�DOWHUQDWLYH�VH�SUH]LQW �vQWU-un tabel de forma:
Tabel 3.1.
Valoarea
caracteristicii 5 VSXQVXO�
înregistrat )UHFYHQ H�DEVROXWH )UHFYHQ H�UHODWLYH
0 1 2 3
x1 = 1 DA
M �QXP UXO�XQLW LORU�
FDUH�SRVHG �caracteristica)
N
Mp =
x2 = 0 NU
(N-M) �QXP UXO�GH�XQLW L�FDUH�QX�SRVHG �caracteristica)
p1N
MNq −=−=
Total N = M + (N - M) p + q = 1
Media:
N
Mp =
Dispersia:
)1(sau 2p
2p ppqp −⋅=⋅= σσ
$EDWHUHD�PHGLH�S WUDWLF :
qpp ⋅=σ
Asimetria
$VLPHWULD�DEVROXW ( As ) MoxAs −=
&RHILFLHQ LL�GH�DVLPHWULH�SURSXúL�GH�.DUO�3HDUVRQ��SHQWUX�VHULL�GH�GLVWULEX LH�XúRU�DVLPHWULFH��
⇒<
⇒>
⇒=−=
0
0
0
as
as
as
as
C
C
C
MoxC
σ
Acest coeficient poate lua valori cuprinse între -�� úL� +1; cu cât este mai mic în valoare
DEVROXW �FX�DWkW�DVLPHWULD�HVWH�PDL�PLF � ÌQ�FD]XO� FkQG�VH�FXQRDúWH�PHGLDQD�VHULHL�� FRHILFLHQWXO�GH�DVLPHWULH� )( asC ′ se poate calcula
XWLOL]kQG�UHOD LD�
σ)(3 Mex
Cas
−=′
3HDUVRQ�PDL�SURSXQH�úL�XQ�Dlt coeficient pentru calculul gradului de asimetrie al unei serii IRUPDW �GLQWU-XQ�QXP U�IRDUWH�PDUH�GH�REVHUYD LL�
=−
=
−=
=′′
∑∑
∑∑
22
2
3
3
32
23
)(
)(
unde ,
σµ
µ
µ
µ
i
ii
i
ii
as
n
nxx
n
nxx
C
Coeficientul propus de Yule:
)()(
)()(
13
13
QMeMeQ
QMeMeQCasY −+−
−−−= ,
unde [ ]1,1−∈asYC
Coeficientul propus de Bowley:
)()(
)()(
19
19
DMeMeD
DMeMeDCasB −+−
−−−= ,
unde [ ]1,1−∈asBC
,QGLFDWRULL�FRQFHQWU ULL Coeficientul de concentrare propus de statisticianul italian Corado Gini:
kigC iG ,1 2 == ∑ ,
unde :
1
∑=
=k
i
ii
iii
nx
nxg
VHULH�VLPHWULF
DVLPHWULH�SR]LWLY
DVLPHWULH�QHJDWLY
Acest coeficient ia valori în intervalul
1,
1
n.
Coeficientul de concentrare propus de R. Struck:
1
12
−−
= ∑n
gnC i
S
Acest coeficient ia valori în intervalul [ ]1,0 .
6(5,,�'(�',675,%8 ,(�%,',0(16,1$/(
$��&DOFXOXO�FX�IUHFYHQ H�DEVROXWH
O serie de distrLEX LH�ELGLPHQVLRQDO �VH�SUH]LQW �vQWU-un tabel de forma:
Tabel 3.2.
Valorile caracteristicii de grupare X
Variantele sau valorile caracteristicii dependente Y
Volumul grupei
(ni.)
Medii pe grupe
)y( i
y1 y2 … yj … ym x1 n11 n12 ... n1j … n1m n1. )y( 1
x2 n21 n22 … n2j … n2m n2. )y( 2
... ... ... … ... … ... ... ... xi ni1 ni2 … nij … nim ni. )y( i
... ... ... … ... … ... ... ... xr nr1 nr2 … nrj … nrm nr. )y( r
Total n.1 n.2 … n.j … n.m ∑=∑==
m
1jj
r
1ii nn ..
y
9ROXPXO��IUHFYHQ D��JUXSHL�i:
.1
i
m
jij nn =∑
=
0 ULPL�PHGLL
• 0HGLLOH�GH�JUXS ( iy ):
∑
∑
=
==m
jij
m
jijj
i
n
ny
y
1
1
• Media pe total:
∑
∑
∑
∑
=
=
=
===
r
ii
r
iii
m
jj
m
jjj
n
ny
y
n
ny
y
1.
1.
1.
1.
sau
,QGLFDWRULL�YDULD LHL
• 'LVSHUVLD�GH�JUXS sau GLVSHUVLD�SDU LDO )( 2iσ :
∑
∑
=
=
−=
m
jij
m
iijij
i
n
nyy
1
1
2
2
)(σ ,
unde: yj �UHSUH]LQW �YDULDQWD�VDX�PLMORFXO�LQWHUYDOXOXL�j al caracteristicii dependente;
iy media grupei i; nij IUHFYHQ HOH�FRUHVSXQ] WRDUH�ILHF UHL�YDULDQWH��LQWHUYDO�GH�YDORUL��GLQ�FDGUXO�JUXSei.
• 0HGLD�GLVSHUVLLORU�GH�JUXS )( 2
/
2
ryσσ =
∑
∑
=
==r
ii
i
r
ii
n
n
1.
.1
2
2
σσ ,
unde: 2iσ - dispersia grupei i;
.in - volumul grupei i.
• Dispersia dintre grupe )( 2
/
2
xyσδ = :
∑
∑
=
=−
==r
ii
r
iii
xy
n
nyy
1.
1.
2
22/
)(
δσ
• DisSHUVLD�WRWDO )( 22
yσσ = :
∑
∑
=
=−
==m
jj
m
jjj
y
n
nyy
1.
1.
2
22
)(
σσ
5HJXOD�DGXQ ULL�GLVSHUVLLORU:
2/
2/
2ryxy σσσ +=
Pe baza regulii de adunare a dispersiilor se pot calcula indicatori statistici cu caracter de P ULPL�UHODWLYH�GH�VWUXFWXU � • Gradul de dHWHUPLQD LH )( 2
/ xyR :
1002
2/2
/ ⋅=σ
σ xyxyR
'DF � )R( x/y !�����DGPLWHP�F �IDFWRUXO�GH�JUXSDUH�HVWH�KRW UkWRU��VHPQLILFDWLY��GHWHUPLQDQW��
SHQWUX�YDULD LD�IDFWRUXOXL�GHWHUPLQDW��Y). • *UDGXO�GH�QHGHWHUPLQD LH:
1002
2/2
/ ⋅=σ
σ ryxyK
$EDWHUHD�PHGLH�S WUDWLF �OD�QLYHOXO�JUXSHL:
2ii σσ =
$EDWHUHD�PHGLH�S WUDWLF �SH�WRWDO:
2σσ =
&RHILFLHQWXO�GH�YDULD LH�OD�QLYHOXO�JUXSHL�
100⋅=i
ii y
vσ
,
unde:
iσ -�DEDWHUHD�PHGLH�S WUDWLF �D�JUXSHL�
iy - media grupei. &RHILFLHQWXO�GH�YDULD LH�SH�WRWDO�
100⋅=y
vσ
,
unde: σ -�DEDWHUHD�PHGLH�S WUDWLF �SH�WRWDO� y - media pe total.
%��&DOFXOXO�FX�IUHFYHQ H�UHODWLve 2�VHULH�GH�GLVWULEX LH�ELGLPHQVLRQDO �VH�SUH]LQW �vQWU-un tabel de forma:
Tabel 3.3.
Valorile caracteristicii
de grupare X )UHFYHQ H�UHODWLYH���� Total
(%) Ponderea grupei
(ni(%)) y1 y2 … yj … ym
x1 n*11 n*12 ... n*1j … n*1m 100 n1(%) x2 n*21 n*22 ... n*2j … n*2m 100 n2(%) ... ... ... … ... … ... ... ... xi n*i1 n*i2 ... n*ij … n*im 100 ni(%) ... ... ... … ... … ... ... ... xr n*r1 n*r2 ... n*rj … n*rm 100 nr(%)
Total 100
• 0HGLLOH�GH�JUXS ( iy ):
1001
*∑==
m
j
ijj
i
ny
y
• Media pe total:
100
1(%)∑
==
r
iii ny
y
• 'LVSHUVLD�GH�JUXS �VDX�GLVSHUVLD�SDU LDO )( 2
iσ :
100
)(1
*2
2∑=
−=
m
i
ijij
i
nyy
σ
• 0HGLD�GLVSHUVLLORU�GH�JUXS )( 2
/
2
ryσσ =
100
(%)1
2
2
i
r
ii n∑
==
σ
σ
• Dispersia dintre grupe )( 2
/
2
xyσδ = :
100
)(1
(%)2
22/
∑=
−
==
r
iii
xy
nyy
δσ
'LVSHUVLD�WRWDO )( 22yσσ = :
2222 δσσσ +== y
,QGLFDWRULL�PHGLL�úL�DL�YDULD LHL�SHQWUX�FDUDFWHULVWLFL�DOWHUQDWLYH 'LVSHUVLD�GH�JUXS ( 2
ipσ ):
)1(sau 22iipiip ppqp
ii−== σσ ,
în care: pi -�UHSUH]LQW �PHGLL�GH�JUXS � qi -�IUHFYHQ HOH�UHODWLYH�DOH�XQLW LORU�FDUH�QX�SRVHG �FDUDFWHULVWLFD�vQ�ILHFDUH�JUXS �
0HGLD�GLVSHUVLLORU�SDU LDOH )( 2
pσ :
∑
∑
=
==r
ii
r
iip
p
N
Ni
1
1
2
2
σσ ,
în care Ni�UHSUH]LQW �QXP UXO�WRWDO�DO�XQLW LORU�REVHUYDWH�vQ�ILHFDUH�JUXS � Dispersia dintre grupe )( 2
pδ :
∑
∑
=
=
−=
r
ii
r
iii
p
N
Npp
1
1
2
2
)(
δ ,
în care p este media caracteristicii alternative pe întreaga colectivitate. 'LVSHUVLD�WRWDO )( 2
pσ :
qpp ⋅=2σ
5HJXOD�DGXQ ULL dispersiilor:
222ppp δσσ +=
9HULILFDUHD�VHPQLILFD LHL�IDFWRUXOXL�GH�JUXSDUH�IRORVLQG�WHVWXO�³)´
2/
2/
ry
xycalculat S
SF = ,
unde:
( )1
1.
2
2/ −
⋅−=
∑=
r
nyyS
r
iii
xy (r -�QXP UXO�GH�JUXSH�
( )rn
nyy
S
r
i
m
jijij
ry −
⋅−=
∑∑= =1 1
2
2/
'DF �)calculat> Ftabelar factorul de grupare este semnificativ 'DF �)calculat< Ftabelar factorul de grupare nu este semnificativ Ftabelar� VH� GHWHUPLQ � vQ� IXQF ie de un anumit nivel de semnifica LH� �GH� H[HPSOX�� ������ úL� GH�
gradele de libertate f1=r-1�úL�f2=n-r.
PROBLEME REZOLVATE
1. Într-un magazLQ�OXFUHD] ����YkQ] WRUL��FDUH�vQ�OXQD�RFWRPEULH������DX�UHDOL]DW�XQ�YROXP�DO�
desfacerilor (mil. lei) astfel: 138,8; 146,0; 150,0; 152,3; 158,2; 163,1; 165,0; 170,4; 176,2; 180,0. Se cere: 1. V �VH�IRUPH]H�VHULD�SULYLQG�YROXPXO�GHVIDFHULORU�SULQ�FHQWUDOL]DUea datelor individuale; 2. V � VH� FDOFXOH]H� YROXPXO� PHGLX� DO� GHVIDFHULORU� GLQ� DFHVW� PDJD]LQ� IRORVLQG� PHGLD�
DULWPHWLF �úL�V �VH�YHULILFH�SULQFLSDOHOH�SURSULHW L�DOH�DFHVWHLD� 3. V �VH�IRORVHDVF �SHQWUX�DFHHDúL�VHULH�úL�DOWH�WLSXUL�GH�PHGLL�úL�V �VH�DUDWH��vQ�FH�Uaport de
P ULPH�VH�DIO �HOH�ID �GH�PHGLD�DULWPHWLF � 4. V �VH�FDOFXOH]H�LQGLFDWRULL�PHGLL�GH�SR]L LH��PHGLL�GH�VWUXFWXU �� 5. V � VH� FDUDFWHUL]H]H� JUDGXO� GH� YDULD LH� DO� GLVWULEX LHL� FDOFXOkQG� vQ� DFHVW� VFRS� LQGLFDWRULL�
VLPSOL�úL�VLQWHWLFL�DL�YDULD LHL��SUHFXP�úL�gradul de asimetrie; 6. V �VH�UHSUH]LQWH�JUDILF�VHULD� Rezolvare
1. 6HULD�VH�IRUPHD] �SULQ�RUGRQDUHD�YDORULORU��YH]L�WDEHOXO������FRORDQHOH���úL����
Notând cu xi termenii individuali, volumul centralizat este ∑=
n
iix
1
.
∑=
10
1iix = 138,8 +146,0 +150,0 +....+180,0 = 1600 mil. lei
2. &DOFXOXO�PHGLHL�DULWPHWLFH�úL�YHULILFDUHD�SULQFLSDOHORU�SURSULHW L�DOH�DFHVWHLD se poate
face cu ajutorul unui tabel de calcul (Vezi tabelul 3.4.).
&DOFXOXO�PHGLHL�DULWPHWLFH�VLPSOH�úL�YHULILFDUHD�SULQFLSDOHORU�SURSULHW L Tabelul 3.4.
Nr.crt. Volumul
desfacerilor (mil. lei)
xxi − axi −
a = 138,8 h
xi ; h = 20
A 1 2 3 4 1 138,8 -21,2 0 6,94 2 146,0 -14,0 7,2 7,3 3 150,0 -10,0 11,2 7,5 4 152,3 -7,7 13,5 7,615 5 158,2 -1,8 19,4 7,91 6 163,1 +3,1 24,3 8,155 7 165,0 +5,0 26,2 8,25 8 170,4 +10,4 31,6 8,52 9 176,2 +16,2 37,4 8,01 10 180,0 +20,0 41,2 9,0 Total 1600,0
∑=
10
1iix
0
)(10
1
xxi
i −∑=
212
)(10
1
axi
i −∑=
80
∑=
10
1i
i
h
x
2.1. Calculul mediei aritmetice simple (vezi coloana 1 din tabelul 3.4.):
16010
1600 === ∑n
xx i �PLO��OHL�YkQ] WRU�
2.2. 9HULILFDUHD�SULQFLSDOHORU�SURSULHW L�DOH�PHGLHL�DULWPHWLFH
a) maxmin xxx <<
1808,138 << x
b) 0)( =−∑ xxi (vezi tabelul 3.4., coloana 2)
Media fiind un cât exact, suma abaterilor pozitive s-a compensat cu suma abaterilor negative:(-54,7) + (54,7) = 0
c) ∑=
⋅=n
ii xnx
1
1600160,010 =⋅ mil. lei
d) axn
axx
n
ii
−=−
=′∑
=1
)(
Alegând arbitrar a = 138,8 - primul termen -�úL�FDOFXOkQG�YDORULOH�xi - a (vezi tabelul ������VH�RE LQH�
2,2110
212)(
1 ==−
=′∑
=
n
axx
n
ii
mil. OHL�YkQ] WRU
deci: axx =′−
160-21,2=138,8 de unde:
axx +′= =x ����������� ����PLO��OHL�YkQ] WRU
e) h
x
n
h
xi
=
∑
/XkQG�DUELWUDU�XQ�K� �����YH]L�WDEHOXO�������FRORDQD����VH�RE LQH�
810
80 ==
=′′∑
n
h
x
x
i
hxx ⋅′′= 160208 =⋅=x �PLO��OHL�YkQ] WRU
3. &DOFXOXO�PHGLHL�S WUDWLFH��DO�PHGLHL�DUPRQLFH�úL�DO�PHGLHL�JHRPHWULFH 3.1.�&DOFXOXO�PHGLHL�S WUDWLFH ( )px (vezi tabelul 3.5., coloana 3):
5,16010
18,25761310
2
== ∑n ip n
xx PLO��OHL�YkQ] WRU
;xx p > 160,5 > 160
&DOFXOXO�PHGLHL�DUPRQLFH��S WUDWLFH�úL�JHRPHWULFH
Tabelul 3.5.
Nr. crt.
Volumul desfacerilor
(mil. lei) ix
1 2
ix ixlg
0 1 2 3 4 1 138,8 0,0072046 19265,44 2,14239
2 146,0 0,0068000 21316,00 2,16425
3 150,0 0,0066670 22500,00 2,17605
4 152,3 0,0065659 23195,29 2,18270
5 158,2 0,0063211 25027,24 2,19921
6 163,1 0,0061312 26601,61 2,21245
7 165,0 0,0060606 27225,00 2,21748
8 170,4 0,0058685 29036,16 2,23045
9 176,2 0,0056753 31046,44 2,24601
10 180,0 0,0055550 32400,0 2,25527
Total 1600,0 0,0628986 257613,18 22,02640
3.2. Calculul mediei armonice ( hx ) (vezi tabelul 3.5., coloana 2)
98,1580628986,0
101
===∑
i
h
x
nx �PLO�OHL�YkQ] WRU
;xxh < 158,98 < 160,0
Deci: ph xxx <<
3.3. Calculul mediei geometrice ( gx ) (vezi tabelul 3.5., coloana 4):
n
n
iig xx ∏
=
=1
; n
xx
n
ii
g
∑== 1
lglg
20264,210
02640,22lg =gx
antilog 2,20264 = 159,5 => gx =�������PLO�OHL��YkQ] WRU
Deci: ag xx < ; 159,5 < 160
5HFDSLWXOkQG�YDORULOH�PHGLLORU�RE LQXWH�DYHP�
0,160== xxa PLO�OHL�YkQ] WRU� px =160,5 PLO�OHL�YkQ] WRU�
hx = 158,98 PLO�OHL�YkQ] WRU� gx = 159,50 PLO�OHL�YkQ] WRU
5HOD LD�GLQWUH�PHGLL�HVWH�
pagh xxxx <<<
158,98 < 159,5 < 160 < 160.5
$EDWHULOH�vQWUH�FHOH�SDWUX�P ULPL�PHGLL�ILLQG�IRDUWH�PLFL��VH�SRDWH�WUDJH�FRQFOX]LD�F �SHQWUX�DFHDVW �VHULH�RULFH�PHGLH�V-DU�IRORVL��HD�P VRDU �VXILFLHQW�GH�FRUHFW�WHQGLQ D�FHQWUDO �úL�GHFL�PHGLD�vQ�DFHVW�FD]�SRDWH�IL�R�YDORDUH�WLSLF �
4. &DOFXOXO�LQGLFDWRULORU�PHGLL�GH�SR]L LH 4.1. Calculul medianei (Me)
• locul medianei: 5,52
110
2
1)( =+=+= n
MeU
• valoarea medianei:
65,1602
1,1632,158
265 =+=
+=
xxMe mil. lei
4.2. Calculul modului )LLQG�VHULH�VLPSO �úL�QHDYkQG�WHUPHQL�FDUH�V �VH�UHSHWH��QX�HVWH�FD]XO� 4.3. Calculul cuartilelor Calculul cuartilelor presupune stabilirea locului cuartilelor U(Q2��úL�DSRL�FDOFXODUHD�YDORULORU�lor:
75,24
11
4
1)( 1 ==+= n
QU
1482
150146
232
1 =+=+
=xx
Q mil. lei
65,1602 == eMQ mil. lei
25,84
)1(3)( 3 =+= n
QU
3,1732
2,1764,170
298
3 =+=+
=xx
Q mil. lei
Potrivit celor trei cuartile, seria se poate structura astfel:
6WUXFWXUD�VHULHL�vQ�IXQF LH�GH�YDORULOH�FHORU�WUHL�FXDUWLOH
Tabelul 3.6.
,QWHUYDO�GH�YDULD LH Structura seriei (%)
138,80-148,00 25
148,00-160,65 25
160,65-173,30 25
173,30-180,00 25
5. &DOFXOXO�LQGLFDWRULORU�GH�YDULD LH�úL�DVLPHWULH PenWUX�FDOFXOXO�LQGLFDWRULORU�GH�YDULD LH�VH�IRORVHúWH�WDEHOXO������
&DOFXOXO�LQGLFDWRULORU�GH�YDULD LH
Tabelul 3.7.
Nr. crt. ix xx i − ( )2xxi −
2ix
100⋅−x
xx i
0 1 2 3 4 5 1 138,8 21,2 449,44 19265,44 - 13,25
2 146,0 14,0 196,00 21316,00 - 8,75
3 150,0 10,0 100,00 22500,00 - 6,25
4 152,3 7,7 59,29 23195,24 - 4,81
5 158,2 1,8 3,24 25027,24 - 1,13
6 163,1 3,1 9,61 26601,61 1,94
7 165,0 5,0 25,0 27225,00 3,13
8 170,4 10,4 108,16 29036,16 6,50
9 176,2 16,2 262,44 31046,44 10,13
10 180,0 20,0 400,00 32400,00 12,50
1600,0
∑=
n
iix
1
109,4
∑ − xx i
1613,18
( )2
1∑
=
−n
ii xx
257613,18
5.1. ,QGLFDWRULL�VLPSOL�DL�YDULD LHL • $PSOLWXGLQHD�YDULD LHL
Aa = xmax - xmin =180-138,8 = 41,2 mil. lei
%75,25100160
2,41minmax% =⋅=
−=
x
xxA
• Abaterile absolute�DOH�ILHF UHL�YDULDELOH�ID �GH�PHGLH��di):
xxd ii −= (vezi tabelul 3.7., coloana 2)
• Abaterile relative (di(%))
100(%) ⋅−
=x
xxd i
i (vezi tabelul 3.7., coloana 5)
5.2. ,QGLFDWRULL�VLQWHWLFL�DL�YDULD LHL • $EDWHUHD�PHGLH�OLQLDU ( )xd �úL�VH�IRORVHVF�GDWHOH�GLQ�WDEHOXO�������FRORDQD��
94,1010
4,1091 ==−
=∑
=
n
xxd
n
ii
x PLO��OHL�YkQ] WRU
• Dispersia ( )2xσ VH�FDOFXOHD] �FX�GDWHOH�GLQ�WDEHOXO�������FRORDQD 3:
( )318,161
10
18,16131
2
2 ==−
=∑
=
n
xxn
ii
xσ
'LVSHUVLD�VH�PDL�SRDWH�FDOFXOD�úL�FX�IRUPXOD�
21
2
2 xn
xn
ii
x −=∑
=σ
Datele necesare VH�J VHVF�vQ�WDEHOXO�������FRORDQHOH ��úL���
318,16116010
18,257613 22 =−=xσ
Deci s-D�DMXQV�OD�DFHODúL�UH]XOWDW�
• $EDWHUHD�PHGLH�S WUDWLF ( )∑ −
=n
xxix
2
σ sH�FDOFXOHD] �H[WU JkQG�U G FLQD�S WUDW �
din dispersie:
70,12318,1612 === xx σσ PLO��OHL�YkQ] WRU�
• &RHILFLHQ LL�GH�YDULD LH (v’, v)
%84,6100160
94,10100 =⋅=⋅=′
x
dv x
%84,7100160
70,12100 =⋅=⋅=
xv xσ
v'�úL�v �������GHFL�VH�SRDWH�DILUPD�F �VHULD�DUH�XQ�JUDG�PDUH�GH�RPRJHQLWDWH�úL�PHGLD�HVWH�UHSUH]HQWDWLY �SHQWUX�vQWUHDJD�VHULH�
6. 5HSUH]HQWDUHD�JUDILF ÌQ�DFHVW�FD]��YDULD LD�QHILLQG�FRQWLQX ��VH�SRDWH�IRORVL�XQ�JUDILF�SULQ�FRORDQH�úL�R�GLDJUDP �
UDGLDO �FX�]HFH�UD]H��YH]L�ILJXULOH������úL�������
180176,2173,4163,1158,2
152,3150146138,8
165
0
20
40
60
80
100
120
140
160
180
200
1 2 3 4 5 6 7 8 9 10
)LJXUD������5HSDUWL LD�VDODULD LORU�vQ�IXQF LH�GH�YROXPXO�GHVIDFHULORU��PLO��OHL�
mil.
lei
020406080
100120140160180
1
2
3
4
5
6
7
8
9
10
)LJXUD������5HSDUWL LD�VDODULD LORU�vQ�IXQF LH de volumul desfacerilor (mil. lei)
2. 3HQWUX�R�VRFLHWDWH�FRPHUFLDO �VH�FXQRVF�XUP WRDUHOH�GDWH�SULYLWRU�OD�YHFKLPHD�vQ�SURGXF LH��
vQVFULVH�vQ�WDEHOXO�������GDWH�FRQYHQ LRQDOH��
5HSDUWL LD�VDODULD LORU�GLQWU-R�VHF LH�GXS �YHFKLPH Tabelul 3.8.
Grupe dH�VDODULD L�GXS �YHFKLPH��DQL�
Sub 5
5-10 10-15 15-20 20-25 25-30 30-35 ���úL�peste
1XP UXO�VDODULD LORU 10 40 60 80 50 30 20 10
Se cere: 1. UHSUH]HQWDUHD�JUDILF �D�VHULHL�IRORVLQG�IUHFYHQ HOH�UHDOH�úL�FHOH�FXPXODWH� 2. vechimea medie a muncitorilor prin caOFXOXO�PHGLHL� DULWPHWLFH� FX� IUHFYHQ H� DEVROXWH��
UHODWLYH�úL�SULQ�FDOFXOXO�VLPSOLILFDW��VSHFLILFkQG�FH�SURSULHW L�DOH�PHGLHL�DULWPHWLFH�VWDX�OD�ED]D�FDOFXOXOXL�PHGLHL�DULWPHWLFH�FX�IUHFYHQ H�UHODWLYH�úL�SULQ�FDOFXOXO�VLPSOLILFDW�
3. LQGLFDWRULL�PHGLL�GH�SR]L ie; 4. LQGLFDWRULL�VLPSOL�úL�VLQWHWLFL�DL�YDULD LHL� 5. indicatorii de asimetrie; 6. PHGLD�úL�GLVSHUVLD�SHQWUX�PXQFLWRULL cu o vechime mai mare decât vechimea medie. Rezolvare
1. 5HSUH]HQWDUHD�JUDILF �D�VHULHL
• +LVWRJUDPD�úL�SROLJRQXO�IUHFYHQ HORU��YH]L�ILJXUD 3.3.); • &XUED�FXPXODWLY �D�IUHFYHQ HORU��YH]L�ILJXUD�������
0
10
20
30
40
50
60
70
80
90
0 5 10 15 20 25 30 35 40 45
vechime (ani)
� ������ ��� ��
)LJXUD������'LVWULEX LD�VDODULD LORU�vQ�IXQF LH�GH�YHFKLPH
0
50
100
150
200
250
300
350
0 5 10 15 20 25 30 35 40
vechime (ani)
QX
PU�
VDOD
ULD
L
)LJXUD������'LVWULEX LD�VDODULD LORU�vQ�IXQF LH�GH�YHFKLPH
2. Calculul vechimii medii În acest scop se vor folosi datele din tabelul 3.9.
Tabelul 3.9.
1XP UXO�VDODULD LORU
)UHFYHQ H absolute cumulate
Grupe de VDODULD L�GXS �
vechime (ani) abs. (ni) ( )
*%in
Centrul de
interval (xi)
ii nx ⋅ ⋅ix ( )*
%in h
axi −
a=17,5 h=5
ii nh
ax⋅
−
cres-F WRU
descres-F WRU
0 1 2 3 4 5 6 7 8 9 0-5
5-10 10-15 15-20 20-25 25-30 30-35 35-40
10 40 60 80 50 30 20 10
3,33 13,33 20,00 26,67 16,67 10,00 6,67 3,33
2,5 7,5
12,5 17,5 22,5 27,5 32,5 37,5
25 300 750
1400 1125 825 650 375
8,325 99,975
250,000 466,725 375,075 275,000 216,775 124,875
-3 -2 -1 0 1 2 3 4
-30 -80 -60
0 50 60 60 40
10 50
110 190 240 270 290 300
300 290 250 190 110
60 30 10
Total
300
∑=
k
iin
1
100
( )∑=
k
iin
1
*%
- 5450
∑=
k
iii nx
1
1816,75
∑=
k
iii nx
1
*(%)
-
40
∑=
⋅−k
ii
i nh
ax
1
- -
2.1. Calculul mediei aritmetice pe baza: • frecven HORU�DEVROXWH
2,18167.18300
5450
1
1 ≅===
∑
∑
=
=k
ii
k
iii
n
nx
x ani/salariat
• IUHFYHQ HORU�UHODWLYH:
( )2,18
1001
*%
≅=∑
=
k
iii nx
x ani/salariat
&DOFXOXO�PHGLHL�IRORVLQG�IUHFYHQWHOH�UHODWLYH�VH�ED]HD] �SH�SURSULHWDWHD�
;
1
1 x
c
nc
nx
k
i
i
k
i
ii
=∑
∑
=
= în acest caz ∑=
=k
iinc
1
2.2. Calculul simplificat al mediei aritmetice:
2,181667,185,175300
40
1
1 ≅=+⋅=+⋅
−
=
∑
∑
=
=ah
n
nh
ax
xk
ii
k
ii
i
ani/salariat
Calculul mediei prin acest procedeu s-D�ED]DW�SH�GRX �SURSULHW L�DOH�PHGLHL�DULWPHWLFH�úL�anume:
( )ax
n
naxx
k
ii
k
iii
−=⋅−
=′
∑
∑
=
=
1
1 , de unde:
( );
1
1 an
naxx
k
ii
k
iii
+⋅−
=∑
∑
=
=
h
x
n
nh
x
xk
ii
k
ii
i
==′′
∑
∑
=
=
1
1 , de unde: hn
nh
x
xk
ii
k
ii
i
⋅=∑
∑
=
=
1
1
&kQG� VXQW� LQWHUYDOH� HJDOH� HVWH� FRQYHQDELO� V se ia a egal cu centrul intervalului cu IUHFYHQ D�PD[LP ��DGLF �a ������úL�SHQWUX�h�R�YDORDUH�HJDO �FX�P ULPHD� LQWHUYDOHORU��DGLF �h = 5 (vezi tabelul 3.9.).
3. &DOFXOXO�LQGLFDWRULORU�PHGLL�GH�SR]L LH� 3.1.�&DOFXOXO�PHGLDQHL��0H��úL�D�FHORUODOWH�PHGLL�GH�SR]L LH: • FX�IUHFYHQ H�DEVROXWH�
5,1502
1300
2
1)( 1 =+=
+=
∑=
k
iin
MeU , deci 15 < Me < 20
5,1780
1105,150515
2
11
1
1
0 =−+=−
+
+=∑
∑ −
=
=
m
m
ii
k
ii
n
nn
hxMe ani
25,752
301
4
1)( 1
1 ==+
=∑
=
k
iin
QU , deci 10< Q1 <15
125,1260
5025,75510
14
1 1
1101 =−+=
−
+
+=∑∑
−
==
u
u
ii
k
ii
n
nn
hxQ ani
5,172 == eMQ ani
75,2253014
31
4
3)(
13 =⋅=
+= ∑
=
k
iinQU deci 20 < Q3 < 25
525,2350
19075,225520
14
3 1
1103 =−+=
−
+
+=∑∑
−
==
v
v
ii
k
ii
n
nn
hxQ ani
• FX�IUHFYHQ H�UHODWLYH�
( )
5,502
1100
2
1
)( 1
*%
=+=+
=∑=
k
ii
n
MeU , deci 15 < Me < 20
( )
( )
( )59,17
67,26
66,365,50515
2
1
*%
1
1
*1
*%
0
%
=−
+=−
+
+=∑
∑ −
=
=
m
m
i
k
ii
n
n
n
hxMei
ani
( )
25,252
101
4
1
)( 1
*
1
%
==+
=∑
=
k
ii
n
QU , deci 10 < Q1 < 15
( ) ( )
( )148,12
20
66,1625,25510
14
1
*%
1
1
*
1
*
01
%%
=−
+=
−
+
+=∑∑
−
==
u
u
i
k
i
n
nn
hxQii
ani
59,172 == eMQ ani
( ) 75,751014
31
4
3)(
1
*3 %
=⋅=
+= ∑
=
k
ii
nQU , deci 20 < Q3 < 25
( ) ( )
( )725,23
67,16
33,6375,75520
14
3
*%
1
1
*%
1
*%
03 =−
+=
−
+
+=∑∑
−
==
v
v
ii
k
ii
n
nn
hxQ ani
'LIHUHQ HOH�GLQWUH�LQGLFDWRULL�FDOFXOD L�FX�IUHFYHQ H�DEVROXWH�úL�FHL�FX�IUHFYHQ H�UHODWLYH�VXQW�
QHVHPQLILFDWLYH�úL�SURYLQ�GLQ�URWXQMLUHD�IUHFYHQ HORU�UHODWLYH� 3.2. Calculul modului (Mo): • FX�IUHFYHQ H�DEVROXWH�
Locul Mo�vO�FRQVWLWXLH�LQWHUYDOXO�FX�IUHFYHQ D�PD[LP �������Mo < 20.
173020
20515
21
10 =
++=
∆+∆∆
⋅+= hxMo ani
• cu frecven H�UHODWLYH�
27,17)67,1667,26()2067,26(
2067,26515
21
10 =
−+−−+=
∆+∆∆
⋅+= hxMo ani
9DORULOH�WHQGLQ HL�FHQWUDOH�FRQVLGHUDWH�FD�P ULPL�WLSLFH�FH�GHILQHVF�DFHDVW �VHULH��SHQWUX�
FDOFXOXO�FX�IUHFYHQ H�DEVROXWH��VXQW� x =18,2 ani; Me=17,5 ani; Mo=17 ani.
4. Calculul indicatorilor�GH�YDULD LH (vezi tabelul 3.10.). 4.1. Calculul indicatorilor simpli
ÌQ�DFHVW�FD]�QX�VXQW�FRQFOXGHQ L�
4.2. Calculul indicatorilor sintetici • $EDWHUHD�PHGLH�OLQLDU �� id ) �VH�RE LQH�GLQ�vQVXPDUHD�YDORULORU�GLQ�FRORDQD����YH]L�WDEHOXO�
3������I U �V �VH� LQ �VHDPD�GH�VHPQ�
5,6300
1956
1
1 ==⋅−
=∑
∑
=
=k
ii
k
iii
x
n
nxxd ani/salariat
• Dispersia ( 2xσ ):
- FDOFXOXO�RELúQXLW�
( )89,67
300
20367
1
1
2
2 ==⋅−
=∑
∑
=
=k
ii
k
iii
x
n
nxxσ
- calculul simplificat:
84,67)5,172,18(25300
820)( 222
1
1
2
2 =−−⋅=−−⋅⋅
−
=∑
∑
=
= axhn
nh
ax
k
ii
k
ii
i
xσ
• $EDWHUHD�PHGLH�S WUDWLF �� xσ )
2,884,672 === xx σσ ani/salariat
• &RHILFLHQ LL�GH�YDULD LH��v’�úL�v) - GDF �VH�SRUQHúWH�GH�OD�DEDWHUHD�PHGLH�OLQLDU �
%7,351002,18
5,6100 =⋅=⋅=′
x
dv x
- GDF �VH�SRUQHúWH�GH�OD�DEDWHUHD�PHGLH�S WUDWLF �
%451002,18
2,8100 =⋅=⋅=
xv xσ
Interpretând valoarea coeficientului de YDULD LH�VH�SRDWH�DILUPD�F �PHGLD�QX�HVWH�VXILFLHQW�GH�
UHSUH]HQWDWLY ��FD�XUPDUH�D�IDSWXOXL�F �VHULD�QX�HVWH�VXILFLHQW�GH�RPRJHQ ��'H�DOWIHO��ILLQG�YRUED�GH�WR L�VDODULD LL��HUD�GH�DúWHSWDW�FD�YHFKLPHD�PHGLH�V �DLE �R�GLVSHUVLH�PDUH��VDODULD LL�vQ�DQVDPEOXl lor V �DSDU LQ �OD�vQWUHJXO�LQWHUYDO�GH�YDULD LH�D�YHFKLPLL�
&DOFXOXO�LQGLFDWRULORU�VLQWHWLFL�DL�YDULD LHL
Tabelul 3.10.
1XP UXO�
muncitorilor
Grupe de VDODULD L�
GXS �
vechime (ani) abs. (ni) ( )*
%in
Centrul de
interval (xi)
xxi −
( ) ii nxx ⋅− ( ) ii nxx .2−
a = 17,5 h = 5
ii nh
ax⋅
− 2
0 1 2 3 4 5 6 7 0-5
5-10 10-15 15-20 20-25 25-30 30-35 35-40
10 40 60 80 50 30 20 10
3,33 13,33 20,00 26,67 16,67 10,00
6,67 3,33
2,5 7,5
12,5 17,5 22,5 27,5 32,5 37,5
-15,7 -10,7 -5,7 -0,7 4,3 9,3
14,3 19,3
-157 -428 -342
-56 215 279 286 193
2464,9 4579,6 1949,4
39,2 924,5
2594,7 4089,8 3724,9
90 160
60 0
60 120 180 160
Total
300
∑=
k
iin
1
100
( )∑=
k
iin
1
*%
-
-
-10 ( )∑ ⋅− ii nxx
1) 20367,0
( )∑ − ii nxx .2
820
∑=
⋅
−k
ii
i nh
ax
1
2
1) 'HRDUHFH� PHGLD� QX� HVWH� XQ� FkW� H[DFW� úL� D� IRVW� URWXQMLW �� QX� YHULILF � SURSULHWDWHD��( )∑ ⋅− ii nxx = 0 .�'LIHUHQ D�ILLQG�IRDUWH�PLF ��-10), rotunjirea medie�QX�YD�LQIOXHQ D�GHFkW�
într-R�P VXU �IRDUWH�PLF �P ULPHD�FHORUODO L�LQGLFDWRUL�
5. Calculul indicatorilor de asimetrie ÌQ� DFHVW� VFRS�YRP�FDOFXOD� FRHILFLHQ LL� GH� DVLPHWULH� SURSXúL�GH�3HDUVRQ� (Cas� úL�&¶as�� úL�Yule:
146,02,8
172,18)( =−=−=x
as
MoxC
σ
256,02,8
)5,172,18(3)(3 =−=−=′x
as
MexC
σ
12
12
qqAS
+−
= , unde q1=Me - Q1; q2 =Q3 – Me
q1 = 17,5 – 12,125 = 5,375 q2 = 33,575 – 17,5 = 6,075
061,0375,5075,6
375,5075,6 =+−=AS
&HL�WUHL�LQGLFDWRUL�GH�DVLPHWULH�FDOFXOD L�LQGLF �R�DVLPHWULH�PRGHUDW �SR]LWLY �
6. 0HGLD�úL�GLVSHUVLD�FDUDFWHULVWicii alternative • Media (p)
N
Mp = ,
unde: M� �QXP UXO�PXQFLWRULORU�FX�R�YHFKLPH�!� x = 18,2 ani; M =110; N �QXP UXO�WRWDO�DO�PXQFLWRULORU��1� �����
3667,0300
110 ===N
Mp
• Dispersia ( 2pσ )
2322,0)3667,01(3667,0)1(2 =−=−= pppσ
3. Într-R�VRFLHWDWH�FRPHUFLDO ��FDUH�DUH�����GH�PXQFLWRUL�V-D�RUJDQL]DW�R�REVHUYDUH�VWDWLVWLF �úL�
s-DX�RE LQXW�XUP WRDUHOH�GDWH�
5HSDUWL LD�PXQFLWRULORU�vQ�IXQF LH�GH�SURGXF LD�RE LQXW ��EXF L�
Tabelul 3.11.
Grupe de muncitori GXS �SURGXF LD�RE LQXW ��EXF��
1XP U�PXQFLWRUL
Sub 120 10 120-140 18 140-160 23 160-180 38 180-200 51 200-220 40 220-240 15 ����úL�SHVWH 5 Total 200
5HSDUWL LD�PXQFLWRULORU�vQ�IXQF LH�GH�YHFKLPH��DQL�
Tabelul 3.12.
Grupe de muncitori GXS �YHFKLPH (ani)
Structura muncitorilor vQ�IXQF LH�GH�YHFKLPH����
Sub 2 4,2 2-4 8,5 4-8 12,9 8-12 30,2 12-18 26,2 18-24 10,2 ���úL�SHVWH 7,8 Total 100,0
Se cere: 1. V �VH�FDUDFWHUL]H]H�VWDWLVWLF�FHOH�GRX �VHULL�IRORVLQG�PHWRGD�JUDILF ��LQGLFDWRULL�WHQGLQ HL
FHQWUDOH��PHGLLOH�GH�VWUXFWXU ��LQGLFDWRULL�GH�YDULD LH úL asimetrie; 2. V �VH�SUHFL]H]H�GXS �FDUH�GLQ�YDULDELOH�HVWH�PDL�RPRJHQ �FROHFWLYLWDWHD�GH�PXQFLWRUL�DL�
VRFLHW LL�FRPHUFLDOH�UHVSHFWLYH� Rezolvare 1. 6HULD�SULYLQG� UHSDUWL LD�PXQFLWRULORU� vQ� IXQF LH�GH�SURGXF LD� LQGLYLGXDO �HVWH�SUH]HQWDW �
FX�IUHFYHQ H�DEVROXWH�úL�HVWH�JUXSDW �SH���LQWHUYDOH�GH�YDULD LH�HJDOH� 3HQWUX�FDUDFWHUL]DUHD� VWDWLVWLF �D�DFHVWHL� VHULL� GH� IUHFYHQ H� VH� YD� IDFH� UHSUH]HQWDUHD� JUDILF �
SULQ�KLVWRJUDP ��SROLJRQXO�úL�FXUED�FXPXODWLY �D�IUHFYHQ HORU��YH]L�ILJ�������úL�������IRORVLQG�FRO���úL�7 din tabelul 3.13. Graficul de concentrare se face analog cu cel din figura 1.4. (vezi capitolul 1).
)LJXUD������'LVWULEX LD�PXQFLWRULORU�vQ�IXQF LH�GH�SURGXF LD�]LOQLF ��EXF L�
0
10
20
30
40
50
60
80 100 120 140 160 180 200 220 240 260 280���� ���� � ��� ������� �
� ���� ���� � ��
10
1823
38
51
40
15
5
0
25
50
75
100
125
150
175
200
225
100 120 140 160 180 200 220 240 260SURGXF LD��EXF��
nr. m
unci
tori
)LJXUD������'LVWULEX LD�PXQFLWRULORU�vQ�IXQF LH�GH�SURGXF LD�]LOQLF ��EXF L� )LLQG�LQWHUYDOH�HJDOH��SHQWUX�PHGLH�úL�GLVSHUVLH�VH�YRU�XWLOL]D�IRUPXOHOH�GH�FDOFXO�VLPSOLILFDW�
ah
n
nh
ax
xk
ii
k
ii
i
+⋅
−
=
∑
∑
=
=
1
1, respectiv 22
1
1
2
2 )( axhn
nh
ax
k
ii
k
ii
i
x −−⋅⋅
−
=∑
∑
=
=σ
Pentru calculul indicatorilor se va folosi tabelul 3.13.
&DOFXOXO�LQGLFDWRULORU�FDUH�FDUDFWHUL]HD] �UHSDUWL LD�FHORU�����GH�PXQFLWRUL� vQ�IXQF LH�GH�P ULPHD�SURGXF LHL�]LOQLFH�RE LQXWH
Tabelul 3.13.
)UHFYHQ H�FXPXODWH
Grupe de muncitori
GXS �
SURGXF LD RE LQXW �
(buc.)
1XP U�
muncitori (ni)
Centre de
interval (xi)
h
axi −
a=190 h=20
ii nh
xx⋅
−
ii nh
ax⋅
− 2
&UHVF WRU Descres-
F WRU
0 1 2 3 4 5 6 7 sub 120 10 110 -4 -40 160 10 200 120-140 18 130 -3 -54 162 28 190 140-160 23 150 -2 -46 92 51 172 160-180 38 170 -1 -38 38 89 149 180-200 51 190 0 0 0 140 111 200-220 40 210 1 40 40 180 60 220-240 15 230 2 30 60 195 20 ����úL�SHVWH 5 250 3 15 45 200 5 Total 200 -93 597 - -
7,18019020200
93 =+⋅−=x buc./muncitor
51,1107)1907,180(20200
597 222 =−−⋅=xσ
28,3351,1107 ==xσ buc./muncitor
%42,181007,180
28,33100 =⋅=⋅=
xv xσ
3H� ED]D� FRHILFLHQWXOXL� GH� YDULD LH� VH� SRDWH� WUDJH� FRQFOX]LD� F � PHGLD� HVWH� UHSUH]HQWDWLY �
pentru cele mai multe valori individuale ale serieL�GHRDUHFH�DP�RE LQXW�XQ�FRHILFLHQW�PLF�GH�YDULD LH�����������FHHD�FH�vQVHDPQ �vQ�DFHODúL�WLPS�F �GDWHOH�VXQW�GHVWXO�GH�RPRJHQH�vQWUH�HOH�
'LQ�DFHODúL�WDEHO�VH�SRW�FDOFXOD�úL�FHLODO L�GRL�LQGLFDWRUL�DL�WHQGLQ HL�FHQWUDOH��YDORDUH�PRGDO �úL�YDORDUH�PHGLDQ �
,QWHUYDOXO�FX�IUHFYHQ �PD[LP ��FXSULQGH�úL�PRGXO�
180 < Mo < 200, de unde:
83,190)4051()3851(
)3851(20180
21
10 =
−+−−+=
∆+∆∆
⋅+= hxMo EXF L
Locul medianei în serie:
5,1002
1200
2
1)( 1 =+=
+=
∑=
k
iin
MeU
ÌQ� LQWHUYDOXO�DO�FLQFLOHD�VXPD�IUHFYHQ HORU�FXPXODWH�HVWH�PDL�PDUH�GHFkW� ORFXO�PHGLDQHL� vQ�
seULH��DGLF � 180 < Me < 200
5,18451
895,10020180
2
11
1
1
0 =−+=−
+
+=∑
∑ −
=
=
m
m
ii
k
ii
n
nn
hxMe EXF L
Deci, x , Me�úL�Mo�VH�J VHVF�vQ�DFHODúL�LQWHUYDO��VHULD�HVWH�XúRU�DVLPHWULF �úL�SXWHP�YHULILFD�
UHOD LD� 3( x -Me) = x -Mo 3(180,700-184,5) =-11,4 x -Mo = 180,7-190,83 =-10,13
ÌQVHDPQ �F �SHQWUX�VHULD�QRDVWU ��FX�R�GLIHUHQ �QHVHPQLILFDWLY ��VH�YHULILF �UHOD LD�GLQWUH�FHL�WUHL�LQGLFDWRUL�DL�WHQGLQ HL�FHQWUDOH�H[LVWHQW �vQ�FD]XO�UHSDUWL LLORU�PRGHUDW�DVLPHWULFH�
Pentru stabilirea gradului de asimetrie vom avea:
30,028,33
13,10
28,33
83,1907,180)(−==
−=
−=
xas
MoxC
σ
34,028,33
4,11
28,33
)5,1847,180(3)(3 −==−=−=′x
as
MexC
σ
12
12
qqAS
+−
= , unde q1=Me - Q1; q2 =Q3 – Me
25,504
201
4
1)( 1
1 ==+
=∑
=
k
iin
QU
140 < Q1 < 160
34,15923
2825,5020140
14
1 1
1101 =−+=
−
+
+=∑∑
−
==
u
u
ii
k
ii
n
nn
hxQ EXF L
75,1502014
31
4
3)(
13 =⋅=
+= ∑
=
k
iinQU
200 < Q3 < 220
375,20540
14075,15020200
14
3 1
1103 =−+=
−
+
+=∑∑
−
==
v
v
ii
k
ii
n
nn
hxQ EXF L
q1 = 184,5-159,34 = 25,16 q2 = 205,375-184,5 = 20,875
093,016,25875,20
16,25875,20 −=+−=AS
În toate cazurile s-DX� RE LQXW� YDORUL� QHJDWLYH� UHODWLY�PLFL� SHQWUX� FRHILFLHQ LL� GH� DVLPHWULH��
FHHD�FH�vQVHDPQ �F �SUHGRPLQ �YDORULOH�PDL�PDUL�úL�FD�DWDUH�LQGLFDWRULL�WHQGLQ HL�FHQWUDOH�VXQW�PDL�UHSUH]HQWDWLYL� SHQWUX� WHUPHQLL� FDUH� SUH]LQW � DEDWHUL� QHJDWLYH� vQ� FRPSDUD LH� FX� FHL� FDUH� SUH]LQW �abateri pozitive ( x < Me < Mo).
2. 6HULD� SULYLQG� UHSDUWL LD� PXQFLWRULORU� GXS � YHFKLPH� HVWH� SH� LQWHUYDOH� QHHJDOH� úL� FX�
IUHFYHQ H�UHODWLYH� 3HQWUX�UHSUH]HQWDUHD�JUDILF �HVWH�QHFHVDU�V �VH�IDF �R�VHULH�GH�FDOFXOH�SUHJ WLWRDUH�
&DOFXOXO�IUHFYHQ HORU�UHGXVH�úL�D�IUHFYHQ HORU�FXPXODWH�pentru seria referitoare la vechime
Tabelul 3.14. )UHFYHQ H�
relative cumulate
*UXSH�GXS �
vechime (ani)
Structura muncitorilor
(n*i(%))
0 ULPHD�
intervalului (hi)
5DSRUWXO�ID �GH�
primul interval
1h
hK i
i =
)UHFYHQ H�
reduse
i
i
K
n *(%)
FUHVF -tor
descres-F WRU
A 1 2 3 4 5 6 0-2 4,2 2 1,0 4,2 4,2 100,0
2-4 8,5 2 1,0 8,5 12,7 95,8
4-8 12,9 4 2,0 6,5 25,6 87,3
8-12 30,2 4 2,0 15,1 55,8 74,4
12-18 26,2 6 3,0 8,7 82,0 44,2
18-24 10,2 6 3,0 3,4 92,2 18,0
24-30 7,8 6 3,0 2,6 100 7,8
Total 100 - 15 - - -
Folosind datele din tabelul 3.14. se constituie: histograma, polLJRQXO� IUHFYHQ HORU� úL� FXUED�
FXPXODWLY ��YH]L�ILJ�������úL�������
)LJXUD������5HSDUWL LD�PXQFLWRULORU�vQ�IXQF LH�GH�YHFKLPH
0
20
40
60
80
100
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
vechime (ani)
stru
ctur
a m
unci
tori
lor
(%)
)vJXUD������5HSDUWL LD�PXQFLWRULORU�vQ�IXQF LH�GH�YHFKLPH
4,2
8,5
12,9
30,2
26,2
10,27,8
0
2
4
6
8
10
12
14
16
0 4 8 12 16 20 24 28 32
vechime (ani)
stru
ctur
a m
unci
tori
lor
(%)
ùL� vQ� DFHVW� FD]� WUHEXLH� FDOFXOD L� LQGLFDWRULL� FDUH� FDUDFWHUL]HD] � VWDWLVWLF� DFHDVW � VHULH�� )LLQG�LQWHUYDOH� QHHJDOH� WUHEXLH� FDOFXODWH� FHQWUHOH� GH� LQWHUYDO� úL� DSRL� VH� YD� DSOLFD� R� IRUPXO � GH� FDOFXO�VLPSOLILFDW�� 'H� UHJXO �� VH� RSWHD] � QXPDL� SHQWUX� IRORVLUHD� XQHL� VLQJXUH� SURSULHW L� bazate pe PLFúRUDUHD�WHUPHQLORU�FX�R�FRQVWDQW ��a" (vezi tabelul 3.15.).
&DOFXOXO�LQGLFDWRULORU�WHQGLQ HL�FHQWUDOH��D�PHGLLORU�GH�SR]L LH�úL�D�LQGLFDWRULORU�GH�YDULD LH�
într-o serie cu intervale neegale
Tabelul 3.15.
Intervale GH�YDULD LH
)UHFYHQ �
reODWLY
( *(%)in )
Centrul intervalului
( ix )
(xi - a) a = 10
*(%))( ii nax ⋅− *
(%)2)( ii nax ⋅− )UHFYHQ H�UHODWLYH�
cumulative
A 1 2 3 4 5 6 0 - 2
4,2
1
-9
-37,8
340,2
4,2 2 - 4 8,5 3 -7 -59,5 416,5 12,7
4 - 8 12,9 6 -4 -51,6 206,4 25,6 8 - 12 30,2 10 0 0 0 5,5,8 12 - 18 26,2 15 5 31 655 82,0 18 - 24 10,2 21 11 112,2 1234,3 92,2 24 - 30
7,8
27
17
132,6
2554,2
100,0 Total 100 - - 226,9 5113,5 -
( )
3,1210100
9,226
100
*(%) ≅+=
⋅−= ∑ ii nax
x ani/muncitor
( )( ) 84,45)103,12(
100
5,5113
10022
*(%)
2
2 =−−⋅=−−−
=∑
axnax ii
xσ
77,684,45 ==xσ ani/muncitor
%551003,12
77,6100 =⋅=⋅=
xv xσ
8 < Mo < 12
25,11)2,262,30()9,122,30(
)9,122,30(48
21
10 =
−+−−+=
∆+∆∆
⋅+= hxMo ani
Locul medianei în serie:
5,502
1100
2
1)( 1 =+=
+=
∑=
k
iin
MeU
8 <Me < 12
( )298,11
2,30
6,265,5048
2
1
*%
1
1
*1
*
0
(%)
(%)
=−
+=−
+
+=∑
∑ −
=
=
m
i
i
n
n
n
hxMe
m
i
k
i
ani
25,254
101)( 1 ==QU
4 < Q1 <8
89,79,12
7,1225,25441 =−+=Q ani
298,112 == eMQ ani
75,751014
3)( 3 =⋅=QU
12 < Q3 < 18
568,162,26
8,5575,756123 =−+=Q ani
155,077,6
25,113,12)( =−=−=x
as
MoxC
σ
44,077,6
)298,113,12(3)(3 =−=−=′x
as
MexC
σ
q1 = Me-Q1=11,298 – 7,89 = 3,488 q2 = Q3 – Me = 16,568 – 11,298 = 5,27
203,0488,32,25
488,32,5
12
12 =+
−=+−
qqAS
$QDOL]kQG� UH]XOWDWHOH� VH� FRQVWDW � F � VHULD� DUH� FDUDFWHU� QHRPRJHQ� GHRDUHFH� FRHILFLHQWXO� GH�
YDULD LH� WRWDO �HVWH�GH������5HIHULQGX-QH� OD�YHFKLPH�� DFHDVW � VLWXD LH�HUD�GH�DúWHSWDW�� VHULD� DYkQG�LQWHUYDOH�QHHJDOH�GH�YDULD LH�úL�IUHFYHQ HOH�GLVWULEXLQGX-se neuniform.
&DOFXOkQG�FRHILFLHQ LL�GH�DVLPHWULH�VH�FRQVWDW �R�DVLPHWULH spre valorile mici (Mo < Me < x ). ÌQ� ILQDO�� VH� SRDWH� VSXQH� F � VHULD� SULYLQG� GLVWULEX LD� SURGXF LHL� �vQ� EXF L�� HVWH� PXOW� PDL�
RPRJHQ ��FDUDFWHUL]DW �GH� LQGLFDWRUL�FX�YDORUL�PLFL�SHQWUX�YDULD LH�úL�DVLPHWULH�úL�HVWH�QHRPRJHQ �SHQWUX�VHULD�SULYLQG�YHFKLPHD�PHGLH��FHHD�FH�QH�SHUPLWH�V �WUDJHP�FRQFOX]LD�F �vQ�PRG�FRUHFW� vQ�vQWUHSULQGHUHD� UHVSHFWLY �SURGXF LLOH� LQGLYLGXDOH�RE LQXWH�QX�GHSLQG� vQ�SULQFLSDO�GH�YHFKLPH�FL�GH�DO L�IDFWRUL�HVHQ LDOL�VSHFLILFL�
4. Într-R�VRFLHWDWH�FRPHUFLDO ��V-DX�vQUHJLVWUDW�FX�SULYLUH�OD�XQ�SURGXV�GH�ED] �GHVWLQDW�YkQ] ULL�
SULQ�XQLW LOH�SURSULL��GDWHOH�QHFHVDUH�GHWHUPLQ ULL�QLYHOXOXL�PHGLX�DO�SURGXFWLYLW LL�PXQFLL�(xi)�úL�DO�consumului specific GH� WLPS�GH�PXQF � �yi���'DWHOH� VH�SUH]LQW � vQWU-R�VHULH�ELGLPHQVLRQDO � vQ� FDUH�FKHOWXLHOLOH�WRWDOH�GH�WLPS�GH�PXQF ��zi��UHSUH]LQW �IUHFYHQ HOH�FRPXQH�SHQWUX�FHOH�GRX �YDULDELOH�
Tabelul 3.16.
Produs Nivelul SURGXFWLYLW LL�
muncii orare �PLL�OHL�RU �
(xi)
Consumul specific de timp GH�PXQF ��RUH-om la 1000 lei
SURGXF LH� (yi)
Cheltuielile totale de timp
GH�PXQF (mii ore-om)
(zi)
A 1 2 3
A 10,0 0,100 11624 B 6,25 0,160 6444 C 7,47 0,134 100 Total 8,65 0,116 18168
Se cere:
1. calculul nivelului mediu al�SURGXFWLYLW LL�PXQFLL�( w = x ); 2. FDOFXOXO�FRQVXPXOXL�PHGLX�GH�WLPS�GH�PXQF �( t = y ).
1RW ��V �VH�VSHFLILFH�FH�IHO�GH�PHGLL�V-DX�XWLOL]DW�úL�GH�FH�" Rezolvare 'DF � FHOH� GRX � FDUDFWHULVWLFL� VXQW�FRQVLGHUDWH� LQWHUGHSHQGHQWH� vQWUH� HOH�� VH� SXQH�SUREOHPD�
DOHJHULL�FHOXL�PDL�SRWULYLW�WLS�GH�PHGLH�SHQWUX�D�S VWUD�DFHHDúL�UHOD LH�GH�LQYHUV �SURSRU LRQDOLWDWH�úL�vQWUH�P ULPLOH�PHGLL�FDOFXODWH�
ÌQWUH� FHOH� GRX � YDULDELOH� DSDU� UHOD LLOH��i
i yx
1= �� SHQWUX� FDUH� VH� YD� RE LQH� R� UHSDUWL LH�
ELGLPHQVLRQDO �FX�IUHFYHQ H�FRPXQH� ( )ii yxn .
3ULPD� VHULH� GH� GLVWULEX LH� D� QLYHOXULORU� SURGXFWLYLW LL� PXQFLL� HVWH� IRUPDW � GLQ� GDWHOH�FRORDQHORU���úL����LDU�FHD�GH-a doua sHULH�GH�GLVWULEX LH�-�D�FRQVXPXULORU�VSHFLILFH�GH�WLPS�GH�PXQF �-� HVWH� IRUPDW �GLQ�GDWHOH� FRORDQHORU� �� úL� ���$VWIHO� SUH]HQWDWH�� FHOH�GRX � VHULL� DX� DFHOHDúL�SRQGHUL�SHQWUX�DPEHOH�FDUDFWHULVWLFL��DGLF �QXP UXO�WRWDO�GH�RP-ore lucrate.
3HQWUX�D�S VWUD�UHOD LD�GH�LQYHUV �SURSRU LRQDOLWDWH�GLQWUH�FHOH�GRX �YDULDELOH�úL�vQWUH�P ULPLOH�PHGLL� SHQWUX� FDOFXOXO� QLYHOXOXL� PHGLX� DO� SURGXFWLYLW LL� PXQFLL� YRP� IRORVL� PHGLD� DULWPHWLF �SRQGHUDW � LDU� SHQWUX� FDOFXOXO� FRQVXPXOXL�PHGLX� GH� WLPS� GH�PXQF � VH� YD� DSOLFD�PHGLD� DUPRQLF �SRQGHUDW �
1. &DOFXOXO�QLYHOXOXL�PHGLX�DO�SURGXFWLYLW LL�PXQFLL
65,818168
10047,7644425,61162410
1
1 =⋅+⋅+⋅===∑
∑
=
=k
iyx
k
iyxi
ii
ii
n
nxxw mii lei/ore-om
2. &DOFXOXO�FRQVXPXOXL�PHGLX�GH�WLPS�GH�PXQF
116,0100
134,0
16444
16,0
111624
10,0
118168
1
1
1 =⋅+⋅+⋅
===∑
∑
=
=k
iyx
i
k
iyx
ii
ii
ny
nyt ore-om/1000 lei
9HULILFDUHD�UHOD LHL�GLQWUH�FHOH�GRX �PHGLL��vQ�PLL�OHL�
;1
yx = ;62,8
116,0
1 =
'LIHUHQ D� SRDWH� SURYHQL� GLQ� URWXQMLUHD� FDOFXOXOXL�� 'HFL�� PHGLD� DUPRQLF � SRDWH� VXEVWLWXL�PHGLD� DULWPHWLF � FkQG� VH� FXQRVF� SURGXVHOH� GH� IUHFYHQ � úL� QX� IUHFYHQ HOH� UHDOH�� 8Q� DVWIHO� GH�exemplu poate fi cel al indiceluL�DUPRQLF�DO�SUH XULORU��YH]L�FDSLWROXO��,QGLFL���
5. 6H�FXQRVF�XUP WRDUHOH�GDWH�FX�SULYLUH�OD�GLVWULEX LD�YkQ] WRULORU�GLQWU-un complex comercial
vQ�IXQF LH�GH�YHFKLPHD�vQ�PXQF �úL�YDORDUHD�YkQ] ULORU�UHDOL]DWH�vQWU-R�V SW PkQ �
Tabelul 3.17.
6XEJUXSH�GH�YkQ] WRUL�GXS �YROXPXO�YkQ] ULORU��PLO��OHL� Grupe GH�YkQ] WRUL�
GXS �YHFKLPH��DQL) sub 190 190-200 200-210 210-220 220
úL�SHVWH
Total
A 1 2 3 4 5 6 sub 10 10-20 ���úL�SHVWH
5 - -
15 12
-
5 35 7
- 8
15
- - 8
25 55 30
Total 5 27 47 23 8 110
Se cere: 1. SROLJRQXO�IUHFYHQ HORU�SULYLQG�UHSDUWL LD�YkQ] WRULORU�GXS �YROXPXO�YkQ] ULORU�SH�WRWDO�úL�
pe grupe de vechime; 2. UHSUH]HQWDUHD�JUDILF �D�QXP UXOXL�YkQ] WRULORU�GXS �YHFKLPH�SH�FHOH trei grupe, folosind
JUDILFXO�SULQ�S WUDW� 3. calculul mediiloU�SH�JUXSH�GH�YHFKLPH�úL�SH�WRWDO� 4. LQGLFDWRULL�VLQWHWLFL�DL�YDULD LHL�SH�ILHFDUH�JUXS �úL�SH�WRWDO� 5. LQWHUSUHWDUHD�JUDGXOXL�GH�RPRJHQLWDWH�SH�JUXSH�úL�SH�WRWDO�� 6. verificarea regulii de adunare a dispersiilor; 7. ce indicatori statistici se pot calcula pe baza UHJXOLL�GH�DGXQDUH�D�GLVSHUVLLORU�úL�FXP�VH�
LQWHUSUHWHD] �VWDWLVWLF�DFHúWL�LQGLFDWRUL� 8. FDOFXOXO� úL� LQWHUSUHWDUHD� GLVSHUVLLORU� SHQWUX� FDUDFWHULVWLFD��YkQ] WRUL� FDUH� VH� DIO � SHVWH�
PHGLD�YkQ] ULORU�SH�WRWDO��
Rezolvare
1. 5HSUH]HQWDUHD�JUDILF �D�GLVWULEX LHL�YkQ] WRULORU��ILJXULOH������úL�������
0
5
10
15
20
25
30
35
40
45
50
170 180 190 200 210 220 230 240
)LJXUD������5HSDUWL LD�YkQ] WRULORU�vQWU-XQ�FRPSOH[�FRPHUFLDO�GXS �YROXPXO�GHVIDFHULORU
0
5
10
15
20
25
30
35
40
45
50
170 180 190 200 210 220 230 240
Total
gr. I
gr. II
gr. III
)LJXUD�������5HSDUWL LD�YkQ] WRULORU�GXS �YROXPXO�GHVIDFHULL�SH�WRWDO�úL�SH�JUXSH�GH�YHFKLPH
2. 5HSUH]HQWDUHD�JUDILF �D�QXP UXOXL�YkQ] WRULORU�GXS �YHFKLPH (vezi figura 3.11.).
)LJXUD�������1XP UXO�GH�YkQ] WRUL�SH�JUXSH�GH�YHFKLPH
55
25
30
sub 10 ani 10-20 ani ���DQL�úL�SHVWH
Scara: 1 cm = 2 persoane
1X
PU�
Yk
Q]
WRUL
9ROXPXO�YkQ] ULORU��PLO��OHL�
1X
PU�
YkQ
]WR
UL
9ROXPXO�YkQ] ULORU��PLO��OHL�
3. Calculul mediilor • PHGLLOH�GH�JUXS :
∑
∑
=
==m
jij
m
jijj
i
n
ny
y
1
1
19525
52051519551851 =⋅+⋅+⋅=y �PLO��OHL�YkQ] WRU
27,20455
821535205121952 =⋅+⋅+⋅=y �PLO��OHL�YkQ] WRU
33,21530
82251521572053 =⋅+⋅+⋅=y �PLO��OHL�YkQ] WRU
• PHGLD�JHQHUDO ( )y :
- independent:
∑
∑
=⋅
=⋅
=m
jj
m
jjj
n
ny
y
1
1
18,205110
82252321547205271955185 =⋅+⋅+⋅+⋅+⋅=y PLO��OHL�YkQ] WRU
- pe baza mediilor de JUXS �
∑
∑
=⋅
=⋅
=r
ii
r
iii
n
nyy
1
1
18,205110
3033,2155527,20425195 =⋅+⋅+⋅=y PLO��OHL�YkQ] WRU.
4. &DOFXOXO�LQGLFDWRULORU�VLQWHWLFL�DL�YDULD LHL� • GLVSHUVLLOH�GH�JUXS � ( )2
iσ :
( )
∑
∑
=
=−
=m
jij
m
jijij
i
n
nyy
1
1
2
2σ
4025
5 ) 195- (205 15) 195- (195 5195)- (185 22221 =⋅+⋅+⋅=σ
83,3555
8 ) 204,27 - (215 35) 204,27 - (205 12) 204,27 - (195 22222 =⋅+⋅+⋅=σ
89,4930
8 ) 215,33 - (225 15 ) 215,33 - (215 7 215,33 - (205 22 )223 =
⋅+⋅+⋅=σ
• PHGLD�GLVSHUVLLORU�SDU LDOH� ( )2σ :
63,40110
3089,495583,352540
1
1
2
2 =⋅+⋅+⋅==∑
∑
=⋅
=⋅
r
ii
r
iii
n
nσσ
• dispersia dintre grupe ( )2
/2
xyσδ = :
( )
06,52110
30)18,20533,215(55)18,20527,204(25)18,205195( 222
1
1
2
2
=⋅−+⋅−+⋅−
=−
=
∑
∑
=⋅
=⋅
r
ii
r
iii
n
nyy
δ
• GLVSHUVLD�WRWDO ( )2σ :
( )( ) ( ) ( )
( ) ( )69,92
110
818,2052252318,205215
110
4718,2052052718,205195518,205185
22
222
1
1
2
2
=⋅−+⋅−
+
+⋅−+⋅−+⋅−
=
−
=
∑
∑
=⋅
=⋅
m
jj
m
jjj
n
nyy
σ
5. Aprecierea gradului de omRJHQLWDWH�SH�JUXSH�úL�SH�WRWDO &DOFXOXO�FRHILFLHQ LORU�GH�YDULD LH�
• pe grupe
32,640211 === σσ PLO��OHL�YkQ] WRU
%24,3100195
32,6100
1
11 =⋅=⋅=
yv
σ
98,583,35222 === σσ PLO��OHL�YkQ] WRU
%93,210027,204
98,5100
2
22 =⋅=⋅=
yv
σ
06,789,49233 === σσ PLO��OHL�YkQ] WRU
%18,310033,215
06,7100
3
33 =⋅=⋅=
yv
σ
• pe total
62,969,922 === σσ PLO��OHL�YkQ] WRU
%69,410018,205
62,9100 =⋅=⋅=
yv
σ
&RPSDUkQG�UH]XOWDWHOH�RE LQXWH�VH�FRQVWDW �F �� • ILHFDUH�JUXS �OXDW �VHSDUDW�HVWH�PDL�RPRJHQ �GHFkW�FROHFWLYLWDWHD�JHQHUDO �GLQ�FDUH�D�IRVW�
H[WUDV � • grupD�D�GRXD�HVWH�PDL�RPRJHQ �GHFkW�FHOHODOWH�GRX � • YDORULOH�PLFL�DOH�FRHILFLHQ LORU�GH�YDULD LH�FDOFXOD L�SH�ILHFDUH�JUXS �úL�SH�WRWDO�DWHVW �XQ�
JUDG� GH� RPRJHQLWDWH� ULGLFDW� DO� JUXSHORU� úL� FROHFWLYLW LL� WRWDOH� úL� GHFL� XQ� JUDG� GH�UHSUH]HQWDWLYLWDWH�FRUHVSXQ] WRU�SHQWUX�PHGLLOH�FDUH�OH�FDUDFWHUL]HD] �
6. Verificarea regulii de adunare a dispersiilor
222 σδσ += 92,69 = 52,06 + 40,63
7. 3H�ED]D�UHJXOLL�GH�DGXQDUH�D�GLVSHUVLLORU�VH�SRW�FDOFXOD�DO L�GRL�LQGLFDWRUL�VWDWLVWLFL� • *UDGXO�GH�GHWHUPLQD LH ( 2
/ xyR )
%16,5610069,92
06,52100
2
22
/ =⋅=⋅=σδ
xyR
• *UDGXO�GH�QHGHWHUPLQD LH ( 2
/ xyK )
%84,4310069,92
63,40100
2
22
/ =⋅=⋅=σσ
xyK
6H� SRDWH� DILUPD� F � ������� GLQ� YDULD LD� WRWDO � D� YROXPXOXL� YkQ] ULORU� HVWH� H[SOLFDW � GH�
YDULD LD�SURGXV �GH�IDFWRUXO�GH�JUXSDUH��YHFKLPHD�– factor dominant deoarece 2xy
R > 50%), restul
GH��������ILLQG�LQIOXHQ D�UHODWLY �D�FHORUODO L�IDFWRUL�QHvQUHJLVWUD L�
8. &DOFXOXO� úL� LQWHUSUHWDUHD� GLVSHUVLLORU� SHQWUX� �YkQ] WRULL� FDUH� VH� DIO � SHVWH� YROXPXO�
PHGLX�DO�YkQ] ULORU�SH�WRWDO" *�Calculul mediilor
• PHGLLOH�SH�JUXS : i
ii n
mw = ,
de unde:
025
01 ==w
1455,055
82 ==w
7667,030
233 ==w
• media pe total: 2818,0110
31 ===n
mw
*�Calculul dispersiilor
• GLVSHUVLLOH�GH�JUXS :
( )ii
wwiw −⋅= 12σ ,
de unde:
( ) ( ) 00101 112
1=−⋅=−⋅= ww
wσ
( ) ( ) 1243,01455,011455,01 222
2=−=−⋅= ww
wσ
( ) ( ) 1789,07667,017667,01 332
3=−=−⋅= ww
wσ
• PHGLD�GLVSHUVLLORU�SDU LDOH ( 2wσ ):
1109,0110
301789,0551243,0250
1
1
2
2 =⋅+⋅+⋅=⋅
⋅=
∑
∑
=
=r
ii
r
iiw
w
n
ni
σσ
• dispersia dintre grupe ( 2
wδ ):
( ) ( ) ( )
( )0914,0
110
302818,07667,0
110
552818,01455,0252818,00
2
22
1
1
2
2
=⋅−+
+⋅−+⋅−=⋅−
=∑
∑
=
=r
ii
r
iii
w
n
nwwδ
• dispersia total ( 2
wσ ):
( ) ( ) 2023,02818,012818,012 =−⋅=−= wwwσ
5HJXOD�DGXQ ULL�GLVSHUVLLORU�VH�S VWUHD] �úL�vQ�FD]XO�FDUDFWHULVWLFLL�DOWHUQDWLYH� 222www δσσ +=
0,2023 = 0,1109 + 0,0914
6. 3HQWUX� XQ� DJHQW� HFRQRPLF� FH� GHVI úRDU � WUHL� DFWLYLW L� V-au înregistrat datele privind
veniturile salariale ale personalului pentru luna mai 2001 (vezi tabelul 3.18.).
5HSDUWL LD�VDODULD LORU�SH�DFWLYLW L��vQ�IXQF LH�GH�P ULPHD�YHQLWXULORU�VDODULDOH�vQ�OXQD�PDL�����
Tabelul 3.18. �vQ���ID �GH�WRWDOXO�JUXSHL�
Grupe duS �VDODULL�
(zeci mii lei)
Activitatea I
Activitatea II
Activitatea III
A 1 2 3 sub 250
10
-
- 250 - 270 12 - -
270 - 290 25 20 - 290 - 310 28 35 - 310 - 330 15 20 20 330 - 350 10 15 47 350 - 370 - 10 23 ����úL�Seste
-
-
10 Total 100 100 100
1XP UXO� GH� VDODULD L�D� vQUHJLVWUDW� XUP WRDUHD� VWUXFWXU �� DFWLYLWDWHD� O� - 20%; activitatea II -
-�����úL�DFWLYLWDWHD�,,,�- 30%. Se cere: 1. V �VH�DSOLFH�UHJXOD�GH�DGXQDUH�D�GLVSHUVLLORU�úL�V �VH�VSHFLILFH�GDF �IDFWRUXO�Ge grupare
este semnificativ sau nu; 2. V �VH�DUDWH�JUDGXO�GH�RPRJHQLWDWH�SH�FHOH� WUHL�DFWLYLW L� úL�SH� WRWDO� úL� V � VH� LQWHUSUHWH]H�
UH]XOWDWHOH�RE LQXWH�SHQWUX�FDUDFWHUL]DUHD�YDULD LHL� 3. V �VH�FDOFXOH]H�QXP UXO�GH�VDODULD L��IUHFYHQ HOH�DEVROXWH��úWLLQG�F �SH�WRWDO�QXP UXO�ORU�D�
IRVW�GH�����úL�V �VH�YHULILFH�SH�ED]D�IUHFYHQ HORU�DEVROXWH�GLVSHUVLD�WRWDO � Rezolvare 1. )LLQG� LQWHUYDOH� HJDOH�� PHGLLOH� úL� GLVSHUVLLOH� VH� SRW� GHWHUPLQD� FX� IRUPXOHOH� GH� FDOFXO�
VLPSOLILFDW��FX�IUHFYHQ H�UHODWLYH�
ah
nh
ay
y
m
jij
j
i +⋅
−
=∑
=
1001
*(%)
( )221
*(%)
2
2
100ayh
nh
aym
jij
j
i −−⋅
−
=∑
=σ
1.1. Pentru activitatea I� VH� YD�H[WUDJH� VHULD� úL� VH� YRU�FDOFXOD�� PHGLD��GLVSHUVLD��abaterea PHGLH�S WUDWLF �(vezi tabelul 3.19.).
&DOFXOXO�PHGLHL�úL�GLVSHUVLHL�VDODULLORU�SHQWUX�DFWLYLWDWHD�,
Tabelul 3.19.
Grupe GXS �
salarii (zeci mii lei)
Structura VDODULD LORU
( )*(%)1 jn
Centrul de interval ( )jy
h
ay j −
a = 300 h = 20
*(%)1 j
j nh
ay⋅
− *
(%)1
2
jj nh
ay⋅
−
A 1 2 3 4 5 230-250 250-270 270-290 290-310 310-330 330-350
10 12 25 28 15 10
240 260 280 300 320 340
-3 -2 -1 0 1 2
-30 -24 -25
0 15 20
90 48 25 0
15 40
Total 100 - - -44 218
200,29130020100
441 =+⋅−=y zeci mii lei/salariat
( ) 56,7943002,29120100
218 2221 =−−⋅=σ
187,2856,7941 ==σ zeci mii lei/salariat
%68,91002,291
187,281 =⋅=v
1.2. Pentru activitatea II (vezi calculele în tabelul 3.20.)
&DOFXOXO�PHGLHL�úL�GLVSHUVLHL�VDODULLORU�SHQWUX�DFWLYLWDWHD�D�,,,-a
Tabelul 3.20. *UXSH�GXS �
salarii (zeci mii lei)
Structura VDODULD LORU
( )*(%)2 jn
Centrul de interval ( )jy
h
ay j −
a = 300 h = 20
*(%)2 j
jn
h
ay⋅
− *
(%)2
2
j
jn
h
ay⋅
−
A 1 2 3 4 5 270-290 290-310 310-330 330-350 350-370
20 35 20 15 10
280 300 320 340 360
-1 0 1 2 3
-20 0
20 30 30
20 0
20 60 90
Total 100 - - 60 190
31230020100
602 =+⋅=y ]HFL�PLL�OHL�VDODULDW��DGLF ���������OHi/salariat
( ) 61630031220100
190 2222 =−−⋅=σ
82,246162 ==σ zeci mii lei/salariat
%95,7100312
82,242 =⋅=v
1.3. Pentru activitatea a III-a vezi calculele în tabelul 3.21.
&DOFXOXO�PHGLHL�úL�GLVSHUVLHL�VDODULLORU�SHQWUX�DFWLYLWDWHD�D����-a Tabelul 3.21.
*UXSH�GXS �
salarii (zeci mii lei)
Structura VDODULD LORU
( )*(%)3 jn
Centrul de interval ( )jy
h
ay j −
a = 300 h = 20
*(%)2 j
j nh
ay⋅
− *
(%)3
2
j
jn
h
ay⋅
−
A 1 2 3 4 5 310-330 330-350 350-370 3���úL�SHVWH
20 47 23 10
320 340 360 380
-1 0 1 2
-20 0
23 20
20 0
23 40
Total 100 - - 23 83
6,34434020100
233 =+⋅=y zeci mii lei/salariat sau 3446000 lei/salariat
( ) 84,31034084,31020100
83 2223 =−−⋅=σ
6,1784,3103 ==σ zeci mii lei/salariat
%11,510084,344
06,173 =⋅=v
1.4. &XQRVFkQG� PHGLLOH� SDU LDOH� úL� SRQGHULOH� VDODULD LORU� DFWLYLW LORU� UHVSHFWLYH� VH� SRDWH�
calcula media pe total:
( ) ( ) ( ) ( )62,317
100
306,34450312202,291
1001
*%
=⋅−⋅+⋅
==∑=
r
iii ny
y zeci mii lei/salariat
1.5.�$SOLFkQG�UHJXOD�DGXQ ULL�GLVSHUVLLORU�� 222 σδσ += VH�FRQVWDW �F �SXWem calcula doar
GRX �GLQWUH�HOH��XUPkQG�FD�GLVSHUVLD�WRWDO �V �ILH�FDOFXODW �SH�ED]D�UHOD LHL�GH�PDL�VXV�
164,560100
30)(310,84 50)(616 20)(794,56
1001
*(%)
2
2 =⋅+⋅+⋅
==∑
=
r
iii nσ
σ
( )
24,378100
30)62,3176,344(50)62,317312(20)62,3172,291(
100
222
1
*(%)
2
2
=⋅−+⋅−+⋅−
=
=−
=∑=
r
iii nyy
δ
222 σδσ += = 560,164 + 378,24 = 938,404
1.6. &DOFXOXO�JUDGXOXL�GH�GHWHUPLQD LH�úL�QHGHWHUPLQD LH:
%31,40100404,938
276,378100
2
22
/ =⋅=⋅=σδ
xyR
%69,59100404,938
164,560100
2
22
/ =⋅=⋅=σσ
xyK
⇒> 2
/2
/ xyxy RK �IDFWRU�GH�JUXSDUH�QHUHSUH]HQWDWLY��FHHD�FH�vQVHDPQ �F �vQ�OXQD�QRLHPEULH�QX�
a existat decât într-R� SURSRU LH� GH� ������� GLIHUHQ H� VSHFLILFH� vQWUH� FHOH� WUHL� DFWLYLW L� vQ� cazul YHQLWXULORU�VDODULDOH��FHHD�FH�HUD�GH�DúWHSWDW�
2. 3HQWUX�DQDOL]D�RPRJHQLW LL��VH�YRU�IRORVL�LQGLFDWRULL�SUH]HQWD L�vQ�tabelul 3.22. (coloana 5).
5HSDUWL LD�SH�FHOH�WUHL�DFWLYLW L�D�VDODULLORU�PHGLL�úL�D�LQGLFDWRULORU�GH�YDULD LH Tabelul 3.22.
AFWLYLW L
Structura
personalului Salariile medii
(zeci mii lei)
Dispersiile
Abaterile medii S WUDWLFH
(zeci mii lei)
&RHILFLHQ LL GH�YDULD LH
A 1 2 3 4 5 I II III
20% 50% 30%
291,2 312,0 344,6
794,56 616,00 310,84
28,187 24,820 17,600
9,68% 7,95% 5,11%
Total 100% 317,62 938,404 30,63 9,64%
ÌQ�H[HPSOXO�DQDOL]DW�VH�SRW�WUDJH�XUP WRDUHOH�FRQFOX]LL� • DWkW� SH� vQWUHDJD� VRFLHWDWH� FRPHUFLDO � FkW� úL� SH� ILHFDUH� DFWLYLWDWH� VHULLOH� VXQW� RPRJHQH��
deoarece s-DX� RE LQXW� YDORUL�PLFL� SHQWUX� FRHILFLHQ LL� GH� YDULD LH��'DF � VH� LHUDUKL]HD] �VHULLOH�GXS �FULWHULXO�YHQLWXULORU�VDODULDOH��RUGLQHD�RPRJHQLW LORU�HVWH��DFWLYLWDWHD�D�,,,-a (5,11%); activitatea a II-D���������úL�PDL�SX LQ�RPRJHQ �SULPD�DFWLYLWDWHD���������FDUH�GHS úHúWH�FX�IRDUWH�SX LQ�JUDGXO�GH�RPRJHQLWDWH�SH total (9,64%);
• DYkQG� FRHILFLHQ L� PLFL� GH� YDULD LH��PHGLLOH� VXQW� vQ� WRDWH� FD]XULOH� FX� XQ� JUDG�PDUH� GH�UHSUH]HQWDWLYLWDWH� ID � GH� YDORULOH� LQGLYLGXDOH� GLQ� FDUH� DX� IRVW� FDOFXODWH�� GHRDUHFH� WR L�FRHILFLHQ LL�GH�YDULD LH�VXQW�VXE�����
3. Seriile din tabelul 3.1���VH�SRW�SUH]HQWD�úL�FX�IUHFYHQ H�DEVROXWH� ùWLLQG�F �QXP UXO�WRWDO�DO�VDODULD LORU�D�IRVW�GH�����úL�DYkQG�UHSDUWL LD�ORU�UHODWLY �SH�DFWLYLW L�
VH�RE LQH�
• activitatea I 80100
20400 =⋅ VDODULD L
• activitatea a II-a 200100
50400 =⋅ salaria L
• activitatea a III-a 120100
30400 =⋅ VDODULD L
$SOLFkQG� JUHXW LOH� VSHFLILFH� GLQ� ILHFDUH� DFWLYLWDWHD� OD� QXP UXO� GH� VDODULD L�� VH� FDOFXOHD] �IUHFYHQ HOH� DEVROXWH�� FDUH� VH� GHWHUPLQ � QXPDL� vQ� QXPHUH� vQWUHJL� UHIHULQGX-se la persoane (vezi tabelul 3.23).
5HSDUWL LD�VDODULD LORU�SH�FHOH�WUHL�DFWLYLW L�vQ�IXQF LH�GH�YHQLWXULOH�VDODULDOH Tabelul 3.23.
*UXSH�GXS �VDODULL�
(zeci mii lei) Activitatea I
Activitatea II
Activitatea III
Total
sub 250 8 - - 8
250-270 10 - - 10
270-290 20 40 - 60
290-310 22 70 - 92
310-330 12 40 24 76
330-350 8 30 56 94
350-370 - 20 28 48
����úL�SHVWH - - 12 12
Total 80 200 120 400
&DOFXOXO�PHGLHL�úL�GLVSHUVLHL�SH�WRWDO��SH�ED]D�IUHFYHQ HORU�DEVROXWH Tabelul 3.24.
*UXSH�GXS �VDODULL�
(zeci mii lei) 1XP UXO�
salDULD LORU ( )jn
Centrul de interval ( )jy
h
ay j −
a = 340 h = 20
jj nh
ay⋅
−
jj nh
ay⋅
− 2
A 1 2 3 4 5 sub 250 8 240 -5 -40 200
250-270 10 260 -4 -40 160
270-290 60 280 -3 -180 540 290-310 92 300 -2 -184 368
310-330 76 320 -1 -76 76
330-350 94 340 0 0 0
350-370 48 360 1 48 48
����úL�SHVWH 12 380 2 24 48
Total 400 - -448 1440
6,31734020400
448
1
1 =+⋅−=+⋅
−
=∑
∑
=
= ahn
nh
ax
yk
ij
k
ij
j
zeci mii lei/salariat
24,938)6,317340(400400
1440)( 222
1
1
2
2 =−−⋅=−−⋅⋅
−
=∑
∑
=
=ayh
n
nh
ay
k
ij
k
ij
j
yσ
Deci s-a ajuns la aceleDúL� YDORUL� úL� SHQWUX� PHGLH� úL� SHQWUX� GLVSHUVLH� FD� vQ� FD]XO� IRORVLULL�IUHFYHQ HORU�UHODWLYH�
PROBLEME PROPUSE
1. Pentru 40 de agen L�HFRQRPLFL�FH�DFWLYHD] �vQ�DFHODúL�GRPHQLX�GH�DFWLYLWDWH� s-au înregistrat
datele privind profitul realizat (mil. lei) în luna august: 40, 63, 59, 52, 62, 50, 45. 51, 54, 50, 48, 55, 59, 47, 57, 53, 46, 55, 64, 53, 42, 51, 53, 56, 52, 58, 49, 54, 60, 46, 41, 47, 42, 44, 63, 58, 49, 55, 50, 53.
Se cere:
1. V �VH�JUXSH]H�FHL����GH�DJHQ L�HFRQRPLFL�SH�LQWHUYDOH�GH�YDULD LH�HJDOH��GXS �SURILWXO�UHDOL]DW� 2. V �VH�UHSUH]LQWH�JUDILF�UHSDUWL LD�RE LQXW �OD�SXQFWXO�SUHFHGHQW� 3. V �VH�FDOFXOH]H�LQGLFDWRULL�WHQGLQ HL�FHQWUDOH�úL�V �VH�LQWHUSUHWH]H�UHOD LD�GLQWUH�HL� 4. V �VH�YHULILFH�UHSUH]HQWDWLYLWDWHD�PHGLHL� 5. V �VH�GHWHUPLQH�LQGLFDWRULL�FDUH�VHSDU �����GLQ�DJHQ LL�HFRQRPLFL�VLWXD L�OD�FHQWUXO�
UHSDUWL LHL� 6. V �VH�P VRDUH�JUDGXO�GH�DVLPHWULH�
2. 3HQWUX�R�XQLWDWH�HFRQRPLF �VH�FXQRVF�GDWHOH�
Tabelul 3.25.
Grupe de muncitori GXS �P ULPHD�
SURGXF LHL��PLL�EXF��
Nr. muncitori
Sub 10 2 10-12 6 12-14 10 14-16 20 16 -18 18 ���úL�SHVWH 6 Total 60
Se cere:
1. V �VH�UHSUH]LQWH�JUDILF�GLVWULEX LD�PXQFLWRULORU�GXS �P ULPHD�SURGXF LHL� 2. V �VH�YHULILFH�GDF �GLVWULEX LD�PXQFLWRULORU�GXS �P ULPHD�SURGXF LHL�HVWH�RPRJHQ � 3. V �VH�FDOFXOH]H�QLYHOXO�FHQWUDOL]DW�DO�SURGXF LHL�SHQWUX�PXQFLWRULL�FX�SURGXF LD�GH�FHO�
SX LQ����PLL�EXF� 4. V �VH�FDOFXOH]H�LQGLFDWRUXO�FDUH�VHSDU �����GLQ�PXQFLWRUL�GH�UHVWXO�PXQFLWRULORU�
3. 3HQWUX�����GH�VDODULD L�DL�XQHL�VRFLHW L�FRPHUFLDOH��VH�FXQRVF�GDWHOH�
Tabelul 3.26.
Grupe GH�VDODULD L��
GXS �YHFKLPH��DQL� 1XP U�
VDODULD L Timpul mediu nelucrat (min.)
Abaterea medie S WUDWLF �SULYLQG�
timpul nelucrat (min.)
A 1 2 3 sub 10 20 30 4,8 10-20 90 45 9,0 ���úL�SHVWH 40 55 6,6
Se cere: 1. V �VH�UHSUH]LQWH�JUDILF�VWUXFWXUD�VDODULD LORU�GXS �YHFKLPH� 2. V �VH�YHULILFH�RPRJHQLWDWHD�SH�JUXSH�úL�SH�WRWDO� 3. V �VH�GHWHUPLQH�vQ�FH�P VXU �YDULD LD�WLPSXOXL�QHOXFUDW�VH�GDWRUHD] �GHRVHELULORU�SULYLQG�
YHFKLPHD�vQ�PXQF �
4. Dintr-XQ�VRQGDM�VWDWLVWLF�GH�����SURSRU LRQDO�VWUDWLILFDW�V-DX�RE LQXW�GDtele:
Tabelul 3.27.
*UXSH�GH�DJHQ L�HFRQRPLFL�GXS �P ULPHD�SURILWXOXL� (mil. lei)
*UXSH�GH�DJHQ L�
HFRQRPLFL�GXS �
P ULPHD�FDSLWDOXOXL�
(mil. lei) sub 10 10-14 14-18 18-22 22-26 ���úL�
peste
Total
A 1 2 3 4 5 6 7 I 5 10 5 - - - 20 II - 10 15 25 15 5 70 III - - 10 20 15 5 50 Total 5 20 30 45 30 10 140
Se cere:
1. V �VH�UHSUH]LQWH�JUDILF�GLVWULEX LD�DJHQ LORU�HFRQRPLFL�GXS �P ULPHD�SURILWXOXL��SH�WRWDO� 2. V �VH�YHULILFH�UHJXOD�GH�DGXQDUH�D�GLVSHUVLLORU�úL�V �VH�DUDWH�GDF �IDFWRUXO�GH�JUXSDUH�HVWH�
semnificativ sau nu; 3. V �VH�FDUDFWHUL]H]H�RPRJHQLWDWHD�SH�ILHFDUH�JUXS �úL�SH�WRWDO��
4. V �VH�P VRDUH�JUDGXO�GH�DVLPHWULH�SHQWUX�JUXSD�D�,,-a; 5. V �VH�YHULILFH�UHJXOD�DGXQ ULL�GLVSHUVLLORU�SHQWUX�FDUDFWHULVWLFD�³DJHQ L�HFRQRPLFL�FX�
profitul mai mic decât 18 mil.lei.
5. 3HQWUX�XQ�HúDQWLRQ�GH�����GH�DJHQ L�HFRQRPLFL��VH�FXQRVF�GDWHOH�
Tabelul 3.28.
Grupe tipice de DJHQ L�HFRQRPLFL�
GXS �FLIUD� de afaceri
Structura DJHQ LORU�
economici (%)
Profitul mediu (mil.lei/ag.ec.)
Dispersia privind P ULPHD�
profitului
A 1 2 3 sub 20 50 14,4 11,52 20-30 30 19,68 11,28 ���úL�SHVWH 20 30,8 5,76
Se cere:
1. V �VH�SUHFL]H]H�vQ�FH�SURSRU LH�FLIUD�GH�DIDFHUL�LQIOXHQ HD] �YDULD LD�SURILWXOXL� 2. V �VH�SUHFL]H]H�FDUH�JUXS �HVWH�PDL�RPRJHQ �� 3. V �VH�UHSUH]LQWH�JUDILF�SURILWXO�WRWDO�vQ�IXQF LH�GH�IDFWRULL�GH�LQIOXHQ �